Mathematical Theory of Elasticity [2 ed.] [Second ed.]

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MATHEMATICAL THEORY OF ELASTICIT1

Mathematical Theory

f!f Elastuzty

1. S. SOKOLNIKOFF Profe88OT oj Mathematic8 University of California L08 Angele8

SECOND EDlTlON

TATA McGttAw.tHLL PUBLISHING COMPANY LTD.

",Bomba,-N_ DeIhl

ACC

. i.. >.J.,\'P

tI()._~ ;3,7 98

CL.IIIO._ ••_.

....

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------.:::::::':'::-:&:-:.1 ~~~ /

-

MATHEMATICAL THEORY Of ELASTICITY

Copyright @ 1956 by the McGraw-Hili Book Company, Inc. Copyright 1946, by the McGraw-Hili Book Company, Inc.

All Rights Reserved. This book. or parts thereof. may not be reproduced ill any form without permission of the publishers.

T M H Edition

Reprinted in India by arrangement with McGraw-Hill Book Company, Inc.

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at Rekha Printers, Mew Delhl.55

PREFACE The theory of elasticity, in its broad aspects, deals with a study of the behavior of those substances that possess the property of recovering their size and shape when the forces producing deformations are removed. In common with other branches of applied mathematics, the growth of this theory proceeded from a synthesis of special ideas and techniques devised to solve concrete problems. This resulted in a patchwork of theories treating isolated classes of problems, determined largely by the geometry of bodies under consideration. The embedding of such diverse theories in a unified structure, and the construction of the analytical tools for calculating stresses and deformations in a strained elastic body, are among the dominant concerns of the mathematical theory of elasticity. This book represents an attempt to present several aspects of the heory of elasticity from a unified point of view and to indicate, along with the familiar methods of solution of the field equations of elasticity, 30me newer general methods of solution of the two-dimensional problems. The first' edition of this book, published in 1946, had its origin in a course of lectures I gave in 1941 and 1942 in the Program of Advanced Instruction and Research in Mechanics conducted by the Graduate School of Brown University. In those lectures I stressed the contributions to the theory by the RuSsian school of elasticians and, in particular, the relatively little--known work of great elegance and importance by N. I. Muskhelishvili. I planned ~o supplement that book by a companion volume dealing with effective methods of attack on the twodimensional and anisotropic problems of elasticity. The developments in the intervening years, however, were so rapid that I was urged to puhlish instead a single volume containing an up-to-date treatment of material presented in the. first edition and supplement it with new topics, in order to give a rounded idea of the current state of the subject. The present edition differs from its predecessor by extensive additions ad ~ Most of the material appearing in the .last three chapters had no counterpart in the first edition. Throughout r bve tried to give " .clea1' indieation of the fronti.,rs of the developments, and 1 have constantlY kept in mind those readers whose principal conrero is with prallw.l appication of the theory. While nfo) volume of this size can lay .... to an _ _ _ve list of referenoos to reeeateh lite~, I bave

vi

PREFACE

selected such references with care so as to give an accurate picture of the present state of the topics treated in this book. I deliberately omitted any dis~on of the theory of shells, because a palatable treatment of the shell theory cannot be made in the space of fewer than 300 pages. The best available treatment of this subject, in my opinion, is given in a Rllssian monograph by A. L. Goldenveiser, Theory of Thin Elastic Shells, .Moscow (1053), 544 pp. The first three chapters, despite their brevity, contain a comprehensive treatment of the underlying theory of mechanics of deformahle media. These chapters are essential to the understanding of the remaining chapters, which can be read independently of one another. Chapter 4 gives an up-to-date treatment of extension, torsion, and flexure of beams, including the deformation of homogeneous and nonhomogeneous beams by loads distributed on their lateral surfaces. Chapter 5 is concerned with two important categories of plane problems of elasticity. It contains an account of the general modes of attack on such problems with the aid of the theory of functions of complex variables. Although a clear indication of the use of such methods in the problems. of transverse deflection of thin plates is made, illustrations are chosen mainly from problems on plates in the states of plane strain or generalized plane stress. Chapter 6, dealing with the three-dimensional problems, is brief for the simple reason that effective general techniques for the analytic solution of such problems still remain to be developed. The most promising approach in this connection appears to be (as in Chap. 5) via the use of general solutions of Navier's equations in terms of harmonic functions. The chapter contains a formulation of thermoelastic problems and an introductory account of the theory of vibrations and propagation of waves in elastic media. Chapter 7 on Variational Methods contains a treattnent of the energy theorems in elasticity and their bearing on the variational methods of solution of elastostatic problems. I have tried to present the variational techniques of Ritz, Galerkin, Trefftz, Kantorovich, and others in a unified way, without resorting to function space methods so as to make matters meaningful to a wider circle of readers. This chapter includeR a discussion of the method of finite differences and relaxation, which are frequently used when analytic methods fail. This volume owes much to the recent contribution to elasticity made by Russian scholars.. Suitable acknowledgment to sources is made throughout this volume, but my chief debt is to Academician N. I. Muskhelishvili, whose unparalleled monograph, "Some Basic Problems of the Mathematical Theory of Elasticity," Moscow (1954), originally published in 1933 and now in the fourth edition, bas left. an indelible imprint. A large part of the tnaterial in this volume was prepared in the (lOUl'l!e

vii

PREFACE'

of the investigations and lectures I gave during my tenure as a Guggenheim Fellow during the academic year 1952-1953. I am pleased to have this opportunity to acknowledge my gratitude to the Guggenheim Memorial Foundation, whose grant enabled me to discuss this book with my colleagues in England and on the Continent. I also wish to repeat an acknowledgment, made in the Preface to the first edition, to the Wisconsin Alumni Research Foundation for a grant-in-aid that facilitated the publication of the predecessor of this volume. I am indebted to Dr. George E. Forsythe, Research Mathematician at the University of California at Los Arigeles, for material on Relaxation Methods in Sec. 125, and to Robert K. Froyd, Research Assistant at the University of California at Los Angeles, for his help in proofreading and preparing the index matter. . I.

S. SOKOLNIKOFF

CONTENTS v

PBml'ACI!l . HISTORICAL SKETCH.

CHApTI!lR

1. 2. 3. 4. 5. 6.

'1. 8. 9. 10. 11.

1.

ANALYSIS OF STRAIN

Deformation Affine Transformations . Infinitesimal Affine Deformations A Geometrical Interpretation of the Components of Strain Strain Quadric of Cauchy Principal Strains. Invariants General Infinitesimal Deformation . Examples of Stmin Notation Equations of Compatibility . Finite Deformations. 2. ANALYSIS OF STRESS Body and Surface Forces. Stress Tensor . Note on Notation and Units. Equations of Equilibrium Transformation of Coordinates . Stress Quadric or Cauchy Maximum Normal and Shear Stresses. Examples of Stress

35 36

39 40 42 45

Mohr's Diagram

3. EQUATIONS 01' ELASTICITY. 20. Hooke's Law . 21. Generalized Hooke's Law 22. Homogeneous Isotropic Media 23. Elastic Moduli for ;sotropic Media. Simple Tension. Pure Shear. Hydrostatic Pressure. 24. Equilibrium Equations 'or an Isotropic Elastic Solid 25. Dynamical Equations o' an Isotropic Elastic Solid . 26. The Strain-Energy Func·,ion and Its Connection with Hooke's Law 27. Uniqueness oC Solution. Remarks on Existence of Solution 28. Saint-Venant's Principle.

CHAPTER

CHAPTIIl1l4.

6 9 12 14 16 20 23 25 25 29

35

CIlAPTER

12. 13. 14. 15. 16. 17. 18. 19.

5 5

ExTI!lNSION, TORSION, AND FIJilIroRm 01' BEAMS.

29. Statement of Problem 30. Extemon of Beams by Longitudinal Forces 31. Beam Strstehed by Its Own Weight

19

63 56 56 5ll 66

67 71 80 81

86 89 91

91

32. Bending of Beams by Termin&! Couples

96 97 100

33: Torsion of & Circular Shaft .

107 is

CONTENTS

34. Torsion of Cylindrical Bars

109 114 36. Torsion of Elliptical Cylinder 120 124 37. Simple Solutions of the Torsion Problem. Effect of Grooves 38. Torsion of a Rectangular Beam and of a Triangular Prism . 128 39. Complex Form of Fourier Series 134 137 40. Summary of Some Results of the Complex Variable Theory 41. Theorem of Harnack. 143 42. Formulas of Schwarz and Poisson 145 43. Conformal Mapping . 147 44. Solution of the Torsion Problem by Means of Conformal Mapping 151 45. Applications of Conformal Mapping 157 46. Membrane and Other Analogies. 165 47. Torsion of Hollow Beams 169 177 48. Curvilinear Coordinates. 49. Torsion of Shafts of Varying Circular Cross Section . 186 50. Local Effects . 190 51. Torsion of Anisotropic Beams . 193 52. Flexure of Beams by Terminal Loads 198 53. Center of Flexure . 204 54. Bending by a Load along a Principal Axis. 208 55. The Displacement in a Bent Beam. 209 56. Flexure of Circular and Elliptical Beams 213 57. Bending of Rectangular Beams. . . 217 58. Conformal Mapping and the General Problem of Flexure; the Cll.rdioid Section. 219 J\9. k-wii."G..f c:.i.',A: + (au + a,,)AIA. + (0113 + (32)A.A s + (au + al3)AaAI.

A aA =

Since for a rigid body transformation IlA vaniRhes for all values of AI, A., As, we must have (l11

au

=

(l"

=

(l ..

= 0,

+ au = a.3 + a32 = au + au =

O.

Hence a necessary and sufficient condition that the infinitesimal transformation (3.1) represent a rigid body motion is (3.3)

lXii

=

(i,j = 1,2,3).

-ai',

In this case, the set of quantities lXii is said to be skew-symmetric. When the coefficients aij are skew-symmetric, the transformation (3.1) takes the form aA I = - a.,A. /lA. = a.IA I /lA. = -a"AI

+ a32A.

This transformation can be written as the vector product of the infinitesimal rotation vector Cd = e."Wj and the vector A, namely I

IlA=CdXA=

if we take \

= -a23 = ~(a32 - au), = - a n = 7\I(a13 - au), a21 = -au = ~(a21 - au).

WI"" (l32

(3.4)

{

w. == W3

==

(l13

The eque.tions representing the rigid body motion can be obtained by observing that Ai = x, - x? and that

Mi = A: - A,

= (x: - xf;) - (x. - xl') "" (x: - Xi) - (x~' = &x, - Ilx2

or Ox;

=

ox~

+ &Ai =

xn

Ilx~

+ (Cd X A),.

I We reeall that when a rigid body rotates with the angular velocity 0, the linear ·velocity ... is ... - 0 X A and 8A - 0 X A at - .. X A, where", _.Q Uis the infinitesimal angle of rotation.

