New Algorithms for
Calculating Hertzian
Stresses,
Deformations, and
Contact Zone
Parameters
Emil W. Deeg
AMP Incorporated
1
ABSTRACT
The complete set of assumptions directly or indirectly affecting Hertz’s contact stress theory is presented. A review
of the derivation of his fundamental formulas leads to a
computer program that does not depend on elliptic integrals nor the conventional numerical method and the
tabulated function of two coefficients introduced by Hertz.
A method permitting systematic characterization and description of combinations of two contacting bodies with
individual, monotone sequentially, or randomly changing
principal radii of curvature is introduced. The validity of
Hertz’s theory for describing fiber-optic PC connections is
tested through a comparison of specific features of the
system with Hertz’s assumptions.
INTRODUCTION
The ongoing miniaturization of electrical contacts and the
introduction of fiber-optic physical contact (PC) connectors
triggered an examination of theories suitable for predicting
local stresses and deformations at the point of contact.
A generally accepted theory serving this purpose was
presented by H. Hertz in 1881 and extended in 1882 2 to
include a hardness definition. Summaries and examples are
found in many textbooks and handbooks.3-8 Application
to electrical connections is still under discussion.9-ll For
fiber-optic PC connections, Hertz’s theory yields design
guidelines if calculations are restricted to the endfaces of
the contacting fibers.12 Analysis of a complete, 3-layer, rotationally symmetric PC connection requires finite element
methods. 13
Although Hertz’s theory provides only one approach to
contact stress theory, its success in estimating allowable
maximum or desired minimum local loads warrants a thorough examination of its features. A particularly convincing
argument is its role in the optimization of design and use of
ball and roller bearings. One of the early reports on this
topic was given in 1901 by F. Heerwagen, who at the time
was in charge of technology at a mine in Spain. In cooperation with an Austrian pump manufacturer, Heerwagen
reduced drastically the failure rate of ball bearings in his
pumps. 14 A systematic, basic engineering study related
to this topic, but addressing different applications, was
© Copyright 2004 by Tyco Electronics Corporation. All rights reserved.
14
E.W. Deeg
AMP Journal of Technology Vol. 2 November, 1992
published in 1921 by H. L. Whittemore and S. N.
Petrenko.15 , a Hertz’s theory has been used also for optimal
selection of abrasives and pressure in grinding and
polishing of brittle materials.16
At present, calculation of Hertzian stresses for ellipsoidal
surfaces relies on two functions of geometric parameters
1,2
presented in tabular form. A revised and extended table
was published by Heerwagen,14 rearranged by Whittemore15b
and is found in most of the references. 3-8 Use of these
numerically defined functions requires time-consuming,
rather tedious calculations of auxiliary quantities. To avoid
them, ellipsoidal surfaces in contact with each other are
frequently approximated by idealized geometries such as
sphere-on-sphere, sphere-on-plane, cylinder-on-cylinder,
and cylinder-on-plane. Improper use of these simplifications can lead to erroneous prediction of the behavior of
the modelled system.
There are also problems of a fundamental nature. The
more-than-a-century-old Hertzian contact stress theory is
being applied mostly based on secondary sources of the
kind mentioned above. By necessity, handbooks, textbooks,
and compendia of formulas can give only incomplete accounts of original publications. It is left to the user to consult the sources in case of questions. For Hertz’s contact
stress theory this is difficult today. Although his collected
works are available in leading libraries, the language barrier remains. But even engineers fluent in today’s technical
German will have to familiarize themselves with concepts
and terminology of the late 19th century. One example is
Hertz’s use of the term Druck for pressure as well as for a
quantity today termed Kraft (= force). Furthermore, his
contemporaries were quite familiar with elliptic functions
and integrals. A case in point is Heerwagen’s study.14 Consequently, Hertz left out details of derivations that now
require special effort by the reader to follow. Finally, the
collected works contain misprints that become obvious
when following Hertz’s derivations step by step but will
lead to errors if not recognized and corrected. Also of
interest, Hertz does not give explicit expressions for the
stress components inside the contacting bodies. Such formulas were derived later by M. T. Huber17, who also found
that the intuitive stress distribution near the contact zone
given by Hertz2 is incorrect. Subsequent papers by
S. Fuchs18 and W. B. Morton et al.19 address the same
topic.c
of the theory in its most general form, several computer
algorithms were written requiring only input of material
constants, applied force and the geometric parameters of
the system. All use readily available software packages
without resorting to traditional computer programming.
