New Algorithms for Calculating Hertzian Stresses, Deformations, And Contact Zone Parameters

New Algorithms forCalculating HertzianStresses,Deformations, andContact ZoneParameters

Emil W. DeegAMP Incorporated

ABSTRACTThe complete set of assumptions directly or indirectly af-fecting Hertz’s contact stress theory is presented. A reviewof the derivation of his fundamental formulas leads to acomputer program that does not depend on elliptic inte-grals nor the conventional numerical method and thetabulated function of two coefficients introduced by Hertz.A method permitting systematic characterization and de-scription of combinations of two contacting bodies withindividual, monotone sequentially, or randomly changingprincipal radii of curvature is introduced. The validity ofHertz’s theory for describing fiber-optic PC connections istested through a comparison of specific features of thesystem with Hertz’s assumptions.

INTRODUCTIONThe ongoing miniaturization of electrical contacts and theintroduction of fiber-optic physical contact (PC) connectorstriggered an examination of theories suitable for predictinglocal 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 toinclude a hardness definition. Summaries and examples arefound in many textbooks and handbooks.3-8 Applicationto electrical connections is still under discussion.9-ll Forfiber-optic PC connections, Hertz’s theory yields designguidelines if calculations are restricted to the endfaces ofthe contacting fibers.12 Analysis of a complete, 3-layer, ro-tationally symmetric PC connection requires finite elementmethods. 13

Although Hertz’s theory provides only one approach tocontact stress theory, its success in estimating allowablemaximum or desired minimum local loads warrants a thor-ough examination of its features. A particularly convincingargument is its role in the optimization of design and use ofball and roller bearings. One of the early reports on thistopic was given in 1901 by F. Heerwagen, who at the timewas in charge of technology at a mine in Spain. In coopera-tion with an Austrian pump manufacturer, Heerwagenreduced drastically the failure rate of ball bearings in hispumps. 14 A systematic, basic engineering study relatedto 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

1

published in 1921 by H. L. Whittemore and S. N.Petrenko. Hertz’s theory has been used also for optimalselection of abrasives and pressure in grinding andpolishing of brittle materials.16

At present, calculation of Hertzian stresses for ellipsoidalsurfaces relies on two functions of geometric parameterspresented in tabular form.

1,2 A revised and extended tablewas published by Heerwagen, 14 rearranged by Whittemore15b

and is found in most of the references.3-8 Use of thesenumerically defined functions requires time-consuming,rather tedious calculations of auxiliary quantities. To avoidthem, ellipsoidal surfaces in contact with each other arefrequently approximated by idealized geometries such assphere-on-sphere, sphere-on-plane, cylinder-on-cylinder,and cylinder-on-plane. Improper use of these simplifica-tions can lead to erroneous prediction of the behavior ofthe modelled system.

There are also problems of a fundamental nature. Themore-than-a-century-old Hertzian contact stress theory isbeing applied mostly based on secondary sources of thekind mentioned above. By necessity, handbooks, textbooks,and compendia of formulas can give only incomplete ac-counts of original publications. It is left to the user to con-sult the sources in case of questions. For Hertz’s contactstress theory this is difficult today. Although his collectedworks are available in leading libraries, the language bar-rier remains. But even engineers fluent in today’s technicalGerman will have to familiarize themselves with conceptsand terminology of the late 19th century. One example isHertz’s use of the term Druck for pressure as well as for aquantity today termed Kraft (= force). Furthermore, hiscontemporaries were quite familiar with elliptic functionsand integrals. A case in point is Heerwagen’s study. Con-sequently, Hertz left out details of derivations that nowrequire special effort by the reader to follow. Finally, thecollected works contain misprints that become obviouswhen following Hertz’s derivations step by step but willlead to errors if not recognized and corrected. Also ofinterest, Hertz does not give explicit expressions for thestress components inside the contacting bodies. Such for-mulas were derived later by M. T. Huber17, who also foundthat the intuitive stress distribution near the contact zonegiven by Hertz2 is incorrect. Subsequent papers byS. Fuchs18 and W. B. Morton et al. 19 address the sametopic.