11

ANALYSIS OF STRAIN

Then the rigid body portion of the infinitesimal affine transformation (2.1) can be written as (3.5)

8x, = 8xt - ",.(Xt - x~) + "'2(X, - xl)., 6x. = 8xg + "'3(X, - xY) - w,(xa - x~), { 6X3 = 6x3 - "'.(X1 - xV + ",,(x. - xi).

The quantities oxf "" x2' - x? are the components of the displacement vector representing the translation of the point PO(x·) (see Fig. 1), while t,he remaining terms of (3.5) represent rotation about the point po. At the beginning of this section, we proposed the problem of separatirig the infinitesimal affine transformation oA, = a;,;Aj into two component transformations, one of which is to represent rigid body motion alone; we have seen that this rigid body motion corresponds to a transformation in which the coefficients are skew-symmetric; that is, lXij = -aj'. Now,any set of quantities a'j may be decomposed into a symmetric and a SKewsymmetric set in one, and only one, way.1 We can thus write aij

=

Yz(ai;

+ aj,) + Yz(ai;

- aji).

Then Eq. (3.1) can be written as

oA,

= a'jA; = [Yz(lXi;

+ aj,) + Yz(lXi;

- aji)]A j ,

or (3.6) where eij = Wij

=

eji -Wji

== ==

+

Yz (ai, aji) , %(aij - (Xji).

The skew-symmetric coefficients "'i; correspond to a rigid body motion, and from (3.4) it can be seen that they are connected with the components of rotation, WI, W2, W3, by the relations

It is clear from Eqs. (3.6) for the transformation of the components of a vector that an infinitesimal affine transformation of the vector Ai can be decomposed into transformation /lA, = wijAi , representing rigid body motion, and into transformation (3.7)

/JA, = evA;,

representing pure deformation. The symmetric coefficients i!;j are called components of the strain tensor, and they characterize pure deformation. We shall investigate the properties of the strain tensor in the next section. 1

See Prob. 1 at the end of this chapter.

12

MATBElUTICAL THEOllT OJ' ELASTICITY

t. A Geometrical Interpretation of tile Componeats of Strain. The geometrical significance of the components of strain 8iJ entering into (3.7) can be readily determined by inserting the expressions (3.7) in the formula (3.2), which then takes the form A aA = Ai aA, = eo/A,A i, or (4.1)

&A

evA,Ai

A=~'

If initially the vector A is parallel to the Xl-axis, so that A = Al and At == A, = 0, then it follows from (4.1) that

OA

A = ell·

(4.2)

Thus, the component ell of the strain tensor rep1'esents the extension, or change in length per unit length, of a vector originally parallel to the Xl-axis. ~a

FIG. 2

Hence, if all components of the strain tensor with the exception of el1 vanish, then all unit vectors parallel to the Xl-axis will be extended by an amount 611 if this strain component is positive and contracted by the same amount if ell is negative. In this event, one has a homogeneous deformation of material in the direction of the :kaxis. A cube of material whose edges before deformation are I units long will become a rectangular parallelepiped whose dimensions in the x1-direction are 1(1 + ell) units and whose dimensions in the directions of the Xr and x,-axes are unaltered. A similar significance can be ascribed to the components e22 and e83. In order to interpret geometrically such strain components as e23, consider two vectors A = etAs and B = eaR. (Fig. 2), initially directed along the Xr and x,-axes respectively . Upon deformation, these vectors become A' = el BAt BI == el &B t

+ e,(A + 6A,) + e, flA., + e, 6B + ea(Ba + 4B.). I

2

We denote the angle between AI and B' by

II and consider the change /2) - II in the right angle between A and B. From the·definition

CII", ... ( ..

13

ANALYSIS OF STRAIN

of the scalar product of A' and B', we have A'B' COIl

e ... A' . B'

+ (A. + M,) aB. + (B. + liB.) M. + B. IiAa,

'" aA I aBI :, A2 liB.

if we neglect the products of the cha.nges in the components of the vectors A and B. To the same a.pproximation, we ha.ve

cos

(4.3)

e=

A'·B' A'B'

V(M 1) ' + (A. + oA.)· + (M.)' V(IlB 1)' + (oB.)· :, (A. aB. + B. IiAs)(A. + 1iA.)-I(B s + IiB.)-1 :, A, IiB z + Bs liA. = liB. + liAs. A.Bs B. A.

+ (B. + liB,)'

Since all increments in the components of A and B have been neglected except liA. and liB" the deformation can be represented as shown in Fig. 3. If we remember that Al

As = BI = B. = 0,

=

,xa

then Eqs. (3.7) yield (4.4)

oB 2

= e••B.,

6B~B'

liAs

= e23A •.

=

cos (~ - a ••)

. = au

liB.

__ ~

_. (/ I

R -..:' .::::::::': ___ ,Q " I I

From (4.3) we have cos 9

~ ~

I I

I I

II

=

liA.

"P ,-;, IiAa

sin a ..

= If; + L =

2e fI,

or Hence a positive value of 2e23 represents a decrease in the right angle between the vectors A and B, which were initially directed along the positive Xr and x.-axes. Again, from (4.4) and Fig. 3 we see that LPOP' :, tan POP' =

1i:

• 2

= 6.s,

, &B

LROR' :, tan ROR' = B: = en.

Since the angles POP' a.nd ROR' a,re·equa.l, it follows tha.t, by rota.ting the parallelogram R'OP'Q' through an a.ngle 623 about the origin, one can obtain the configuration shown in Fig. 4. Obviously it represents a. slide or a shear of the elements parallel to the zlxrpiane, where the amount of slide is proportional to the distance x, of the element from the xlxrplane.

14

AlA'l'BEMATICAL 'l'BEORY OF ELASTICITY

A similar interpret.ation can obviously be made in regard to the components e12 and e31. It is dear that the areas of the rectangle and the parallelogram in Fig. 4 are equal. Likewise an element of volume originally cubical is deformed into a parallelepiped. and the volumes of the cube and parallelepiped are equal if one disregards the products of the changes in the linear elements. Such deformation is called pure shear. The characterization of strain presented in Secs. 3 and 4 is essentially due to Cauchy. It should be noted that the strain components e'i refer to the chosen set of coordinate .r2 axes; if the axes are changed, the eij will, in general, assume different values. FIG. 4 5. Strain Quadric of Cauchy. With eac):; point PO(XO) of a continuous medium, we shall associate a quadric surface, the quadric of deformation, which enables one to determine the elongation of any vector A

= e,(x,

- x?)

that runs from the point PO(XO) to some point P(x). Now if a local system of axes Xi is introduced, with origin at the initial point pO(XO) of the vector A and with axes parallel to the space-fixed axes, then formula (4.1) characterizing the extension e = 8A/ A of A can be written as (5.1) eA' = e,jXixi. We consider the quadratic function (5.2)

and constrain the end point P(x) of the vector A, as yet unspecified, to lie on the quadric surface (5.3)

2G(Xl, x., x.)

= ±k',

where k is any real constant and the sign is chosen so as to make the surface real. Comparison of (5.3) with (5.1) leads to the relation (5.4)

and the strain quadric takes the form (5.5)

From (3.4) we see that the extension of anllline througk PO(XO) ill irwer8elg along the line frem the point (~O), at wkick the Btrain is being Bttldied, to a peint (x) on the qttadric

~ to tkufjtlare of the radim vector tkat TUm

15

ANALYSIS OF STRAIN

surface. Accordingly, the maximum and minimum elongations Will be in the directions of the axes of the quadric (5.5). We refer the quadric surface of deformation (5.5) to a new coordinate system x~, x;, x;, obtained from the old by a rotation of axes. Let the directions of the new coordinate axes be specified relative to the old system Xi by the table of direction cosines

x:

_ _x_' I~_:_ x~

111

x~

z.,

X~

I

131

112

113

122

I •• 133

1,2

in which l;j is the cosine of the angle between the x;- and the xraxes. old and the new coordinates are related by the equations

x, = lllx~ x. = lux; x, = lux;

The

+ 121X; + lux;, + lux; + 1,,,:1:;, + 1.,,:1:; + laax;,

or, more compactly, (5.6)

Xi

It i5.retJ,dily shown that the

=

in~'erse

l...z~.

trnnsformtJ,tion is of the formJ

The well-known orthogonality relations between the direction cosines can be written in the form (5.7)

x:

When the quadric surface (5.5) is referred to the coordinate system, a new set of strains e;j is determined and (5.5) is replaced hy the new equation of the surface, namely, e~p:xi =

±k2•

The right-hand member of (5.5), however, has a geometrical meaning that is independent of the choice of coordinate system (± k' = eA'); consequently (5.8) In other words, the quadratic form e;jXaj is invariant with respect to an orthogonal transformation of coordinates. Equations (5.6) and (5.8) together yield

e;tl...lf1iX~ 1

See Prob. 5 at the end of this chapter.

= e~~:.xft,

16

MATHEMATICAL THEOBY OF ELASTICITY

and since the x~ are arbitrary, e~=~.

(5.9)

Similarly it can be shown that (5.10)

6«/1

= UJ~J'

A set of quantities e;J transforming according to the law (5:9 ) is said to represent a cartesian tensor of rank 2. We shall meet several jjuch tensors in the subsequent discussion. Differentiating 2O(xl, X2, X3) = e,j%;%J and noting from (3.1) that for a pure deformation ~A, = e;;A j = e;j%}, we find that (5.11)

aG

-a x,

=

e;j%J

==

~A,.

But~ are the direction ratios of the normal v to the quadric ~rface (5.5)

"x.

at the point (x,), and it follows that the vector 6A is directed along the normal to the 2 Af"--~v plane tangent to the surface eifX;%j = ±k (see Fig. 5). This property of the 8t~aiU quadric will prove useful in. the Ilext sectIon, ~\:..""~ ~"" ~~ ~"" 1>~",~\. 'U:ea.. 'If. t,ru._ quadric surface and their signifieance for the FIG. 6 deformation. S. Principal Strains. Invariants. We seek now the direcfion ratios of the lines through (X O) whose orientation is left unchanged by jlhe deformation SA, == e;;A;. If the direction of the vector A is not aH,ered by the strain, then ~A and A are parallel and their components are pfoportional.l Therefore

It should be noted that e = aA i is the extension of each cowponent of A Ai and is thus the extension of A itself, or e = M/ A. Equation (5.1) then shows that the extension e is given by the expression e = e;j%f£;/ A 2. We return now to aA. == e;;AJ. from which it is seen that (6.1)

or (6.2) ThhI set (6.2) of three homogeneous equations in the u"mowns AI ptlI!S6IiIIteI a nonvanishing solution if, and only if, th", determJ.nant of the t In other WOrds, the directions we seek are those of the axes m the quadric (6.6); that., we wek the directions yielding the extreme values at t-he.elQaptiOlIIlS•

17

ANALYSIS OF 8'l'IlAIN

coefficients of the AJ is equal to zero; that is,-

lev -

(6.3)

elql == 0,

or en-e en en \ ell en - e en = O. \ en e" en - e

We prove next that the three roots el, eo, ea of this cubic equation in the elongation e are all real. Let the three directions determined by the three numbers ej be given i

by the vectors l A.