One of them is presented here.
REVIEW OF HERTZ’S CONTACT STRESS
THEORY
Basic Assumptions
The decision to rejector accept an established theory for
modelling a specific physical system requires a comparison
of the features of the system with the complete set of assumptions or axioms on which the theory is based. For the
Hertzian contact stress theory, the fundamental
assumptions2,d are:
(a) at the point of contact the shape of each of the contacting surfaces can be described by a homogeneous
quadratic polynomial in two variables;
(b) both surfaces are ideally smooth;
(c) contact stresses and deformations satisfy the
differential equations for stress and strain of homogeneous, isotropic, and elastic bodies in equilibrium;
(d) the stress disappears at great distance from the
contact zone;
(e) tangential stress components are zero at both
surfaces within and outside the contact zone;
(f) normal stress components are zero at both surfaces
outside the contact zone;
(g) the stressg integrated over the contact zone equals
the force pushing the two bodies together;
(h) the distance between the two bodies is zero within
but finite outside the contact zone;
(i) in the absence of an external force, the contact zone
degenerates to a point.
While not directly identified in his papers, Hertz alerts the
reader to some assumptions generally accepted by his
contemporaries. In the first paragraph of reference 1, he
considers the contact force to be of the nature of what
would be now termed a (Dirac) d -distribution:
In the theory of elasticity it is assumed that deformations are
caused by forces acting upon the interior and by forces acting upon
the surface of a body. Both kinds of forces may become infinitely
high in isolated, infinitely small portions of the body but so that
their integrals over these portions are finite. If we then surround
the point of discontinuity by a closed surface that is very small
These considerations and the need to model stresses, deformations as well as shape and size of the contact zone in
fiber-optic PC connections led to a study of Hertz’s papers.
The results are summarized below. To make efficient use
d
a
This study distinguishes conceptually and experimentally between friction under
load and maximum safe load. The mechanics of electrical connections make a
similar distinction. Whittemore’s report, if consulted and interpreted properly,
will add valuable information to the ongoing discussion.
b
Whittemore credits Dr. L. B. Tuckerrnan for preparing the table of coefficients
for Hertz’s theory.
c
Although restricted to contacts sphere/sphere and sphere/p late, respectively,
these papers provide essential input to the ongoing discussion on applicability of
Hertz’s theory to connector mechanics.
AMP Journal of Technology Vol. 2 November, 1992
Reference 2, pp. 155-157 is preferred here because of the more concise formulation than that given in reference 1. Secondary assumptions of importance are
identified in the text below.
e
According to Hertz2, this assumption is the result of expanding around the
point of contact the equations of the contacting surfaces and neglecting higher
than second-order terms.
f
To this statement should be added “as commonly used by mechanical engineers
during the late 19th century.”
g
Hertz calls this force Gesamtdruck (total pressure), which according to today’s
terminology leads to a dimensionally incorrect end result for the contact stress,
E.W.
Deeg 15
compared to the overall dimensions of’ the body but very large compared to the dimension of the part in which the forces act, we can
consider the deformations outside of this surface independently
from those inside. At the outside the deformations depend on the
shape of the entire body, the distribution of the forces in general
and the finite integrals of the force components at the points of discontinuity. Inside they depend only on the distribution of the forces
acting there: pressures and deformations are infinitely high
compared to those outside.
As is common in mathematical descriptions of’ physical systems, the above assumptions are idealizations. The terms
ideally smooth, homogeneous, isotropic, elastic bodies, equilibrium, at great distance indicate this. For each concrete
case, the degree of approximation of the physical system by
the theory must be considered and the effect on the results
assessed accordingly. Frequently, comparing qualities is
sufficient to determine the utility of the theory. If this
approach is not possible or its feasibility appears questionable, quantitative comparisons are necessary. Examples of
such critical evaluations are already given by Hertz1,2 and
by Whittemore15. Neglecting these considerations has led
to misinterpretations and occasionally even unjustified
rejection of the theory.