These considerations and the need to model stresses, de-formations as well as shape and size of the contact zone infiber-optic PC connections led to a study of Hertz’s papers.The results are summarized below. To make efficient use

a This study distinguishes conceptually and experimentally between friction underload and maximum safe load. The mechanics of electrical connections make asimilar 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 coefficientsfor Hertz’s theory. Although restricted to contacts sphere/sphere and sphere/p late, respectively,these papers provide essential input to the ongoing discussion on applicability ofHertz’s theory to connector mechanics.

of the theory in its most general form, several computeralgorithms were written requiring only input of materialconstants, applied force and the geometric parameters ofthe system. All use readily available software packageswithout resorting to traditional computer programming.One of them is presented here.

REVIEW OF HERTZ’S CONTACT STRESSTHEORY

Basic AssumptionsThe decision to rejector accept an established theory formodelling a specific physical system requires a comparisonof the features of the system with the complete set of as-sumptions or axioms on which the theory is based. For theHertzian contact stress theory, the fundamentalassumptions are:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

at the point of contact the shape of each of the con-tacting surfaces can be described by a homogeneousquadratic polynomial in two variables;

both surfaces are ideally smooth;

contact stresses and deformations satisfy thedifferential equations for stress and strain of homo-geneous, isotropic, and elastic bodies in equilibrium;

the stress disappears at great distance from thecontact zone;

tangential stress components are zero at bothsurfaces within and outside the contact zone;

normal stress components are zero at both surfacesoutside the contact zone;

the stress integrated over the contact zone equalsthe forceg pushing the two bodies together;

the distance between the two bodies is zero withinbut finite outside the contact zone;

in the absence of an external force, the contact zonedegenerates to a point.

While not directly identified in his papers, Hertz alerts thereader to some assumptions generally accepted by hiscontemporaries. In the first paragraph of reference 1, heconsiders the contact force to be of the nature of whatwould be now termed a (Dirac) d -distribution:

In the theory of elasticity it is assumed that deformations arecaused by forces acting upon the interior and by forces acting uponthe surface of a body. Both kinds of forces may become infinitelyhigh in isolated, infinitely small portions of the body but so thattheir integrals over these portions are finite. If we then surroundthe point of discontinuity by a closed surface that is very small

d Reference 2, pp. 155-157 is preferred here because of the more concise formu-lation than that given in reference 1. Secondary assumptions of importance areidentified in the text below.e According to Hertz2, this assumption is the result of expanding around thepoint of contact the equations of the contacting surfaces and neglecting higherthan second-order terms.f To this statement should be added “as commonly used by mechanical engineersduring the late 19th century.”g Hertz calls this force Gesamtdruck (total pressure), which according to today’sterminology leads to a dimensionally incorrect end result for the contact stress,

AMP Journal of Technology Vol. 2 November, 1992 E.W. Deeg 15

15 a ,

2 d ,

14

c

c

compared to the overall dimensions of’ the body but very large com-pared to the dimension of the part in which the forces act, we canconsider the deformations outside of this surface independentlyfrom those inside. At the outside the deformations depend on theshape of the entire body, the distribution of the forces in generaland the finite integrals of the force components at the points of dis-continuity. Inside they depend only on the distribution of the forcesacting there: pressures and deformations are infinitely highcompared to those outside.

As is common in mathematical descriptions of’ physical sys-tems, the above assumptions are idealizations. The termsideally smooth, homogeneous, isotropic, elastic bodies, equi-librium, at great distance indicate this. For each concretecase, the degree of approximation of the physical system bythe theory must be considered and the effect on the resultsassessed accordingly. Frequently, comparing qualities issufficient to determine the utility of the theory. If thisapproach is not possible or its feasibility appears question-able, quantitative comparisons are necessary. Examples ofsuch critical evaluations are already given by Hertz1,2 andby Whittemore15. Neglecting these considerations has ledto misinterpretations and occasionally even unjustifiedrejection of the theory.

For the fiber-optic PC connections discussed in references12 and 13, such a comparison is summarized in Table 1.The result is that two PC fiberends in contact with eachother meet Hertz’s assumptions with satisfactory approxi-mation. However, if the entire system comprising fiber,adhesive layer and ferrule is to be analyzed, (c) and (g) arenot valid. Furthermore, if the connection is not alignedaxially, (a) and (e) may be violated. If one or both of theconnecting fibers are recessed, (a) is invalid. In the lattercase, the ferrules, not the fiber endfaces, establish theinitial contact, which may also cause violation of (i).