In this notation, formula (6.1) becomes, (or any root 1

e1 A j

1

ej.A •.

=

2

We mUltiply both sides by Ai and sum over j, getting 1 z 1 1I (6.4) 61A j A j = ej.A.A,. 2

2

Similarly, from e.A; = e;.A. we have 12

12

12

12

e.AJA; = e,.A;A. = e.;A.A; = e;.A.A;,

(6.5)

wherej and k have been interchanged and the symmetry of e;. exploited. Comparsion of (6.4) and (6.5) shows that 1

2

(el - e.)AjA; = O.

(6.6)

Now if we assume tentatively that (6.3) has complex roots, then these can be written where 'El' E I , e. are real.

If e, = El - iE. is substituted for e in (6.2), 2

it will be found that the resulting solutions A; ...

O-j -'

ib; are the complex

1

conjugates of AJ ... a, + ibj , where the latter are obtained by putting e = el = E, + iE~. Therefore 1 2

A;A; == (a; = af

+ ibs)(a; - Wj) + iii + al + bf + bi + b: ~ O.

Hence it follows from (6.6) that e, - e, ... 2iEs = 0, or E. "" 0, and the roots I!i are all real. l The indexi over It. indicates not the jth compo_t but rather the jth vector and ita depeude_ ul)OIl the root 'I of the determinantal equation (6.S).

18

IlATBE¥ATICAL 'rHEOltY OJ' EL.\STICITY

From (6.6) it follows that, if the roots el and e, are distinct, then 1 1I

1

2

A,A, = A • A = 0, j

so that the corresponding directions are orthogonal. These directions A are called the principal directioml of strain, and the strains e" which are i

the extensions of the vectors A in the principal directions, are termed the principal strains.

We have seen that at any point (XO) there are three mutually perpen-

.i

dicular directions (assuming, for the moment, that the eo are distinct) that are left unaltered by the deformation; consequently the vectors i

i

i

i

A, the deformed vectors A + liA, and 8A are collinear. But (5.11) shows that liA is always normal to the quadric surface (5.5), and therefore the principal directions of strain are also normal to the surface and must be the three principal axes of the quadric ei.jXixi = eA 2. If some of the principal strains Ii; are equal, then the associated directions become indeterminate but one can always select three directions that are mutually orthogonal. If the quadric surface is a surface of revolution, then one 1

direction A, say, will be directed along the axis of revolution and any two I

mutually perpendicular vectors lying in the plane ·normal to A may be taken as the other two principal axes. If e1 = e2 = ea, the quadric is a sphere and any three orthogonal lines may be chosen as the principal axes. We recall that 61,62, e, are the extensions of vectors along the principal axes, while ell, e22, e" are the extensions of vectors along the coordinate axes. If the coordinate axes Xi are taken along the principal axes of the quadric, then the shear strains eu, e23, e3l disappear from the equation of the quadric surface and the latter takes the form

The cubic equation (6.3) can be written in the form (6.7)

where 6 1, "2, ". are the sums of the products of the roots taken one, two, and three at a time: (6.8)

"I {

+ + + +

= e1 e2 e. "" 6, U2 = ese. esel ele., " .... 'lete••

,fty expanding the determinant (6.7), we see that these expressions "Can • be written as

19

ANALYBIS Ql!' STBAlN " ... 611

+ 6.1 + 6 • ., + eu6'u + ellell -

iJs = 61tBaa

(6.8)

I

eil - eft - ef.

I I

1

en enl eu en eu ell = e.. e.. + eu en + e12 e22' ". = eueue,s + 2elle..e31 - en4. - ene:l - eaaef.,

I

eu

eu

e12

The expressions for '" and ". can be written compactly by introducing the generalized Kronecker deUa, B''!o~:::, which we now define. If the subscripts p, q, r, . . . are distinct and if the superscripts i, j, k, . . . are the same set of numbers as the subscripts, then the value of 0:/':::: is defined to be + 1 or -1 according as the subscripts and superscripts differ by an even or an odd permutation; the value is zero in all other cases. We can now rewrite the formulas (6.9) in the form {} = (6.10)

(i = 1, 2, 3),

eii,

1

""

fl. =

21 B'Jqe",.eq;,

(i, j, p, q = 1, 2, 3),

". =

~ o~:repie ojerk,

(i, j, k, p, q, r = 1, 2, 3).

Since the principal strains, that is, the roots e" e., e. of (6.7), have a geometrical meaning that is independent of the choice of coordinate system, it is clear that ", "" and ". are invariant with respect to an orthogonal transformation of coordinates. [Note that this invariance could have been used to derive expressions (6.8) from (6.9).] The quantity " has a simple geometrical meaning. Consider as a volume element a rectangular parallelepiped whese edges are parallel to the principal directions of strain, and let the length!! of these edges be h, I" l.. Upon deformation, this element becomes again a" rectangular parallelepiped but with edges of lengths ll(l + el), l2(1 + e.), l.(l + e.). Hence the change oV in the volume V of the element is

OV = I,Z.za(l

+ el)(1 + e2)(1 + e.)

= 1Il.l.(el

plus terms of higher order in

~l

+ ell + e.) eo.

- hZ.z.

Thus

BY

+ e, +61 - " - V'

and the first strain invariant" represents ;he expansion of;i unit ~olume

20

MA.THEMATICAL TllEORY OF ELASTICITY

due to strain produced in the medium, For this reason , ill CAlled the cubical dilatation or simply the dilatation, PROBLEMS 1. Determine the principal directions by finding the extremlll values of

Note that the x,/ A - P, are the direction cosines so that e - 6;/PiPj, }'Iaximize this Bubject to the constraining condition 1. 1I. Refer the quadric of deformation to a set of principal axes, and dismISS the nature of deformation when the quadric is an ellipsoid and when it is a hyperbOloid, Draw appropriate figures and note that if 61 > 0, e. > 0, e. < 0, then, depending on the direction of the vector A from the origin of the quadric, one must consider the surfaces e1~ + es4 - le,lz: - ±kt ,

.,V, -

'1. General Infinitesimal Deformation. In the preceding ~ections, we have discussed the infinitesimal affine transformation (3,7), which carries the vector Ai into the vector A: ... Ai + 6A" where (7.1)

6A,

",;-+2lX;i- + -a.;--21%;;) = a.;Aj = (- Ai = (e;j

+ "',j)A;;

the 6;J and "'ii were constants and so small that their produ()ts could be neglected in comparison with their first powers, Now we does not differ essentially from that just considered. The only difference is in the regions in which the medium experiences compression and tension. 18. Maximum Normal and Shear Stresses. Mohr's Diagram. We have shown in the preceding section [Eq. (17.4)1 that the component N

°

i

of the stress vector in the direction ", normal to the surface element, is inversely proportional to the square of the radius vector A\' to the stress quadric. The extreme values of the radius vector lie along the axes of

50

MATHEMATICAL THEORY OF ELASTICITY

the quadric. Hence the extreme values of N, which we have denoted by Ta, are the extreme values of the normal components of the stress vector acting at po as the sutface element assumes different orientations. These extreme values are obviously of moment in the study of failure of materials. In some theories of failure it is also important to know the

T1, T2,

extreme values of the shearing component 8 of f and the directions " associated with them. These are easily determined. If we direct the coordinate axes at po along the principal directions of stress, the components T12, T23, and 1'13 vanish and 1'11 = 'T1, 'T22 = 'T2, 'T33 = 'T.. From the basic relation

.

we then have (18.1)

T.

= 'T.Va,

and since we get (18.2) But from Fig. 9

8 2 = I'fl' - N', and on substituting in this formula from (18.1) and (18.2) we obtain (18.3) It is clear from (18.3) that if the directions" are taken along the axes of the stress quadric so that V,

± 1, ± 1, = ± 1,

=

V2

1', =

V3

= Va = 0,

V3

=

V,

PI

=

V.

= 0, = 0,

then 8 = O. This merely verifies the known fact that the planar elements normal to the principal directions of stress are free from shear. Thus the minimum (zero) values of 181 are associated with the principal directions. To determine the directions associated with the maximum values of 181, we maximize the function in the right-hand member of (18.3), subject to the constraining relation II;Vi = 1. The simplest way of doing this is to use the method of La.grange multipliers a.nd seek the free extremum of the function F = 8 1 - X... I',. This leads to the three equations,

iJF iJv;

=

° ,

51

ANALYSIS OF STRESS

in X and "i which, together with the relation Vi"i = 1, serve to determine the desired directions. We dispense with the elementa.ty computa.tions and record the final results in the accompanying table, the last column of which gives the values of INI associated with the extreme values of lSI. TABLE OF EXTREMAL VALUES OF

0 0 :1:1 0

±V2 2

0

±l

±l

0 0

0

V2

~/2

±T

±V2 2

181""

Va

"

VI -~

0

8

0 0 0

INI IT.I hi [TIl

±T

i

~~lr2

- Ta!

~~[T2

+ TIl

± V2

\

~ih

- Ttl

Hlra

+ Ttl

I

~~ITI

- T,I

~21Tl

+ ral

2

±~ 2

0

If 1'3 < T. < 1'1, SO that 1'1 is the maximum value of Nand 1'3 is its minimum value, then the maximum value of lSI is,

We see from the table that the maximum shearing stress acts on the surface element containing the X2 principal axis and bisecting the angle between the X,- and xa-axes. If 1'2 = 1'3, there will be infinitely many directions associated with the surface elements that are subjected to a maximum shearing stress. We summarize the main results of this section in the following theorem: THEOREM: The maximum shearing stress is equal to one-half the difference between the greatest and least normal stresses and acts on the plane that bisects the angle between the directions of the largest and smallest principal stresses. The results of this .section can be further illuminated by constructing a diagram proposed' by o. Mohr. If we rewrite Eqs. (18.2) and (18.3) in the form

N =

8'

+ N2

'1",,,1 + T."~ + T3111, + T~vi + TM,

". Ti"~

lOtto Mohr, ZiviJ•."iew- (1882), 1>. 113. See also his book Technische Mecbanik, 2d ed. (1914).

52

MATHE,MATICAL THEORY OF ELASTICITY

recaJI that

vI + vi + vi

VI, we obtain

= 1, and solve for the

(18,4)

We are assuming that '1"8

< '1", < '1""

so that T, - T2 > 0 and T, - T3 > 0, and since clude from the first of Eqs. (18.4) that (18.5)

S'

+ (N

1'; is nonnegative, we con-

- T.)(N - Ta) 2:: O.

We consider now the space of the variables (S, N) and plot in the cartesian SN-plane (Fig. 12) the values of S as ordinates and those of N as abscissas.

s

Q

FIG. 12

The equation (18.6)

S' + (N - 'I",)(N -

'1".)

= 0

represents a circle 0, with center on the N-axis and passing through the points ('1"2, 0), (T3, 0). Hence the region defined by (18.5) is ~rior to the circle (18.6) and includes its boundary. Further, '1". - '1", > 0, TI - '1"1 < 0, and we conclude from the second of Eqs. (18.4) that (18.7)

8" + (N - 'I".)(N - Tl) ~

o.