Fundamental Formulas h8
Hertz uses a Cartesian system of coordinates {x,y,z] with
origin at the initial point of contact of the two bodies and
the z-axis oriented parallel to the applied force. This requirement introduces an additional assumption if, for more
general reasons, the physical system is already described in
differently oriented Cartesian coordinates or in a different
coordinate system. The slanted fiber-optic PC connection is
an example. For optical reasons and because of its global
symmetry, the z-axis would be chosen to coincide with the
optical fiber axis. In contrast, the z-axis used for a Hertzian
analysis of the connection must be oriented parallel to the
direction of the slant.
Table 2. List of symbols. The third and fourth column show
those used in this paper.
For the fiber-optic PC connections discussed in references
12 and 13, such a comparison is summarized in Table 1.
The result is that two PC fiberends in contact with each
other meet Hertz’s assumptions with satisfactory approximation. However, if the entire system comprising fiber,
adhesive layer and ferrule is to be analyzed, (c) and (g) are
not valid. Furthermore, if the connection is not aligned
axially, (a) and (e) may be violated. If one or both of the
connecting fibers are recessed, (a) is invalid. In the latter
case, the ferrules, not the fiber endfaces, establish the
initial contact, which may also cause violation of (i).
Table 1. Example for assessing applicability of Hertz’s
contact stress theory. The fundamental assumptions of the
theory are compared with specific features of fiber-optic PC
connections described in references 12 and 13,
In the following each of the two bodies in contact with each
other is identified by the subscript i (= 1, 2). To distinguish
between the two principal radii of curvature of the surfaces
of the two bodies at the contact point, a second subscript j
(= 1,2) is introduced. The theory requires input of three
sets of data:
(1) a one-member set of the force p pushing the bodies
together;
(2) a four-member set of elastic constants, which within
the theory can be reduced to a two-member set;
(3) a five-member set of geometric parameters describing shape and relative orientation of the contacting
bodies.
The two elastic constants Young’s modulus E and Poisson’s
ratio v enter the final formulas in combination as the
h
16
E.W. Deeg
For a summary of Symbols and abbreviations, see Table 2.
AMP Journal of Technology Vol. 2 November, 1992
“Hertz coefficient” i
Four geometric parameters characterize the shape of the
contacting surfaces. They are two principal radii of curvature ril and ri2 for each body i with rij > 0 if the center of
curvature is inside body i. The theory uses the r ij in the
functions
The fifth geometric parameter is an angle w in the {x,y}
plane
Within the same body, i.e. for the same index i, the planes
{z,ril] and {r, i2 } are normal to each other.
Hertz describes the surfaces by
rotates the {x,y} coordinates so that C l = C2 = C and obtains for the distance e normal to the {x,y} plane between
points {X l,yl} = (x2,y2}j
From assumption (h) and the nature of e as a distance, it
follows that the conic described by equation (4) is an
ellipse, which includes the case of two contacting spheres,
i.e. RI = R2 = 0. Contact between cylinders requires
additional considerations and is discussed below.
Introducing an auxiliary angle W
through
yields
To describe deformations of the two bodies under load,
Hertz assigns to each of them a separate Cartesian system,
which at infinity is attached to the corresponding body. Under zero load the new systems coincide with the primary
system. A compressive force p applied parallel to the z-axis
causes a displacement of the secondary systems relative to
the primary system but so that, according to Hertzl, “the
plane z = 0 in each of them is infinitely close to the surfa ce
of the corresponding body and, thus, can be taken as the
surface itself. The direction of the z-axis is the direction of
the normal to said surface.”k
Because the two surfaces described by equation (3) are
pushed towards each other in the z-direction, the circumference of the contact zone is an ellipse. The lengths of its
semiaxes a and b are different from but their directions
coincide with the directions of the semiaxes of an ellipse of
equation (4), with e = constant. To link the force p and the
geometry of the contact ellipse, Hertz introduces a function
P(x,y,z), in reference 2 without explanation and in reference
1 through an electrostatic model]
with u being the positive root of
With the abbreviations of equations (2), it is
k
i
Hertz uses Kirchhoffs elastic constants. For this paper they were converted to
the now commonly accepted Young’s modulus and Poisson’s ratio.
j
This approach follows Hertz’s earlier article. i In the later one, Hertz derives
explicit equations for the surfaces yielding equations (5a) and (5b).2
AMP Journal of Technology Vol. 2 November, 1992
These statements introduce the requirement that inside the contact zone
deformations in the z-direction are small compared to the body dimension in
this direction.