Table 1. Example for assessing applicability of Hertz’scontact stress theory. The fundamental assumptions of thetheory are compared with specific features of fiber-optic PCconnections described in references 12 and 13,

Fundamental FormulasHertz uses a Cartesian system of coordinates {x,y,z] withorigin at the initial point of contact of the two bodies andthe z-axis oriented parallel to the applied force. This re-quirement introduces an additional assumption if, for moregeneral reasons, the physical system is already described indifferently oriented Cartesian coordinates or in a differentcoordinate system. The slanted fiber-optic PC connection isan example. For optical reasons and because of its globalsymmetry, the z-axis would be chosen to coincide with theoptical fiber axis. In contrast, the z-axis used for a Hertziananalysis of the connection must be oriented parallel to thedirection of the slant.

Table 2. List of symbols. The third and fourth column showthose used in this paper.

In the following each of the two bodies in contact with eachother is identified by the subscript i (= 1, 2). To distinguishbetween the two principal radii of curvature of the surfacesof the two bodies at the contact point, a second subscript j(= 1,2) is introduced. The theory requires input of threesets of data:

(1)

(2)

(3)

a one-member set of the force p pushing the bodiestogether;

a four-member set of elastic constants, which withinthe theory can be reduced to a two-member set;

a five-member set of geometric parameters describ-ing shape and relative orientation of the contactingbodies.

The two elastic constants Young’s modulus E and Poisson’sratio v enter the final formulas in combination as the

h For a summary of Symbols and abbreviations,


see Table 2.

h8

“Hertz coefficient” Introducing an auxiliary angle W through

Four geometric parameters characterize the shape of thecontacting surfaces. They are two principal radii of curva-ture ril and ri2 for each body i with rij > 0 if the center ofcurvature is inside body i. The theory uses the rij in thefunctions

yields

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 ob-tains for the distance e normal to the {x,y} plane betweenpoints {Xl,yl} = (x2,y2}

j

From assumption (h) and the nature of e as a distance, itfollows that the conic described by equation (4) is anellipse, which includes the case of two contacting spheres,i.e. RI = R2 = 0. Contact between cylinders requiresadditional considerations and is discussed below.

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. Un-der zero load the new systems coincide with the primarysystem. A compressive force p applied parallel to the z-axiscauses a displacement of the secondary systems relative tothe primary system but so that, according to Hertzl, “theplane z = 0 in each of them is infinitely close to the surfa ceof the corresponding body and, thus, can be taken as thesurface itself. The direction of the z-axis is the direction ofthe normal to said surface.”k

Because the two surfaces described by equation (3) arepushed towards each other in the z-direction, the circum-ference of the contact zone is an ellipse. The lengths of itssemiaxes a and b are different from but their directionscoincide with the directions of the semiaxes of an ellipse ofequation (4), with e = constant. To link the force p and thegeometry of the contact ellipse, Hertz introduces a functionP(x,y,z), in reference 2 without explanation and in reference1 through an electrostatic model]

with u being the positive root of

With the abbreviations of equations (2), it is

k These statements introduce the requirement that inside the contact zonedeformations in the z-direction are small compared to the body dimension inthis direction.1 P is the potential of an ellipsoidal shell of zero thickness and zero extension in

Hertz uses Kirchhoffs elastic constants. For this paper they were converted to the z-direction (principal semiaxis c = 0), The justification given in reference 1the now commonly accepted Young’s modulus and Poisson’s ratio. cannot hide its axiomatic introduction. A concise review of potentials ofj This approach follows Hertz’s earlier article. i In the later one, Hertz derives ellipsoids, still written in a form that provides access to their use by Hertz’sexplicit equations for the surfaces yielding equations (5a) and (5b).2 contemporaries, is given by R von Mises.20

AMP Journal of Technology Vol. 2 November, 1992 E . W . D e e g 1 7

i

i

The body-specific material constants are then introduced The force p compresses the two bodies within the contactby two auxiliary functions 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) yieldsThe functions P and IIi are used mostly to prove the valid-ity of the theory within the set of assumptions (a) through(i)m. A slightly modified P(x,y,0) yields the final expressionsfor the shape of and the deformation and stress at the con-tact zone. To that end the displacements x i, h i and z i in the

Comparing equations (1 lb) and (12b) yields

directions x, y, and z, respectively, are written as

These equations contain as unknowns only the twosemiaxes of the circumference of the contact ellipse. Intro-ducing 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 termsfor the stress components. Of particular interest is the dif-ference of the deformations in the z-direction in the planez = 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 thetheory this paper concentrates on the derivation of the final formulas.