Thus the region defined by (18.7) is a closed region, interior to the circle C, (Fig. 12), whose equation is S'

+ (N -

'I".)(N - "1) -

o.

ANALYSIS OF STRESS

Finally the third of Eqs. (18.4) yields the result tha.t (18.8)

S2

+ (N -

Tl)(N - ,.,)

~

0,

since T. - "1 < 0 and T3 - T2 < O. The region defined by (18.8) is exterior to the circle C. (Fig. 12), with center on the N-axis and passing through the points (Tl' 0), (T2, 0). It follows from inequalities (18.5), (18.7), and (18.8) that the admissible values of Sand N lie in the crescent-shaped regions (shaded in Fig. 12) bounded by the circles C1, C" and Ca. The maximum shearing stress S, as is clear from Fig. 12, is represented by the greatest ordinate 0' Q of the circle C2, and hence

s _ Tl mu:: -

- Ta. 2

To determine the orientation of the surface elements that support this stress, we make use of formulas (18.4). The value of N, corresponding to Sma_ (shown as 00' in Fig. 12), is N = Tl

+ ,.., 2

and the substitution of this value and Smax = ~h - 7",) in (18.4) yields vi = vi = ~, vi = O. These coincide with the values appearing in the table on page 51. PROBUMS Discusa the Mohr circle diagram for the case where T, = n, and detennine the orientation of surface elements experiencing extreme shearing stresses. Consider also the case where 1"1 = 1"2 = Ta.

19. Examples of Stress. This section contains ·several examples closely paralleling those in Sec. 8. As in that section, we prefer to use the unabridged notation. a. Purely Norma}, Stress. If for every plane passing through a point

f is normal to the plane, that is, if it is directed along the normal" or opposite to it, then for any choice of rectangular coordinates

PO(;Z;O) the stress vector

and T"" = T .. = ,."" = 0, The stress quadric in this case is a sphere whose equation is a;1

+ y2 + Zl.=

±k2 _ _• T_

Any set of orthogonal axes that pass through the point po may be taken . as principal axes of the quadric. This case corresponds to hydrostatic pressurei£,."" is negative.

MATHEMATICAL THEORY OF ELA.STlCITY

b. Simple Tension Dr CompresBioo. A state of simple tension or compression is characterized by the fact that the stress vector for one plane through the point is normal to that plane and the stress vector for any plane perpendicular to this one vanishes. Hence if the x'-, y'-, and z'-axes coincide with the principal axes of streM, then the stress quadric (17.3) has the equation Transforming to any other orthogonal coordinate system x, y, z with the aid of (17.12), we obtain the following stress components: 'T.. = 'T,lflJ 'T.. = 'T,llll21,

'TIIJI = 'T,l~" 'T.. = 'T ,l"l",

r ••

= 'T1Z111

'T•• = 'T,l31l11,

where lll' l", l31 are the direction cosines of the x'-axis relative to the axes x, y, z. A positive value of 'TI represents tension, and a negative represents compression. c. Shearing Stress. Consider a streM B quadric (19.1) which is a hyperbolic cylinder whose elements are parallel to the z'-axis and which represents a shearing stress of magnitude'T. Equation (19.1) takes the form

o

'Txt -

FIG. 13

ryl

=

±kl ,

when the axes are rotated through an angle of 45° about the z'-axis. A comparison of this equation with the general equation of the streM quadric (19.2) when the latter is referred to the principal axes of stress shows that we must have 'Tu = 0,

Tu

==

- TW

= 1".

Thus, the shearing streM is equivalent to tension acr088 one plane and compression of equal magnitude acr088 a perpendicular plane. This can also be shown geometrically by considering the equilibrium of the element PBO (Fig. 13). Hence the stress on the face BO is a pure shear of magnitude l' - -'T"" = +'T... This type of shearing stress would tend to slide planes of the material originally perPendicular to the y'-axis in a direction parallel to the :r!-axis and planes of the material originally perpendicular to the :r!-axis in a direction parallel to the y'-axis. d. Plane St.r_. If one of the principal stresses vauishes, then the 1ItNI8II quadric becomes a cylinder whose base is a eonic, the stress conic.

55

ANALYSIS OJ' STR1!l88

A state of stress, in this case, is said to be plcM. The base of the cylinder lies in a plane containing the directions of the non vanishing principal stresses. For example, if this plane is perpendicular to the ,-axis, the equation of the quadric is T•..x·

+ T..Y· + 2T.,xy

=

±k·.

For simple tension in the x-direction, the stress conic reduces to the pair of lines x ==

±kt ± -_. T ...

~

For the case of shear, the stress conic is a rectangular hyperbola %JI =

kl ± _. 2T..,

If the stress conic is a circle, there is equal tension or compression in all directions in the plane of the circle.

CHAPTER

3

EQUATIONS OF ELASTICITY

20. Hooke's Law. It has already been noted that the treatment contained in Chaps. 1 and 2 is applicable to all material media that can be represented with sufficient accuracy as continuous bodies; this chapter will be concerned with the characterization of elastic solids. The first attempt at a scientific description of the strength of solids was made by Galileo. He treated bodies as inextensible, however, since at that time there existed neither experimental data nor physical hypotheses that would yield a relation between the deformation of a solid body and the forces responsible for the deformation. It was Robert Hooke who, some forty years after the appearance of Galileo's Discourses (1638), gave the first rough law of proportionality between the forces and displacements. Hooke published his law first in the form of an anagram" ceiiinosssttuu" in 1676, and two years later gave the solution of the anagram: "ut tensio sic vis," which can be translated freely as "the extension is proportional to the force." To study this statement further, we discuss the deformation of a thin rod subjected to a tensile stress. Consider a thin rod (of a low-carbon steel, for example), of initial cr088sectional area ao, which is subjected to a variable tensile force F. If the stress is assumed to be distributed uniformly over the area of the cross section, then the nominal8tre8s T = F I ao can be calculated for any applied load F. The actual stress is obtained, under the assumption of a uniform stress distribution, by dividing the load at any stage of the test by the actual area of the cross section of the rod at that stage. The difference between the nominal and the actual stress is negligible, however, throughout the elastic range of the material. If the nominal stress T is plotted as a function of the extension e (change in length per unit length of the specimen), then for some ductile metals a graph like that in Fig. 14 is secured. The graph is very nearly a straight line with the equation (20.1)

T = Ee

until the stress reaches the proportional limit (point P in Fig. 14). The position of this point, however, depends to a considerable extent upon the sensitivity of the testing apparatus. The constant of proportionality E is known as Young's modulus.

EQUATIONS OF ELASTICITY

57

In most metals, especially in soft and ductile materials, careful observation will reveal very small permanent elongations which are the results of very small tensile forces. In many metals, however (steel and wrought iron, for example), if these very small permanent elongations are neglected (less than 1/100,000 of the length of a bar under tension), then the graph of stress against extension is a straight line, as noted above, and practically all the deformation disappears after the force has been removed. The greatest stress that can be applied without producing a permanent deformation is called the elastic limit of the material. When the applied force is increased beyond this fairly sharply defined limit, the material exhibits both elastic and plastic properties. The determination of this limit requires successive loading and unloading by ever larger forces U until a permanent set is recorded. For many materials the proportional limit is very nearly equal to the elastic limit, and the distinction between the two is sometimes dropped, particularly since the former is more easily obtained. When the stress increases beyond the elastic limit, a point is reached (Yon the graph) at which the rod suddenly Strain stretches with little or no increase in the load. The stress at point Y is called FIG. 14 the yield-point stress. The nominal stress T may be increased beyond the yield point until the ultimate stress (point U) is reached. The corresponding force F = Tao is the greatest load that the rod will bear. When the ultimate stress is reached, a brittle material (such as a high-carbon steel) breaks suddenly, while a rod of some ductile metal begins to "neck"; that is, its crosssectional area is greatly reduced over a small portion of the length of the rod. Further elongation is accompanied by an increase in actual stress but by a decrease in total load, in cross-sectional area, and in nominal stress until the rod breaks (point B). The elastic limit of low-carbon steels is about 35,000 lb per sq in.; the ultimate stress is about 60,000 lb per sq in. .Hard steels may be prepared with an ultimate strength greater than 200,000 lb per sq in. We shall consider only the behavior of elastic materials subjected to stresses below the proportional limit ; that is, we shall be concerned only with those materials and situations in which Hooke's law, expressed by Eq. (20.1), or a generalization of it, is valid.' I In order to give the reader Ilome feeling regarding the magnitude of deformations with which the theory of elasticity de&ls, note that a l-in.-long rod of iron with proportionallimit of 25,000 Ib per sq in., a yield point of 30,000 lb per sq in., and Young'swod-

58

Il.\THEMATICAL THEORY OF ELASTICITY

Some materials subjected to tensile tests have an extremely small range of values of extensions e for which the law (20.1) is valid. In this case, the stress-strain curve above the proportional limit may have the appearance indicated in Fig. 100. In the process of loading and unloading specimens made of such materials, the same curve PQ may be traced out, and if there is no residual deformation, the material is elastic with the stress-strain law of the form

T = I(e), where' is a single-valued nonlinear function. More frequently, however, the loading-unloading diagram has the appearance shown in Fig. 15b. In this diagram the curve 0 A is associated with the loading of the specimen and AB with the unloading. In this instance there is a residual T

T

A

Q

e

CI

lal

Ibl FIG. 15

deformation, represented by DB, which characterizes the plastic behavior. For plastic materials the relationship between T and e is no longer one-toone, and after repeated loadings and unloadings a saw-tooth pattern indicated in Fig. 15b may be obtained. A natural generalization of Hooke's law immediately suggests itself, namely, one can invoke the principle of superposition of effects and assume that at each point of the medium the $train components 6>i are linear functions of the stress components Tij. Such a generalization was made by Cauchy, and the resulting Jaw is known as the generalized Hooke'BlaW. We discuss it in the following section. 21. Genera1ized Hooke'S Law. We saw in the preceding chapters that the state of stress in continuous media is completely determined by the stress tensor Tij, and the state of deformation by the strain tensor e;i' We shall now assume that when an elastic medium is maintained at a muof 10,000,000 lb per aq in. will eloncate under & load of 13,000 lb per aq in. about 0.0004 in. Even if the rod is loaded to the yield point, the determiD&tioa of the extensioa wiD require very refined m _ t s .

59

EQUATIONS OF ELA8T1CITY

hed temperature there is a one-to-one analytic relation Tij

= Fij(en,

en, ..• ,en),

(i, j

= 1, 2, 3)

between the Tij and eoj and that the Tij vanish when the strains eo; are all zero. This last assumption implies that in the initial unstrained state the body is unstressed. Now, if the functions Fij are expanded in the power series in e;; and only the linear terms retained in the expansions, we get (i, j, k, I = 1, 2, 3).