1
P is the potential of an ellipsoidal shell of zero thickness and zero extension in
the z-direction (principal semiaxis c = 0), The justification given in reference 1
cannot hide its axiomatic introduction. A concise review of potentials of
ellipsoids, still written in a form that provides access to their use by Hertz’s
contemporaries, is given by R von Mises.20
E.W. Deeg
17
The body-specific material constants are then introduced
by two auxiliary functions
The force p compresses the two bodies within the contact
zone. This results in a deformation of the surfaces so that
two points located inside each body at {xl,yl}] = {x2,y2}
approach each other in the z-direction by
Some of their properties are discussed by Hertz.l
which with equation (4) yields
The functions P and IIi are used mostly to prove the validity of the theory within the set of assumptions (a) through
(i)m. A slightly modified P(x,y,0) yields the final expressions
for the shape of and the deformation and stress at the contact zone. To that end the displacements x i, h i and z i in the
directions x, y, and z, respectively, are written as
Comparing equations (1 lb) and (12b) yields
These equations contain as unknowns only the two
semiaxes of the circumference of the contact ellipse. Introducing the ratio k = b/a and setting l = b2 · t2 in equations
(llc) and (11d) but l = a2 · t2 in equation (11e) yields
and subsequently introduced in the corresponding terms
for the stress components. Of particular interest is the difference of the deformations in the z-direction in the plane
z = 0. Equation (l0b) yields
or
and after some algebraic rewriting
with
where k is the root of the transcendental equation
m
These proofs can be carried out without difficulties and are not repeated here.
Frequently used is — 2 P = O. To facilitate understanding of the features of the
theory this paper concentrates on the derivation of the final formulas.
18
E.W. Deeg
AMP Journal of Technology Vol. 2 November, 1992
To find the quantity known today as Hertzian stress the
normal stress component ZZ is calculated for the contact
zone where z = 0, which in equation (8b) yields u = 0.
With
infinite. In this case, a is determined by the global shape of
the bodies and not by the conditions existing at the contact
area. Thus, the cylinder/cylinder contact is actually outside
the validity range of the theory.”
To make the theory accessible to numerical application,
Hertz expresses a and b by
it is
n
where the dependency on the auxiliary angle W and the
integrals I(k) and J(k) of equations (15a) and (15b) are
included in the two coefficients f and g. Because of equation (16) they depend only on .W Hertz states in reference
2, page 182
For u = 0 the derivative ≠ P/≠z assumes the indeterminate
form 0 · ` Some rewriting involving the solution of the cubic equation (8b) and application of de l’Hospital’s rule
yields for the normal stress (ZZ)Z=0 within the contact zone
which for {x,y} = {0,0] is the Hertzian stress
If the contact is formed by two cylinders, Hertz considers in
reference 2, p. 186/87 the contact ellipse for a Æ ` . This
requires also that p Æ `¥ to maintain a finite value for the
force per unit length. He finds B = (rl + r2)/2 with ri being
the finite radius of curvature of body i. It is also u << a2
which permits a to be moved in front of the integral in
equation (1le). Finally he replaces the indeterminate form
p/a = ` / ` by an arbitrary constant 4p' /3 where p' is the
force per unit length of the cylinder. The formulas for the
small semiaxis and the stress Z z are under these assumptions
and
The compression a would now become logarithmically
n
Indices i = 1, 2 are left off here, The expressions apply equally to both bodies.
AMP Journal of Technology Vol. 2 November, 1992
. . . the integrals in question can all be reduced to complete elliptic
integrals of the first kind and their derivatives after the modulus.
Thus, they can be found via Legendre’s tables without introducing
new quadrature. However, the calculations are extensive and
p
therefore I calculated the table given below.
His table, which is included here in Table 3, contains to
three decimal points the coefficients f and g for ten values
of .W They are found in footnotes in references 1 (p. 164)
and 2 (p. 182); according to the latter, “interpolation between these values will probably always offer sufficient
accuracy.”