To find the quantity known today as Hertzian stress the infinite. In this case, a is determined by the global shape ofnormal stress component ZZ is calculated for the contact the bodies and not by the conditions existing at the contactzone where z = 0, which in equation (8b) yields u = 0. area. Thus, the cylinder/cylinder contact is actually outsideWith the validity range of the theory.”

To make the theory accessible to numerical application,Hertz expresses a and b by

where the dependency on the auxiliary angle W and theintegrals I(k) and J(k) of equations (15a) and (15b) areincluded in the two coefficients f and g. Because of equa-tion (16) they depend only on W. Hertz states in reference2, page 182

For u = 0 the derivative P/z assumes the indeterminateform 0 · ` �

��

the integrals in question can all be reduced to complete elliptic

bic equation (8b) and application of de l’Hospital’s ruleintegrals of the first kind and their derivatives after the modulus.Thus, they can be found via Legendre’s tables without introducing

yields for the normal stress (ZZ)Z=0 within the contact zone new quadrature. However, the calculations are extensive andtherefore I calculated the table given below.

His table, which is included here in Table 3, contains tothree decimal points the coefficients f and g for ten valuesof W. They are found in footnotes in references 1 (p. 164)

which for {x,y} = {0,0] is the Hertzian stress and 2 (p. 182); according to the latter, “interpolation be-tween these values will probably always offer sufficientaccuracy.”

If the contact is formed by two cylinders, Hertz considers inreference 2, p. 186/87 the contact ellipse for a Æ ` . Thisrequires also that p ` ¥ to maintain a finite value for theforce per unit length. He finds B = (rl + r2)/2 with ri beingthe finite radius of curvature of body i. It is also u << a2

which permits a to be moved in front of the integral inequation (1le). Finally he replaces the indeterminate formp/a = ` / ` by an arbitrary constant 4p' /3 where p' is theforce per unit length of the cylinder. The formulas for thesmall semiaxis and the stress Zz are under these assump-

Hertz leaves it to the reader to convert the improper inte-grals in equations (14) and (15) to integrals with finite up-per limits. He also does not describe how he reduced theseintegrals named here H(k), I(k), and J(k) to complete ellip-tic integrals of the first kind and their derivatives after themodulus. An attempt to reconstruct his approach is shownin Appendix A. However, without knowing exactly the re-sources Hertz used, it cannot be claimed that his methodwas the same. One must keep in mind that compendia wetake for granted21 23 were not yet printed.

tionsNEW ALGORITHM FOR HERTZ’S CONTACTSTRESS THEORY

General Remarks

andTo use Hertz’s theory in today’s environment it is essentialto describe by a condensed code the set of geometric pa-rameters of the system to be modelled. This is particularlyimportant for systematic studies of ordered or random

The compression a would now become logarithmicallyo 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 byhigh but finite ratios r l /r z in the regular formulas gives satisfactory approxima-tions.

Indices i = 1, 2 are left off here, The expressions apply equally to both bodies. P A similar statement is found in reference 1, p, 164, footnote 1.

AMP Journal of Technology Vol. 2 November, 1992 E . W . D e e g 19

it isn

n

p

. . .

-

Æ

i i

Some rewriting involving the solution of the cu-

Table 3. Factors f and g for calculating semiaxes a and b ofquadruple of principal radii

the contact ellipse. The program described in this paperyields values listed in columns fEWD and gEWD.

The angle w, the fifth element of the set of geometric pa-rameters, 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. Eachordered quadruple now depends on the integers m andn, i. e.

sequences of the rij. Such sequences occur e.g. during de-sign optimization or in determining the effect of variationsof processing conditions on shape and size of the contactzone and on Hertz stress. The methodology is describedbelow in the section on characterizing combinations of con-tacting surfaces; examples are found in reference 12.

It is also essential to streamline the present method fornumerical evaluation of Hertz’s formulas using personalcomputers or workstations and software not requiring con-ventional programming. A simple approach would be towrite the table of coefficients f and g and the conventionalmethod onto disk. More desirable would be to write com-puter programs incorporating algorithms for completeelliptic integrals The most natural approach, however,would utilize Hertz’s fundamental formulas directly andintegrate them without transformation to elliptic integrals.This method and a corresponding computer program aredescribed below.