(21.1)

The coefficients Cijkl, in the linear forms (21.1), in general will vary from point to point of the medium. If, however, the C,jkl are independent of the position of the point, the medium is called elastically homogeneous. Henceforth we confine our attention to those media in which the (;,;kl do not vary throughout the region under consideration. The law (21.1) is a natural generalization of Hooke's law, and it is used in all developments of the linear theory of elasticity.' Inasmuch as the components Tij are symmetric, an interchange of the indices i and j in (21.1) does not alter these formulas, so that Moreover, we can assume, without 1088 of generality, that the C'j'" are also symmetric with respect to the last two indices. For if the constants C:jkl and C:;kl are defined by the formulas

=

c:;>1

72(Cijkl

+ Ci;a),

C~;.l:l = 72(Cijld -

then, clearly, c:;>1 = the sum

O:jlk

and

C:;kl =

Cij1cl

CijlJ:),

-C:;'k'

= C~ikl

ThuB

Cijkl

can be written as

+ C~;kh

in which the C:j11 are symmetric and the C;;kl are skew-symmetric with respect to k and 1. Accordingly, the law (21.1) can always be written in the form Tij

=

C:;kleH

+ C:;kle",.

However, the double sum in the second term of this expression vanishes inasmuch as ekl = eU. and C:;kl = Thus,

-C:;11'

Tij

==

C~.i.l:,eA:l,

where the C#kl are symmetric with respect to the first two and the last two indices. , It is Important to note that the generaliJled Hooke'alaw (21.1) is akIo uaed in some in'\"e8tiptionB. where the Btraina are finite, in the llen&e of Sec. 11. For many materiall a linear relationship (21.1) holds for an appreciable range of values of the 8,ar forms in (21.2) can be written as follows, the law (21.1) is written in the form

(i,1, Ie, l - 1, 2, 3), valid in all coordinate systems, then it follows from the tensor characte. of the f';; a.nd e.. that the are components of .. tensor of rank 4. Consequently, 'mder a transformation of coordinates from the system X to X', the transform according to the

ott

ctl

law (4)

If the

ttl are invariant (so that c'ft - etl) under a given coordinate tr-ansform.ation,

then the tranafo1'lllation Characteril!eB the nature of elastic symmetry. The ~ IigurIJ:c

ing in the law (4) arethedirectionoosines ..ppearinginthetablesofthlslWK!tion,~_ the systeJns X a.nd X' are cartesian.

63

EQUATIONS OF ELASTICITY

(~: ~ ~

(21.5)

o

0

0

Cn

Co.

c..

u. ~)

Cu c" 0 0 0 coo Such materials as wood, for example, have three mutually orthogonal planes of elastic symmetry and are said to be or.tftoiJ:J!,pic. In considering such materials, we shall choose the axes '(;r co()rdinates so that the coordinate planes coincide with the planes of elastic symmetry. Tn this case, some of the coefficients Co; exhibited in the array (21.5) vanish, Besides the symmetry with respect to the xlx.-plane, expressed by (21.5), the elastic constants co; must also be invariant under the transformation of coordinates defined by the following table of direction cosines, Xl

X.

X.

-1

0

0

- -- - - - - --I

Xl

,

1 0 0 x., 0 0 x. This change of coordinates is a reflection in the x.x.-plane and leaves the and unchanged with the following exceptions:

r,

e,

e~

e~

= -e5,

= -e•.

From (21.5) we have TI

=

CUel

+ Cl2e. + cue. + cue•.

This becomes or

Tl = CUel + Cl2ll. + Clae. - cue., from which it follows that C16 = O. By cOlliudering in a similar way the transformed expressions for T2, • • • ,TO, we find thaV

c•• = c.. = Cn = Cu = Cn = CO2 = co. = O. Thus, for orthotropic media the matrix of the c;, takes the following form.

(21.6)

(

~:: ~:: ~:: ~ ~

Caa 0 0 0 0 C44 0 OOOOc" o 0 0 0 0 Cu

o

C..

D

Note that elastic symmetry in the ztzrplane and in the "'o2'rplane intplies elastic ~ in the """'rplane. 1

64

MATHEMATICAL THEORY OJ' ELASTICITY

If the coefficients Cij are symmetric, that is, C'I == Cj;, (i, j == 1, 2, . . . ,6), we see that there are 13 e88ential constants in the array (21.5) and 9 in (21.6). This symmetry has not been assumed, however, in establishing the forms of the arrays of coefficients (21.5) and (21.6), nor will it be used in the next section, where the law (21.2) is specialized to that for an isotrop;" medium. It is worth noting that the statement of the law (21.2) is not devoid of inconsistency. In the process of formulating the notion of the components of strain e,j, it was assumed that the components of displacement u, are functions of the coordinates (XI, X" Xa) of the body in its undeformed state; that is, Lagrangian coordinates were used. On the other hand, Eulerian coordinates were employed in defining the components of the stress tensor To;; that is, it was assumed that the T" are functions of the coordinates (xi, X;, X;) of the stressed (and hence deformed) medium. Of course, if the displacements U; and their derivatives are small, then the values of T,iCx) and Ti;(X') cannot differ by a great deal. As an indication of the order of approximation involved here, note that, if x;. = x. + 'Uk, then (21.7)

Or.; = Or,;

ax.

ax:

ax; aXk

Hence, in writing aaT,; =

x.

=

Ora':,

Xk

an; (a k! + au!) axi

ax.

=

aTij ax~

+ Or;; aU!. axl ax.

we assume that the displacement derivatives

are small compared with unity. In what follows, it will be assumed that both the components of strain eoj and the components of stress Ti; are functions of the initial coordinates (Xl, X" xa). REFERENCES FOR COLLATERAL .READING A. E. H. Love: A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, London, Sees. 60-65, pp. 92-100. Chap. VI of Love's treatise is given to a discussion of the eqUilibrium Of nonisotropic elastic solids and contains further references on the subject. Voigt's Lehrbuch der Kristallphysik is a standard treatise on the subject. L. Lecornu: Theorie mathematique de ]'eiastieite, Memorial des sciences math6matiques, Gauthier&-Villars & Cie, Paris, pp. 12-1S. Contains a discussion of the theory of Poincare regarding the number of elastic constants in the generalized Hooke's law. PROBLEMS

1. Are the principal axes of strain coincident with those of stress for an anisotropic medium with Hooke's law expressed by Eq. (21.2)? For a medil1m with one plane of elastic symmetry? For an orthotropie mediull1 ? Him: Take the eoordinate axes along the principa.l "as of strain @() that e. - e. _ .... - o.

55

EQUATIONS OF ELASTICITY

I. Show directly from the generalized Hooke's law (Eq. (21.2)) that in an isotropio body the principal axes of strain coincide with those of stress. Hint: Take the coordinat.. a.nd p. were introduced by G.

Tij.

p, "F 0 L&m~

and 3A + 21' "F O. and are called the

Lame constants. We .have shown that the stress-strain law for isotropic media involves no more than two elastic constants. The fact that no further reduction is posiible is physically obvious from the simple tensile tests, but an

67

EQUATIONS OF ELASTICITY

analytic proof of this, utilizing the properties of isotropic tensors, can be constructed. 1 If the axes x; are directed along the principal axes of strain, then eoa = en = 612 = O. But from (22.3) we see that in this case'T23, 'Tn, and T1S also vanish. Hence the axes X; must lie along the principal axes of stress, and we have the result that the principal axes of stress are coincident 'With the principa~e~.of strain if the medium is isot;opiC: -This property was used by Cauchy to define the isotropic elastic medium. Henceforth no distinction will be made between the principal axes of strain and those of stress, and such axes will be referred to simply as the principal axes. 23. Elastic Moduli for Isotropic Media. Simple Tension. Pure Shear. Hydrostatic Pressure. In order to gain some insight into the physical significance of elastic constants entering in formulas (22.3), we consider the behavior of elastic bodies subjected to simple tension, pure shear, and hydrostatic pressure. Assume that a right cylinder with the axis parallel to the Xl-axis is subjected to the action of longitudinal forces applied to the ends of the cylinder. If the applied forces give rise to a uniform tension T in every cross section of the cylinder, then (23.1)

Tll

=

T

= const,

T ••

= Ta. =

T12

= To.

= T31 = O.

Since the body forces are not present, the state of stress determined by (23.1) satisfies the equilibrium equations (15.3) in the interior of the cylinder, and equations (13.3) show that the lateral surface of the cylinder is free of tractions. The substitution from (23.1) in (22.5) yields the appropriate values of strains, namely,' (23.2)

{

+ p.)T + 2p.)'

-XT

(X p.(3X

ell

=

e12

= e23

= e31 =

0,

which clearly satisfy the compatibility equations (10.9). Accordingly, the state of stress (23.1) actually corresponds to the one that can exist in a deformed elastic body. Noting that e.. -X ell = 2(X + p.)' we introduce the abbreviations (23.3)

" .., 2(X

+ p.)'

1 H. Jeffreys, Cartesian Tensors (1931). • The integratioIl of Eqs. (23.2), yielding the displacements See. 30.

u"

is carried out in

MATHEMATICAL THEORY OF ELASTICITY

Then Eqs. (23.2) can be written in the form 6n =

(23.4)

I

E T,

-(T

"If" T

6u = 6aa =

= -.,.6u,

{

612 = e2. = 681 = O. H the stress T represents tension, so that T > 0, then a tensile stress will produce an extension in the direction of the axis of the cylinder and a contraction in its cross section. Accordingly, for T > 0, we have ell > 0, e" < 0, eaa < O. It follows that E and 11 are both positive. Physical interpretations of the elastic moduli E and 11 are easily obtained. It follows from the first of the formulas (23.4) that the quantity

represents the ratio of the tensile stress T to the extension by the stress T. Again, from (23.4), it is seen that 22

°

(39.5)

Co

=

ao

:f

Then

. c.e"" L i--..

=~

+

L. \,~ - i ~) (cos k8 + i sin k8) k-l

=~

+

i (~ + ~) (C~8

+

..

i

k(J - i sin klJ)

i-I

L

(a. cos klJ

+b

k

sin k8).

k-I

Let h(8) and !t(8) be a pair of real functions, each of which can be expanded in Fourier series in the interval (0, 2r), and form the complex function ft(8) + if,(O). Then (39.6)

f1(0)

+ ifz(O)

..

L c.e

=

ik

',

, _ _ flO

where c.. =

i..102r

[!tel)

+ if2(t)]e-int dt,

If we set

c..

= "Y.

(n =

+ i3..,

0, ±1, ±2, ...).

137

1IIXTENSION, TOllSION, AND .B'LEXUlUIl O.B' BEAMS

where ')'. and &.. are real numbers, then /I(D)

+ ifJ(D)

..

=

l .. --. . L

(')'k

+ ilik)(COB kD + i sin kD)

(')'t

COB kfJ - li. sin kfJ)

1:-- -

+i =

')'0

+

..

.