Hertz leaves it to the reader to convert the improper integrals in equations (14) and (15) to integrals with finite upper limits. He also does not describe how he reduced these
integrals named here H(k), I(k), and J(k) to complete elliptic integrals of the first kind and their derivatives after the
modulus. An attempt to reconstruct his approach is shown
in Appendix A. However, without knowing exactly the resources Hertz used, it cannot be claimed that his method
was the same. One must keep in mind that compendia we
take for granted21 23 were not yet printed.
NEW ALGORITHM FOR HERTZ’S CONTACT
STRESS THEORY
General Remarks
To use Hertz’s theory in today’s environment it is essential
to describe by a condensed code the set of geometric parameters of the system to be modelled. This is particularly
important for systematic studies of ordered or random
o
See Hertz'S introductory remarks to his first paper. A translation is given above
in the section on Basic Assumptions. We found that simulating a cylinder by
high but finite ratios r il /r i z in the regular formulas gives satisfactory approximations.
P
A similar statement is found in reference 1, p, 164, footnote 1.
E . W . D e e g 19
Table 3. Factors f and g for calculating semiaxes a and b of
the contact ellipse. The program described in this paper
yields values listed in columns f EWD and gEWD .
quadruple of principal radii
The angle ,w the fifth element of the set of geometric parameters, is included as index.
TO characterize monotone sequences of the rijj let
and
describe the contacting part of the surface of body 1, and
describe the contacting part of the surface of body 2. Each
ordered quadruple now depends on the integers m and
n, i. e.
sequences of the rij. Such sequences occur e.g. during design optimization or in determining the effect of variations
of processing conditions on shape and size of the contact
zone and on Hertz stress. The methodology is described
below in the section on characterizing combinations of contacting surfaces; examples are found in reference 12.
It is also essential to streamline the present method for
numerical evaluation of Hertz’s formulas using personal
computers or workstations and software not requiring conventional programming. A simple approach would be to
write the table of coefficients f and g and the conventional
method onto disk. More desirable would be to write computer programs incorporating algorithms for complete
elliptic integrals The most natural approach, however,
would utilize Hertz’s fundamental formulas directly and
integrate them without transformation to elliptic integrals.
This method and a corresponding computer program are
described below.
Characterization of Combinations of Contacting
Surfaces
At the point of contact the surface of each of the two bodies i (= 1, 2) is defined by two principal radii of curvature rij
with j (= 1, 2). Thus, regarding their shape the pair of
contacting bodies can be characterized by an ordered
q
In connection with the work described here several such programs were written
and tested. References 21 through 24 were consulted and used. Of particular
help was Hastings’ polynomial approximation of complete elliptic integrals. 25
20
E.W. Deeg
and the q w ,m,n can be combined in a M x N matrix
It is frequently desirable to have the central element of G
serve as reference describing two spherical surfaces in contact. The remaining elements q w ,m,m would then describe
geometrically deviating combinations. Because of the form
in which the rij are expressed in equations (22a) and (22b),
this can be accomplished by choosing M and N as odd numbers. Depending on the values chosen for the members of
the set {w , rl0 , r20 , D rll, D r12, D r21, D r22, M, N] various
combinations of ellipsoidal and spherical surfaces can be
described.
For simulations of random variations of the surface shapes,
the monotone increasing sequences for m and n in
equations (22a) and (22b) are replaced by randomized
sequences. The MATLAB program given in Appendix B
serves this purpose. It generates four sets of random numbers mj and nj, enters them into equations (22), computes
the rij and plots bar charts of their distribution.
Direct Quadrature of Hertz’s Basic Expressions
The earliest point in Hertz’s derivations where modern
computational techniques could be applied without violating his fundamental concepts is found in reference 1,
pp. 164/65, or in reference 2, pp. 181 and 183. The
AMP Journal of Technology Vol. 2 November, 1992
corresponding equations above are (14) and (15). The
transformation
printing of the entire program, the summary shown in
Table 8 is convenient. It fits on an 8.5 x 11 sheet.