Characterization of Combinations of ContactingSurfacesAt the point of contact the surface of each of the two bod-ies i (= 1, 2) is defined by two principal radii of curvature rij

with j (= 1, 2). Thus, regarding their shape the pair ofcontacting bodies can be characterized by an ordered

q In connection with the work described here several such programs were writtenand tested. References 21 through 24 were consulted and used. Of particularhelp was Hastings’ polynomial approximation of complete elliptic integrals.25

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 Gserve as reference describing two spherical surfaces in con-tact. The remaining elements q w ,m,m would then describegeometrically deviating combinations. Because of the formin which the rij are expressed in equations (22a) and (22b),this can be accomplished by choosing M and N as odd num-bers. Depending on the values chosen for the members ofthe set {w , rl0 , r20 , D rll, D r12, D r21, D r22, M, N] variouscombinations of ellipsoidal and spherical surfaces can bedescribed.

For simulations of random variations of the surface shapes,the monotone increasing sequences for m and n inequations (22a) and (22b) are replaced by randomizedsequences. The MATLAB program given in Appendix Bserves this purpose. It generates four sets of random num-bers mj and nj, enters them into equations (22), computesthe rij and plots bar charts of their distribution.

Direct Quadrature of Hertz’s Basic ExpressionsThe earliest point in Hertz’s derivations where moderncomputational techniques could be applied without violat-ing his fundamental concepts is found in reference 1,pp. 164/65, or in reference 2, pp. 181 and 183. The


21

corresponding equations above are (14) and (15). Thetransformation

printing of the entire program, the summary shown inTable 8 is convenient. It fits on an 8.5 x 11 sheet.

Table 4. Data input for the MathCAD program. The linesseparate the three sets of data. Note the changed symbols r ij

changes the integrals in equations (14a), (14b), (15a), and to rij etc.(15b) to

For 0 < k # 1 these integrals converge absolutely. Theirvalues can be found e.g. by Simpson’s rule or Romberg’smodification of the trapezoidal rule.26 Both can be accessedthrough one of the MathCAD versions. With these, as wellas other software packages, time-consuming, traditionalprogramming can be avoided. MathCAD version 2.5 waschosen here because of the transparency of its symboliclanguage, its ease of access and the very reasonable hard-ware requirements.r The symbols used for the variables inthe program had to be adjusted to MathCAD terminologyand differ from those in the text. Both are included inTable 2.

The program is listed in Tables 4 to 7. It follows essentiallythe sequence of formulas above. Because the results of theroot operator are influenced by the seed value and to opti-mize for speed, three different seed values Kl, K2, and K3are pre-selected, each for a specific range of the auxiliaryangle W The conditional statements “if [. . .]” in the pro-gram identify the proper range and, without operatorassistance, select the proper seed value. The program cal-culates semiaxes a and b of the contact ellipse, area Q ofthe 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. Ifdesired, other areas of interest, for instance those of fibercore and fiber cladding12, can be plotted in the same dia-gram. For plots in Cartesian coordinates of the contactzone the parameter t with 0 # t 2 p and S : = 75,s := 0..S, t s := 2 p ·s/S is introduced in

The value S := 75 gives satisfactory resolution of the plotsbut can be reduced to increase speed. To avoid repeated

MathCAD 2.5 requires IBM PC, PC/XT, PC/AT or compatibles with 512KRAM, floppy or hard disks and one of the common graphics adapters. Numeri-cal coprocessor, although recommended, are not required. It works in MS-DOSor PC-DOS and supports a wide range of common printers and plotters.

Table 5. Definition of functions for converting primary inputdata to intermediate quantities used by the program. Thefour-member set of material constants {El, E2, vI, v2] isreduced to the two-member set {01, 02], the five-memberset of geometric parameters {rl1, r12, r21, r22, w ] is re-duced to the four-member set {R, Rl, R2, W ].

Table 6. Functions for computing the ratio k = b/a of thesemiaxes of the contact ellipse. The lower part of the tableshows the pre-selected seed values K1, K2, K3 and theconditional statements for their optimal selection.