L k--

.-1L [(')'. +. ')'-.) + L +

(lit COB kfJ

+ ')'. sin kIJ)

110

cos kIJ - (0. - L.) sin kill

[(lit

i

L

k)

COB kIJ

+ ('Y' -

+ ilio

'Y .....) sin kIJl .

• -1

Hence

where

.

.-1I. (a. cos kfJ + b. sin kfJ), !,(IJ) = Y2a~ + L (a~ cos kIJ + b~ sin kfJ), .-1

!1(fJ) = %ao

%ao = 'Yo, Y2a~

=

lio,

+

a. = 'Y'

+ 'Y-.,

+

a~ = li. L., (k = 1,2,3, .

b. = -li. + li ....., bk = 'Y. - 'Y ....., .).

It follows from these formulas that the representation of a complex function f1(IJ) + i!2(fJ) in a series of the type (39.3) is unique, since the representation of the functions !1(IJ) and f2(IJ) in series of the type (39.1) is unique. REFERENCES FOR COLLATERAL READING D. Jackson: Fourier Series and Orthogonal Polynomials, Mathematical Association of America, Chicago, Chap. 1. I. S. Sokolnikoff: Advanced Calculus, McGraw-Hill Book Company, Inc., New York, Chap. XI. E. T. Whittaker and G. N. Watson: Modem Analysie, Cambridge University Press, London, p. 170. K. Knopp: Theory and Application of Infinite Series, Blackie &: Son, Ltd., Glasgow. p.356. H. S. Carslaw: Fourier Series and Integrals, The Macmillan Company, New York. R. V. Churchill: Fourier Series and Boundary Value Problems, MoGraw-Hill Book Company, Inc., New York.

to. Summary of Some Results of the Complex Variable Theory. We shall need in our subsequent work some theorems from the theory of

138

lIlATHEMATICAL THroaT Oll' ELASTlCI'l'Y

functions of a complex variable. In this section, 80Ille of the more familiar results will be stated without proof, and the prooflil of the less familiar ones will be outlined. A detailed discussion can be found in the reference books listed at the end of the next section . . It will be recalled that a single-valued function I(A) = u(x, y)

+ w(x, y)

of a complex variable A = x + iy is called analytic, or holo1rwrphic, in a given region R if it posselilSes a unique derivative at every point of the region R. Points at which the function I(A) ceases to have a derivative are termed the singular pointe of the analytic function. The necessary and sufficient conditions for the analyticity of the function I(A) are given by the well-known Cauchy-Riemann equations

(40.1)

av

au

ax

ay

-= --,

where it is assumed that the partial derivatives involved are continuous functions of x and y. It is known that, if I(A) is analytic in the region R, then not only do the first partial derivatives of u and v exist in R, but also those of all higher orders. It follows from this observation and from (40.1) that the real and imaginary parts of an analytic function satisfy the equation of Laplace; that is, V2u = 0,

V'v =

o.

The following theorem is basic to all considerations of the theory of analytic functions: CAUCHY'S INTEGRAL THEOREM: If ICA) is continuous in the closed region' R bounded by a simple closed contour C, and il I(A) is analytic at every interior point of R, then

fe f(A) dA =

O.

This theorem can easily be extended to the case of multiply connected regions to yield another. THEOREM: If f(A) is continuous in the closed, muUiply connected region R bounded by the exterior simple contour Co and by the interior simple contours C" C2 , • • • ,Cn , then the integral of f(A) over the exterior contour Co is equal to the sum of the integrals over the interior contours, whenever f(3) is analytic in the interior of R. The integration over all the contoura is performed in the same direction. The following numerical results are worth noting: H n is an integer and ; = a is a fixed point that lies either within or 'The term "continuous in a closed region" is used to mean that the function is continuous up to and on the boundary.

EXTENSION, TORSION, AND FLEXURE OF BEAMS

139

without the simple closed contour C, then ifn-F-l. If the point a is outside the contour 0, then the truth of the formula follows from Cauchy's Theorem, whatever be the value of n; if it is within, then the result follows from elementary calculations. If the point a is within the contour, then an !llementary calculat,ion gives

=2ri. Jc( ~. ~ - a This latter formula, in conjunction with Cauchy's Integral Theorem, can be used to establish Cauchy's Integral Formula. CAUCHY'S IN'l'EGRAL FORMULA: If a = a is an interior point of the regicm R bounded by a contour C, then (40.2)

J.

(fW da

21rtjca- a

=

f(a),

whenever f(i) is continuou; in the closed region R and analytic at every interior point of R. If the variable of integratlon in (40.2) is denoted by \, and if a is any point interior to R, then (40.2) becomes (40.3)

f(a)

= _!_. ( fer)

dr.

z...t}cr-a

CalCUlation of the derivative from the formula (40.3) yields

!'Ca)

=

_!_ [ fCr) dr ,

z...i Jc (r -

a)2

and, in general, j+(t) and 4>-(t) a8 a approache8 an arbitrary point t on C from the interior and exterior of C, respectively, are:

4>+(t) ... (40.7)

4>-(t)

e

r

! F(t) + ~ F(r) dr, Z,...z}cj-t

2 -

.! F(t) + ~

r

F(r) ds.

Z?rt}cs-t

Z

The im'i!"0per integrals in (40.7). are interpreted in the sense of Cauchy's principtil values. 3 We shall make use of integrals of Cauchy's type to represent analytically some functions that are useful in tlie theory of elasticity. However, it must be noted that such representation is not unique, so that the same function can be represented by different integrals of Cauchy's type. AB an illustration, consider a contour C that contains in the interior the point ! == 0, and let us determine the analytic fanction that vanishes at every point of the region R enclosed by C. If we choose in (40.4) the density 'J. Plemelj, M0nat8hefte fUr Mathematik und Physik, vol. 19 (1908), pp. 205-210. A detailed discussion of these fQrmulas under restrictions somewhat less severe than those made by Plemelj is contained in Chap. 2 of N. I. Muskhelishvili's Singular Integral EqUAtions (1953). • Usually .. is restricted to lie in the interval 0 < a S 1, because for a > 1 the condition (40.6) implies that F'(r) - 0, 80 that F{r) - const. • H an arc L of length 2e with l' - t as the mid-point is deleted from C, then the F(r) dr over the remaining curve C - L becomes proper and the ptinci-

integral (

Jc-£!' - t

pat value of (

.

F{r) dt is defined as lim

Jcr -t

.

....0

f.

F{t) dr. C-Ll'-t

EXTENSION, TORSION, AND FLEXURE OF BEAMS

function F(1") = 0, then

4>1(~)

1

143

Also, if we choose F(r) = l/r, then

55 O.

(

dr

= 2,..i jo W· - 4) = 0

4>1(3)

lor every position of the point a in the region R.l Hence if we add this integral to an integral of Cauchy's type that defines an analytic function >1'(3), we shall obtain another integral of Cauchy's type that defines the same analytic function >1'(3). It follows from these remarks that no conclusion can be drawn concerning the equality of the density functions F1(t} and F.(r} from the equality of the two integrals

..!.. (

F,(f) dr =

27rzjor-a

J.... (

F.(r) dr

27rzjor-3

for all values of 3 in the interior of C. We shall see, however, that if some additional restrictions are imposed on the density functions and on the contour C, then the equality will obtain. This is the subject of the next section, which contains a discussion of the Theorem of Harnack. 41. Theorem of Harnack.' In considering the applications of the theory of functions of a complex variable to problems in elasticity, we shall most frequently deal with the region bounded by the unit circle, that is, the region 131 ~ 1. In order to avoid a possible misinterpretation of the formulas, we shall draw the unit circle in the complex t-pJane, where r = ~ + i." (~ and." being real). The boundary of the unit circle 111 ~ 1 will be denoted by the letter 'Y, and the points on the boundary 'Y by 0" = eit . 3 All functions of the argument 8 will be assumed to be periodic, so that 1(8 + 211") = f«(J). THEOREM: Let f«(J) and !p(8) be continuous real functions of the argument 8 (defined on the unit circle 'Y); if

r

_!_ f(8) dO" 27rij~0"-t

(41.1)

for

all,'al~8

=..!.

r 1"(8) du

2rij..,u-r

of r interior to 'Y, then f(8}

==

1"(8).

If the point r i8 exterior to oy, and if the equality (41.1) is true for all values 'F or

1

r(r -

i) -

1

'hi

r

1, 1 h

Hr _ II - if; ence dr

lor

1.

This corollary follows at once from Harnack's Theorem when we consider the results of adding and subtracting the equalities in question. REFERENCES FOR COLLATERAL READING

E. C. Titchmarsh: The Theory of Functions, Oxford University Press, New York, 2d ed., pp. 64--101, 399-428. W. F. Osgood: Lehrbuch der Funktionentheorie, Teubner Verlagsgesellschaft, Leipzig, vol. 1. t. Gol1l"5at: Cours d'analyse, Gauthiers-Villars &; Cie, Paris, vol. 2. E. Picard: LeQOns sur quelques types simples d'equatioDB aux dbivees partielles. Gauthiers-Villars &; Cie, Paris.

42. Formulas of Schwarz and Poisson. We have already seen that the determination of the torsion function 'P(x, y) and its conjugate function ",(x, y) are special cases of the fundamental boundary-value problems of Potential Theory-the so-called problems of Dirichlet and Neumann. These problems occur also in other branches of applied mathematics, notably in hydrodynamics and in electrodynamics. While the solution of the two-dimensional problems of Dirichlet and Neumann for special types of boundaries is likelY' to present serious calculational difficulties, it is possible to write down general formulas for the case when the boundary of the region is a circle. We shall give a derivation of formulas associated with the names of Schwarz and Poisson that solve the problem of Dirichlet for a circular region.

146

MATHEMATICAL THEORY OJ' ELASTICITY

Consider a region bounded by a circle, which we can take, without loss of generality, to be a unit circle with center at the origin. As in the preceding section, we denote the boundary of the circle ItI = 1 by 'Y and any point on the boundary by u = e;9. Let it be required to determine a harmonic function u(~, 1/), which on the boundary of the circle 'Y assumes the values (42.1)

u

I" = f(9),

where f(9) is a continuous real function of 8. Denote the conjugate harmonic function by Vet, 1/); the function 71) is determined to within an arbitrary constant from the knowledge of the function u(~, 1/) [see (35.2)J. Then the function

va,

F(r) = u(~, 1/)

+ iv(~, ,,)

is an analytic function of the complex variable r = ~ + i1/ for all values of r interior to Irl = 1. If we assume that F(r) is continuous in the closed region Irl ~ 1, then we can rewrite the boundary condition (42.1) in the form l F(u)

(42.2)

+ P(If)

= 2/(8)

on 'Y.