Table 4. Data input for the MathCAD program. The lines
changes the integrals in equations (14a), (14b), (15a), and
(15b) to
For 0 < k # 1 these integrals converge absolutely. Their
values can be found e.g. by Simpson’s rule or Romberg’s
modification of the trapezoidal rule.26 Both can be accessed
through one of the MathCAD versions. With these, as well
as other software packages, time-consuming, traditional
programming can be avoided. MathCAD version 2.5 was
chosen here because of the transparency of its symbolic
language, its ease of access and the very reasonable hardware requirements.r The symbols used for the variables in
the program had to be adjusted to MathCAD terminology
and differ from those in the text. Both are included in
Table 2.
The program is listed in Tables 4 to 7. It follows essentially
the sequence of formulas above. Because the results of the
root operator are influenced by the seed value and to optimize for speed, three different seed values Kl, K2, and K3
are pre-selected, each for a specific range of the auxiliary
angle W The conditional statements “if [. . .]” in the program identify the proper range and, without operator
assistance, select the proper seed value. The program calculates semiaxes a and b of the contact ellipse, area Q of
the contact zone, compression a l and a 2 for each body,
total compression a = a l + a 2, and Hertzian stress Zmax.
It also plots the circumference of the contact zone. If
desired, other areas of interest, for instance those of fiber
core and fiber cladding 12, can be plotted in the same diagram. For plots in Cartesian coordinates of the contact
zone the parameter t with 0 # t # 2 p and S : = 75,
s := 0..S, t s := 2 p ·s/S is introduced in
separate the three sets of data. Note the changed symbols r ij
to rij etc.
Table 5. Definition of functions for converting primary input
data to intermediate quantities used by the program. The
four-member set of material constants {El, E2, vI, v2] is
reduced to the two-member set {01, 02], the five-member
set of geometric parameters {rl1, r12, r21, r22, w ] is reduced to the four-member set {R, Rl, R2, W ].
Table 6. Functions for computing the ratio k = b/a of the
semiaxes of the contact ellipse. The lower part of the table
shows the pre-selected seed values K1, K2, K3 and the
conditional statements for their optimal selection.
The value S := 75 gives satisfactory resolution of the plots
but can be reduced to increase speed. To avoid repeated
r
MathCAD 2.5 requires IBM PC, PC/XT, PC/AT or compatibles with 512K
RAM, floppy or hard disks and one of the common graphics adapters. Numerical coprocessor, although recommended, are not required. It works in MS-DOS
or PC-DOS and supports a wide range of common printers and plotters.
AMP Journal of Technology Vol. 2 November, 1992
E.W. Deeg
21
Table 7. Definition of functions for data output.
Table 8. Summary of input, selected intermediate, and output data. The box marked PLOT contains the contour of the
contact ellipse in the {x,y} plane according to equation (28).
based. It shows also that for each concrete case the degree
of approximation between the features of the physical system to be modelled and the entire set of Hertz’s assumptions must be compared. Neglecting these considerations
can lead to misinterpretation of theoretical predictions and
to unjustified rejection of the theory. The review of Hertz’s
two original publications also reveals expressions that can
be written as computer algorithms requiring only input of
material constants, applied force, and geometric parameters describing the contacting surfaces. The programs make
use of readily available software packages without resorting
to traditional computer programming and permit application of the theory in its most general form. Among several
programs developed and tested for this purpose, one was
selected because of the transparency of its terminology,
ease of access, and modest hardware requirements. Written in MathCAD version 2.5, the program is described in
detail.
Optimizing design or studying the effect of variations of
manufacturing conditions on shape, size, deformation, and
stress in the contact zone requires large sequences of
systematically or randomly modified principal radii of curvature of the contacting surfaces. Such sequences can be
characterized and described by a matrix whose elements
are ordered quadruples of the four radii of each individual
combination. For monotone sequences of the principal
radii of curvature, the elements are algebraically related
and by proper selection of their number the central
element of the matrix represents a sphere-on-sphere contact which serves as reference.
REFERENCES
1. H. Hertz, “Über die Berührung fester elastischer
Körper,” Gesammelte Werke (P. Lenard, ed.), Bd. 1,
(J.A. Barth, Leipzig, 1895) pp. 155-173. Originally published in Journal f. d. reine u. angewandte Mathematik
92,156-171 (1881).