AMP Journal of Technology Vol. 2 November, 1992 E.W. Deeg

r

#

based. It shows also that for each concrete case the degreeTable 7. Definition of functions for data output. of approximation between the features of the physical sys-

tem to be modelled and the entire set of Hertz’s assump-tions must be compared. Neglecting these considerationscan lead to misinterpretation of theoretical predictions andto unjustified rejection of the theory. The review of Hertz’stwo original publications also reveals expressions that canbe written as computer algorithms requiring only input ofmaterial constants, applied force, and geometric parame-ters describing the contacting surfaces. The programs makeuse of readily available software packages without resortingto traditional computer programming and permit applica-tion of the theory in its most general form. Among severalprograms developed and tested for this purpose, one wasselected because of the transparency of its terminology,

Table 8. Summary of input, selected intermediate, and out- ease of access, and modest hardware requirements. Writ-put data. The box marked PLOT contains the contour of the ten in MathCAD version 2.5, the program is described incontact ellipse in the {x,y} plane according to equation (28). detail.

SUMMARYThis paper could be subtitled “What if Heinrich Hertzwould have had a personal computer?” The high-speedcomputing tools we are accustomed to did not exist whenhe published his theory about a century ago. To make histheory accessible for engineering applications, he devel-oped a numerical method with mathematical tools knownto his contemporaries. If Hertz had had access to today’scomputing tools, he most likely would not have calculatedthe “little table which, in most instances makes the ratherextensive arithmetic superfluous” (reference 1, p. 164).

A review of his papers leads to a clear identification of thecomplete set of basic assumptions on which his theory is

Optimizing design or studying the effect of variations ofmanufacturing conditions on shape, size, deformation, andstress in the contact zone requires large sequences ofsystematically or randomly modified principal radii of cur-vature of the contacting surfaces. Such sequences can becharacterized and described by a matrix whose elementsare ordered quadruples of the four radii of each individualcombination. For monotone sequences of the principalradii of curvature, the elements are algebraically relatedand by proper selection of their number the centralelement of the matrix represents a sphere-on-sphere con-tact which serves as reference.

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25. C. Hastings, Jr., Approximations for Digital Computers,(Princeton Univ. Press, Princeton).

26. L.W. Johnson and R.D. Riess, Numerical Analysis, 2nded., 323 (Addison-Wesley, Reading, MA, 1982).

Emil W. Deeg recently retired from his position as ProjectManager, 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 Silikat-forschung in Wuerzburg, West Germany. As original con-tributor and throughout his more than 25-year career asR&D manager and executive for several international cor-porations he authored or co-authored more than 70 articlesin physics, ceramics, glass science and engineering, and abook on glass in the laboratory. He holds over 40 patents inthe same fields. Dr. Deeg served as a member of the Inter-national Commission on Glass (1963–1981, offices held), asa member of the International Commission for Optics(1964–1966), and as a consultant to the NASA SpacelabProgram (1971–1978). He is a fellow of the American Ce-ramic Society, was inducted into the Hall of Fame of Engi-neering, Science and Technology, and is listed in Who’sWho in the Worldj Who’s Who in Finance and Industry, andother biographical publications. Dr. Deeg joined AMP in1984 as Project Manager, Materials Engineering.

APPENDIX AWith q being the modulus, the complete elliptic integral ofthe first kind is

Setting q = 1 – k2 in equations (25) and (26) but q = 1 –1 /k2 in (27), the denominator of integrals H(k), I(k) andJ(k) is

and the integrals in equations (26) and (27) assume theform

Mathematical Functions (Dover, New York, 1972).

AMP Journal of Technology Vol. 2 November, 1992 E . W . D e e g 2 3

and the first derivative of K(q) after the modulus is

H(k) is already of the form K(q). Equation (A2) in (Al) APPENDIX Byields MATLAB program for generating random distributions of

radii of curvatures rij.Ml= input(’upper limit of m1 Ml = );N1 =input(’upper limit of n1 N1 = );M2=input(’upper limit of m2 M2 = );N2= input(’upper limit of n2 N2 = );xl =input(’seed for m1-sequence xl = );yl =input(’seed for n1-sequence yl = );x2=input(’seed for m2-sequence x2 = );y2= input(’seed for n2-sequence y2 = );rl0=input(’base radius, endface #1 r10 = );r20=input(’base radius, endface #2 r20 = );dr11 = input(’r11 step size drll = );dr12=input(’r12 step size dr12 = );

Comparing equations (A4) and (A5) yields the functional dr21 =input(’r21 step size dr21 = );relationships for I(k) and J(k) as stated by Hertz: 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


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