If we multiply both members of Eq. (42.2) by

2~ du ,.' where I n u -,

is any

point interior to 'Y, and integrate over the circle 'Y, we obtain the formula

f

f

f

~ F(u) + ~ P(If) du = ~ f(9) 2," "u-r 2?rt "u-r "ITt 'YU-l ' which by the Theorem of Harnack is entirely equivalent to (42.2). The first of the integrals in the left-hand member of (42.3), by Cauchy's In,egral Formula, is equal to F(r), while the second is equaJ2 to F(O). Let (42.3)

au

au

1 We define Pm - Fm and Fil) = F(,}. It is possible to prove that, if f(9} satisfies Holder's condition, then the function F(!) given by (42.6) will be continuous in the closed region Irl :::; 1. We recall that a function f(9} is said to "".tisfy Holder's condition (or a Lipschitz condition) if, for any pair of valu,," 6' 841rl 8" in the interval in question, If(9") - f(e') 1 :::; Mle" - 9'la,

where M and '" are positive constants. This condition is less restrictiw than tbe requirement of the existence of a bounded derivative. • Since F(i) - F(O) '(11) - '(0)

+

1 p'(O) ;;

+ F'(O)t + ~ F"(O)i' + ... , and since on IiI - 1 f - 1/", 1 1 + 21 1'''(0) ;; + ... and term-by-term integration gives

the deIIired result upon noting that 1 /, a" 2ri 'Y ,,"(IT - t) - 0, - 1,

if n > 0,

if 11

-

O.

EXTENSION, TORSION, AND FLEXURE OF BEAMS

147

p(O) = ao - ibo; then (42.3) becomes

P(I) =

('12.4)

~

?N,

1

f(O),. du - ao

~u-,

+ ibo.

If we set I = 0 in (42.4), we obtain ao

+ t'b

1 = ---:;

0

'In

and hence 2a o =

(42.5)

~ 1I"t

1

1 ~

f( 9) du -

f(9) du

'Y

a

u

=~ 11"

ao

-

+' t'b

0,

f2T f(O) dO. Jo

The quantity bo is left undetermined, as one would expect, since the funcis determined to within an arbitrary real constant. tion Inserting the value of ao from (42.5) in (42.4), we have

va, ., )

(42.6)

F(I) =

~

7rZ

=

1

f(O) dO' -

~u-I

J:_.1 27rt

_

0, the vectors At and

A~

tend to coincide with the tangents to C

EXTENSION, TORSION, AND FLEXURE OF BEAKS

149

at P and to 0' at P', respectively, and hence l arg ~ is the angle of rotation of the element of are A8' relative to A8. It follows immediately from this statement that, if C, and C2 are two curves in the I-plane that intersect at an angle 'F, then the corresponding curves C~ and C~ in the rplane also intersect at an angle 'F, since the tangents to these curves are rotated through the Bame angle. A transformation that preserves angles is called conlcmnal, and thus one can state the following theorem. THEOREM: The mapping performed by an analytic lunction "'(!) is conIcmnal at all points 01 the I-plane where ",'(!) ;>6 o. We shall be concerned, for the most part, with the mapping of simply connected regions, where the mapping is one-to-one and hence ",'(!) ;>6 o. The regions Rand R' may, however, be finite or infinite. It should be noted that if the region R is finite and R' is infinite, then the function "'(!) must become infinite at some point a of the region R; otherwise we could not have a point in the region R that corresponds to the point at infinity in the region R'. It is possible to show that at such points the function "'(!) has a simple pole, so that its structure in the neighborhood of the point is

"'(!) = ! ~ a

+/(!),

where c is a constant and I(!) is analytic at all points of the region R. Other types of singularities cannot be present, since the mapping is assumed to be one-to-one. If both regions Rand R' are infinite, and if the points at infinity correspond, then the function "'(!) has the form "'(!) = c! + I(!) , where c is a constant and I(!) is analytic in the infimte region R. We recall that a function is said to be analytic in an infinite region R if, for sufficiently large !, it has the structure

ae+~+~+··· +~+

!

r

r

...

.

Let there be given two arbitrary simply connected regions Rand R', each of which is bounded by a simple closed contour that can be represented parametrically by' ~

= W),

"

= ,,(t),

(0 ::; t ::; t,). la it possible to find the mapping function a = "'(!) that will map the region R on R' conformally and in such a way that the mapping is conl.Note that thia ltatemeDt _ell that ~ ,.. 0 at P. 'Sinee tile curves..." ~ we mllBt have E(O) -2w(l)w(f).

But thU! is a special case of the problem treated in Sec. 42, amI a reference to (42.4) shows that

f! If

~ I(r) = .!, ,

n

or I(t) =

(44.4)

w(er)w(lt) dcr - ao .,2 er-t

w(u)w(lt) dcr 21r ., er - t

+ ib o

+ const.

Noting that, on the boundary 'Y of the unit circle It I = 1, er = flo' and hence It == COl == l/er, one sees that the integral (44.4) can be written as (44.5)

I(J) ==

'fJ

+ i'#t =

;..

J., C4("}w:.Yer)

dcr

+ const.

The formula (44.5) gives us at once the torsion function 'fJ alld its conjugate.p, so that the solution of the torsion problem is reducllli to quadratures. H the numerator C4(er)w(ljer), of the integrand, hapJ)eD8 to be a rational function of er, then the integral can be evaluated with the aid of the theorems on residues. It is not difficult to express' the torsional rigidity D directly in terms of the functionJ(t). From (34.10) (44.6)

D

==". IS

ff

".10

(Xl

+ liS) dx dy +".

ff (x ~ - ::)dxey II

+ ".Do,

'The oa.lculatioDe leading to formulaa (44.7), (44.8), III1If (44.10) alii due to N. L Muakheilillhvili. Bee, for example, hill paper "Sur Ie probl~ de tGmiou. dee c,.Jiadns 6Iaatiqu.. isotropea," AID ~ I'fNIk ~ ~ . . lMtai, _. II, vol 9 (l929), pp. 295-300,

EXTENSION, TORSION, AND J'LEXURE OF BEAMS

153

where 1. is the polar moment of inertia of the area bounded by C and Do =

[f (x ~: -

1/

~=) dx dy =

ff [~

(X9) - :X (1/9)] dxdll·

An application of Green's formula to this integral gives Do= where rf = x'

+ y2.

Jer ",(xdx + ydy)

= _ ( .pd!.r2 ,

Je

2

But on the contour C

r' = Ai = ",(.,.)'-;'(It)

and

'" = %(f(.,.)

+ 1(1t)];

hence

D. = -~~

(44.7) Also

I. =

ff

(x'

loy (f(.,.) + 1(1t)] d["'(fT)'-;'(It)].

+ y2) dx dy =

ff

+

[:y (x"1l)

a:

(xyl)] dx dy

= - !cXY(Xdx - ydy). But x = A + A, 2 and we find that

But

Ie AidA

=

0,

and

!.e I' d~

fe A"a da = fe AI d(%a')

= -

where we make use of integration by parts. polar moment of inertia 10 in the form (44.8)

Ie = -

i fe

A2adA = -

ii

=

0

'

fe alA dA, HenCe we can write the

[';'(11)]2",(.,.) dc,,(fT).

If "'(.,.) is a rational function, then the integrands of (44.7) and (44.8) can be easily evaluated with the aid of theorems on residues and the expression for the torsional rigidity D Can be obtained in closed form. We may note that the shear components T.. and T.. of the stress tensor can likewise be exp,_J directly in terms of the functions F(A) and f(I).

154

MATHEMATICAL THEORY OF ELASTICITY

It follows from the formulas (34.4) and (35.1) that

,

Tn -- IT..

=

/let

(orp. orp iJx - , oy

orp + l.-0'" = /let [-iJx aX

-

y -

-

l X -

.)

tX

.(

. )]

lY ..

But.

and we get (44.9)

Since F(3)

=

F[w(.I»)

=

f(.I), we have

= 1'(.1) t!l

F'(3)

d3

f'(.I) _1_. w'(.I)

=

Hence (44.9) becomes (44.10)

1".. -

.

l1",.

[f'(.I) = 1t0l w' (.I) -

._(;:)]

lW,

.

This formula is extremely useful in calculating the components of shear. If the mapping function is written in the form

L .-0 ~

5 = w(.I) =

an.ln,

then it is not difficult to give a formal solution of the torsion problem in terms of the coefficients a.. We have (44.11)

w(u)w

0) i i Lb.u· + Lb.u-·, . amu m

=

m=O

anu-'

n=O

e

e

n-O

n-l

=

where (44.12)

b. =

Lan+,.a,.

,-0

Upon inserting the expression (44.11) in (44.15), it is seen tha,

.

or (44.13)

f(.I) = rp

+ it/! =

i

.-0Lb.l"",

155

EXTENSION, TORSION, AND FLEXURE OF BEAMS

where Cauchy's Integral Formula and Eq. (41.2) have been used. The expression (44.13) for the complex stress function '(J + i", was derived by R. M. Morris! by a different method and was used to obtain formal solutions of the problem of torsion for those cases in which the complex constants an are known. A formal expression involving the constants an can be given for the torsional rigidity D = !l(Io + Do) [see (44.6»). Equation (44.8) for the moment of inertia 10 can be written as

~

10 = -

i

[w(o-)w

G) ]

w (;) dw(o-).

1: ano-', it follows that ~

Now, from w(o-)

=

.-0

wG) dw(er)

1: ~

=

i

cner n dl),

,, _ _ GO

where

1: 1: ~

c,

=

(n

+ r)a,+rdr,

r- 0

(44.14)

~

C_n =

rdn+..a"

(n = 0, 1, 2, . . . ).

r-O

From this relation and from (44.11), we get GO

(I)

10

= 31

/:7 (1:

bmo- m +

".-0

00

1: bmo---m) (1: ".-1

110

C.O-'

+

n-O

l

c_.cr-n) dl).

n-l

Since 0- = eiB , we see that the integral of every term involving 0-" (n vanishes and we are left with

10 = Similarly we can write

~ [boro +

~

0)

i (bnc_" + bnc,,) l .-1

1 R. M. Morris, "The Internal Problems of Two-dimensional Potential Theory," Matllemati.che Annalen, vol. 116 (1939), pp. 374-400; vol. 117 (1939), pp. 31-38.

156

IlATHBMA.TICAL THEORY OJ' ELASTICITY

Combining the expressions for I. and D., we get finally (44.15)

D=

~ [boCo +

.-1I

+ c._,.b. -

(c.Ii.

1

2OO.Ii.. )

lu!, an illustration of the application of the foregoing procedure, we consider a beam whose crOl!8 section is the cardioid

r = 2c(1 (r' = Xl + y2, tan a = y/x). suitable mapping function is

~t

a=

+ cos a)

is readily verified that in this case a c(l - I)',

so that the only nonvanishing coefficients a.. are ao = c,

aJ = -2c,

The nonvanishing constants b" and bo = 6c

c_, = -2c',

2

b, = 6c',

,

Co

=

Cft

are easily found to be

-4c', c,

=

b. = -6c',

c', c,

=

2c'.