SUMMARY
This paper could be subtitled “What if Heinrich Hertz
would have had a personal computer?” The high-speed
computing tools we are accustomed to did not exist when
he published his theory about a century ago. To make his
theory accessible for engineering applications, he developed a numerical method with mathematical tools known
to his contemporaries. If Hertz had had access to today’s
computing tools, he most likely would not have calculated
the “little table which, in most instances makes the rather
extensive arithmetic superfluous” (reference 1, p. 164).
A review of his papers leads to a clear identification of the
complete set of basic assumptions on which his theory is
22
E.W. Deeg
2. H. Hertz, “Über die Berührung fester elastischer Körper und über die Härte,” Gesammelte Werke
(P. Lenard, ed.), Bd. 1, (J.A. Barth, Leipzig, 1895) pp.
174-196. Originally published in Verhdlg. Ver. Bef.
Gewerbefl., Berlin, Nov. 1882.
3. A.E.H. Love, A Treatise on the Mathematical Theory of
Elasticity, 4th ed. (1927), (Dover, New York, 1944) p.
195.
4. S.P. Timoshenko and J.N. Goodier, Theory of Elasticity
2nd ed. (McGraw-Hill, New York, 1951).
5. Hütte, Des Ingenieurs Taschenbuch, Bd. I, 28. Auflage
(Ernst & Sohn, Berlin, 1955), p. 964.
6. R.C. Juvinall, Engineering Considerations of Stress,
Strain and Strength (McGraw-Hill, New York, 1967)
p. 371.
AMP Journal of Technology Vol. 2 November, 1992
7. R.J. Roark and W.C. Young, Formulas for Stress and
Strain, 5th ed. (McGraw-Hill, New York, 1982) Table
33, p. 518.
8. W. Griffel, Handbook of Formulas for Stress and Strain
(F. Ungar Pub]. Co., New York, 1966) pp. 238-244.
9. E.A. Kantner and L.D. Hobgood, “Hertz stress as an
indicator of connector reliability, ” Connection Technology, 5, 14-22 (March 1989).
10. R. Mroczkowski, Concerning "Hertz Stress” as a Connector Design Parameter, Order No. 82818 (AMP Incorporated, Harrisburg, PA, 1991).
11 H.S. Fluss, “Hertzian stress as a predictor of contact
reliability,” Connection Technology, 6, 12-21 (Dec.
1990).
12. E.W. Deeg and T. Bolhaar, “Contact zone and Hertzian stress in fiber-optic connections with spherical or
ellipsoidal fiber endfaces, ” AMP Journ. of Technol., 2,
29-41 (1992).
13. E.W. Deeg, “Effect of elastic properties of ferrule materials on fiber-optic physical contact (PC) connections,” AMP Journ. of Technol., 1,25-31 (1991).
14. F. Heerwagen, “Kugellager, Erfahrungen aus dem Betriebe und Beiträge zur Theorie.” Zeitschr, Ver. Dtsch.
Ingen., 45, 1701-1705 (1901).
15. H.L. Whittemore and S.N. Petrenko, Friction and carry
ing capacity of ball and roller bearings, Technol. Paper
Natl. Bureau of Standards No. 201, 18 (Governmt.
Printing Office, Washington, DC, 1921).
23. J. Hoüel, Recueil de fomules et de tables numeriques
(Gauthier-Villars, Paris, 1901).
24. L.V. King, On the Direct Numerical Calculation of Elliptic Functions and Integrals (Cambridge Univ. Press,
Cambridge, 1924).
25. C. Hastings, Jr., Approximations for Digital Computers,
(Princeton Univ. Press, Princeton).
26. L.W. Johnson and R.D. Riess, Numerical Analysis, 2nd
ed., 323 (Addison-Wesley, Reading, MA, 1982).
Emil W. Deeg recently retired from his position as Project
Manager, Technology at AMP Incorporated in Harrisburg,
Pennsylvania. He is a consultant for the Company.
Dr. Deeg holds a physics diploma and a Dr. rer. nat.
(magna cum laude) from Julius-Maximilians-Universitaet,
with thesis work at the Max-Planck-Institut fuer Silikatforschung in Wuerzburg, West Germany. As original contributor and throughout his more than 25-year career as
R&D manager and executive for several international corporations he authored or co-authored more than 70 articles
in physics, ceramics, glass science and engineering, and a
book on glass in the laboratory. He holds over 40 patents in
the same fields. Dr. Deeg served as a member of the International Commission on Glass (1963–1981, offices held), as
a member of the International Commission for Optics
(1964–1966), and as a consultant to the NASA Spacelab
Program (1971–1978). He is a fellow of the American Ceramic Society, was inducted into the Hall of Fame of Engineering, Science and Technology, and is listed in Who’s
Who in the Worldj Who’s Who in Finance and Industry, and
other biographical publications. Dr. Deeg joined AMP in
1984 as Project Manager, Materials Engineering.
16. E.W. Deeg, Unpublished, 1970/76.
17. M.T. Huber, “Zur Theorie der Berührung fester elastischer Körper,” Ann. Phys., 14, 153-163 (1904).
18. S. Fuchs, “Haupspannungstrajektorien bei der
Berührung einer Kugel mit einer Platte,” Physikal.
Ztschrft. 14, 1282-1285 (1913).
19. W.B. Morton and L.J. Close, “Notes on Hertz’s theory
of the contact of elastic bodies, ” Phil. Msg. (Ser. 6), 43,
320-329 (1922).
20. Ph. Frank and R. v. Mises, Die Differentialgleichungen
und Integralgleichungen der Mechanik und Physik, Bd. 1,
pp. 608-611, 2nd. ed. (M. S. Rosenberg, New York,
1943). (Photolithographic copy of the original edition
published by F. Vieweg & Sohn, Braunschweig, 1930).
21. E. Jahnke and F. Erode, Tables of Functions, 4th ed.
(Dover, New York, 1945).
APPENDIX A
With q being the modulus, the complete elliptic integral of
the first kind is
Setting q = 1 – k 2 in equations (25) and (26) but q = 1 –
1 /k2 in (27), the denominator of integrals H(k), I(k) and
J(k) is
and the integrals in equations (26) and (27) assume the
form
22. M. Abramowitz and I.A. Stegun, cd., Handbook of
Mathematical Functions (Dover, New York, 1972).
AMP Journal of Technology Vol. 2 November, 1992
E.W. Deeg 23
H(k) is already of the form K(q). Equation (A2) in (Al)
yields
and the first derivative of K(q) after the modulus is
Comparing equations (A4) and (A5) yields the functional
relationships for I(k) and J(k) as stated by Hertz:
24
E.W. Deeg
APPENDIX B
MATLAB program for generating random distributions of
radii of curvatures r ij.
,
Ml = );
Ml= input(’upper limit of m1
N1 = ,);
N1 =input(’upper limit of n1
,
M2=input(’upper limit of m2
M2 =, );
N2 = );
N2= input(’upper limit of n2
,
xl =input(’seed for m1-sequence
xl = );
,
yl = );
yl =input(’seed for n1-sequence
,
x2 = ,);
x2=input(’seed for m2-sequence
y2= input(’seed for n2-sequence
y2 = );
,
rl0=input(’base radius, endface #1
r10 = ,);
r20=input(’base radius, endface #2 r20 = ,);
drll = );
dr11 = input(’r11 step size
,
dr12=input(’r12 step size
dr12 =, );
dr21 =,);
dr21 =input(’r21 step size
dr22=input(’r22 step size
dr22 = );
rand(’uniform’);
rand(’seed’,xl) ;ml= 1 +round(l0*rand( l, Ml));
rand(’seed’,yl);nl = 1 +round(l0*rand( l, Nl));
rand(’seed’,x2) ;m2=l+round(10* rand(l,M2));
rand(’seed’,y2) ;n2=l+round(10* rand(l,N2));
rll=rl0+(ml -( Ml+l)/2)*drl l,pause,
r12=r10+(m2-(M2+l)/2)*dr12,pause,
r21=r20+(nl-(Nl+l)/2)*dr21,pause,
r22=r20+ (n2-(N2+ l)/2)*dr22,pause,
subplot,
subplot(221),bar( rll),subplot(222),bar( r12),
subplot(223),bar( r21),subplot(224),bar(r22),pause
AMP Journal of Technology Vol. 2 November, 1992