The complex stress function is (see also Sec. 58) (44.I6)

f{t) = 'P

+ i+ =

.

i

I b..s.-0

=

ic'{6 - 41

+ to),

while (44.17)

It should be noted that the method outlined above is readily applicable whenever the mapping function w{t) for the region R is a polynomial or whenever the mapping can be approximated with sufficient accuracy by a polynomial. If the mapping function is known 88 a power series in I, then a formal solution can be given in terms of the coefficients a.. of the mapping function. If the mapping function is known in a closed form, then it may be easier to proceed directly from Eq. (44.5) rather than expand w{t) in a power series and then deal with the resulting infinite series (44.12) to (44.15). The reader will verify that formula (44.4) can be used in this case to obtain the result (44.16) with no calcu1&tion&l effort. The problem of the cardioid is of some interest inasmuch 88 it indicates an approximate behavior of a checked beam. PROBLEM Obtain the IIOlutioa fIf the tOIIIioD problem for a cardioid by utiWlioc formula (2(x 2

+ y2) + k,

(i = 0, 1, 2, . . . ,n),

on Ci ,

where the k, are the integration constants. The value of one of these constants, say ko, can be specified arbitrarily,' but the remaining n constants k. must be determined so that the function tp(x, y) =

which can be written as (47.4)

tp(x, y) =

l l

P

(X, Y)

P,(XG.IIO)

P

(x, v)

P.(z •• vo)

(a/

dx X

+ a) ,,'I' dy , uy

(lJift lJift) - ax - - dy , ay

ax

is single-valued' throughout the region R. If the region R is simply connected, the only requirement that the integral of the form (47.5)

F(x, y) =

f

P(Z,Y)

Po(:tO,'II0)

[M(x, y) dx

+ N(x,

y) dy]

define a single-valued function F(x, y) is-that M(x, y) and N(x, y) be of class C' in R and that throughout the region R aM

(4.7.6)

8jj

aN =

ax'

But if F(x, y), defined by (47,5), is to be single-valued in a mUltiply connected domain, then in addition to (47.6) we must demand that the integrals

1.

0,

(M dx

+ N dy)

vanish when evaluated over each interior contour forming the boundary of R. Since '" in (47.4) is a harmonic function, the condition (47.6) is. t t

See rema.rlts in the paragraph following Eq. (35.3). We recall that the displacement to - ",,(x, 1/), and to is a single-valued function.

EXTENSION, TORSION, AND FLEXURE OF BEAMS

171

clearly, satisfied and ",(x, y) will be single-valued in R if, and only if, (47.7)

(i

~

1, 2, . . . ,n).

Thus the constants k; (i = 1, 2, . . . , n) in (47.3) must be chosen so that the solution of the Dirichlet problem (47.8)

in R, on C,

V'", = 0 {

'" = ,%(x'

+ y2) + k,

(i = 0, 1,2, . . . ,n)

satisfies the set of n conditions (47.7). The value of ko, as we have already remarked, can be assigned arbitrarily. If the problem is rephrased in terms of the Prandtl stress function

w "'" ",(x,

y) - ,%(x'

+ yO),

the system (47.8) leads to the new system, (47.9)

{

V'w

= -2

w=

in R, on Ci,

k,

(i = 0, 1, 2, . . . ,n)

and the definition of w yields,

Accordingly, the set of conditions (47.7) becomes, fe; (:; dx - : : dY)

+ !C, (y dx

- x dy) = O.

The second of the line integrals in this formula is numerically equal to twice the area Ai enclosed by C" and the first can be written as

r (ow ~ _ oXoW ddsY) ds Jc,dv' r dw ds

Jei oy ds

=

Thus, the set of formulas (47.7) ip equiv.alent to the set

A.. dw ds = -2A,

(47.10)

'j'e, dv

(i = 1,2, . . . ,n).

The formula (34.10) for the calculation of the torsional rigidity D is still available, and we have (47.11)

D

=

p.

g

(x>

+ y' + x ~ -

Y ::) dx dy =p.

U-(x: +y~:)dXdY,

172

MATHEMATICAL THEORY OF ELASTIGTY

where we make use of the relations

B",= _(X+B't'), ax

and

ay

obtained in Sec. 35. The integration in (47.1l) is performed over the multiply connected region R. The right~hand member of (47,11) can be rewritten as D = p.

ff r

= 2p.

t-x (.rit) - :1; (yit)) dT dy

2it -

g

+" dxdy

+ p.ic it(ydx -

xdy),

where we make use of Green's Theorem, and the subscript C on the line integral means that the integration is to be performed in appropriate directions over all the contours C. (i = 0, 1, 2, . . . , n). Now if we chOO8e the value of it over the contour Co to be zero (that is, ko = 0) and note the boundary conditions in (47.9), we have D = 2p.

g

ifl n

itdxdy

+ p.

But

~Ci (ydx

- xdy)

=2

kl

fCi (ydx -

JJ dxdy

=

xdy).

2A.,

Ai

where Ai is the area enclosed by the contour Ci, and we have D = 2p.

fl

JI A

it dx dy

+

2p.k,A••

The expression for the t.wist.ing moment M is (47.12)

M = 2p.a

(If 't' dx dy + R

i

k.A.).

i-I

It will be recalled that the curves it(x, y) = const determine the lines of shearing stress (sec Sec. 35), and it follows from the boundary conditions in (47.9) that one can obtain a solution for the torsion problem of a hollow shaft from the solution of the torsion problem of a solid shaft by deleting the portion of material contained within the curve it(x, y) = const. Thus, in the discussion of the problem of torsion for an elliptic cylinder in Sec. 36, it was shown that the lines of shearing stress are similar ellipses, concentric with the ellipse x2 y! iii + Ii' = 1, representing the cross section of the cylinder.

Accordingly, if we delete

EXTENSION, TORaIO:!,;, AND FLEXURE OF BEAMS

173

t.he portion of materi.a.l contained within the elliptical cylindel'

as + 11'b' =

x'

(47.13)

(1 - k)·.

(0


.. + 2,,) "" X by A, but there is a fundamental distinction in the two sets of stressed states. In the plan~strain problem, the 11,,,, and 'Tall are independent of the xa~coordi­ nate, whereas in the problem of plane stress these functions may depend on XI. Since the variable Xa may appear as a parameter in all equations of this section, the problem is not truly two-dimensional. However, following the idea of Filon,1 it is possible to modify the system of Eqs. (67.3) to (67,5) in such a way that the resulting two-dimensional system corresponds to a physical problem of great practical interest. Consider a cylinder with the generators parallel to the xs-axis and with bases in the planes Xa = ±h (Fig. 50). We shall term such cylinder a plate if its height 2h is small compared with the linear dimensions of the cross section. The bases of the cylinder are the faces of the plate, and the plane Xa = 0 is the middle plane of the plate. Let us suppose that the faces of the plate are free of bo}lplied loads and all external surface forces act on the edge of the plate, that is, on the lateral surface of the cylinder. Moreover, we shall suppose that the forces acting on the edge lie in the planes parallel to the middle plane and are symmetrically distributed with respect to it. If the components of external surface forces acting on an element au "'" 2h ds of the edge are T a2h ds, the ve(ltor T" is the stress vector applied to the contour C bounding the middle plane. We shall further suppose that the comt L. N. O. Funn, PhilOS{)phical Tramactitmll of the Royal Society (London) (A), vol. 201 (1900), pp, 6.'-155; Quarkrly JourIlal of Applied Mat1temati~, Oxford &riea, I (19!lO), pp, 289-299,

255

TWO-DUUilNSlONAL ELASTO$TATIC PROBLEMS

ponent Fa of the body force vanishes and the ,components F. are symmetrical with respect to the middle plane. Under these hypotheses, the points of the middle plane will undergo no displacement in the directiOll of the xraxis, and 'if the plate is thin, the displacement '1£, will be small. Indeed, the symmetry of distribution of external forces implies that the mean value of '1£3 with respec~ to the thickness of the plate is precisely zero. For thin plates the mean values Uo of the displacements Ui give as

FIG. 50

useful information as that furnished by the with the average values (67.6)

U.(XI, X2) e

u..

This suggests dealing

;~ f~h U,(XI, X2, X3) dx"

where, as we already noted, U3 = 0. Since the faces of the plate are assumed free of external loads, (67.7)

T13(XI, X2,

±h) =

T23(XI, X2,

±h) =

T33(X1,

X2, ±h)

= 0,

and these equations together with the equilibrium equation (67.8)

1"13.1

+

1"23.2

+ T33.3 =

0,

demand' that 1".8•• (X" X2, ±h) = O. The fact that T3. and its derivative with respect to x. vanish on the faces of the plate suggests that T33 can differ from zero but slightly throughout the plate if h is small. This justifies us in assuming that T33 e 0. The remaining equilibrium equations T

1 From

al.l

+

Td.1

+

1"aI.3

+ Fa = 0,

(67.7) we conclude that "1O.1(X1, Xo, ±h) - ,.....(X1, X" ±h) = 0, and since (67.8) is valid throughout the plate, it rollows upon setting x. = ±h in (67.8) that ,.....(X" Xt, ±h) - O.

256

MATHEMATICAL THEORY OF ELASTICITY

upon integration with respect tGXI between the limits -It and (67.9)

1 2ii

fA~Io

(T.. l.1

+

1'... 1

+

1'01.1

+1&, yield

+ F ..) dxa = u.

Since

by (67.7), we can write (67.9) in the form (67.10)

1'.. ,.,

+ 1'..... + P.. = 0,

where T..#(X" x.) ""

(67.11) P,,(XI, X2)

E!

~ J:A T. #(X" x., X3) dXa, 2~ F .. (XI, x., X3) dx.,

J:.

are the mean values of T«# and F... If we form the mean values in the stress-strain relations (67.3) we get three equations, (67.12) with i '"' ZAp/ex + Z,,) and J "" '11...... These, together with two equations (67.10), serve to determine the five unknown mean values 11.. (XI, X2) and 1'«#(XI' x,). The substitution from (67.12) in (67.10) yields two equations of the Navier type, (67.13)

1'v"1l..

a;; + + (A- + 1') .,,--"x..

f!,

l' ..

(XI, x.) = 0,

from which the average displacements 11.. can be determined when the values of the 11.. are specified on the contour. The system of equations involving the average stresses 1'..~ can be got by deducing the corresponding Beltrami-Michell compatibility equations. It turns out to bel (67.14) where 9 1 = 1'11 + Ttt. This equation, together with the equilibrium equations (67.10), suffices to determine the mean stresses 1'«# when the boundary conditions on the edge are given in the form,

For integrating these equations with respect to X3 between the limits -It I

Compare with Eq. (66.8).

TWO-DIMENSIONAL ELASTOSTATIC PROBLEMS

257

and +h and dividing by 2h yields, (67.15)

nnC,

where '/'",(8) dB are the components of applied force acting OJ{ the element of.arc dB of the contour C. The two-dimensional boundary-value problem consisting I)f the lIystem of Eqs. (67.10), (67.14), and (67.15) is known as the problem in generalized plane stress. 1 68. Plane Elastostatic Problems. The discussion of the plane-deformation problem in Sec. 66, and of the generalized plane-strell!s problem in Sec. 67, shows that their mathematical formulations are idtlntical. The relevant differential equations and %2 boundary conditions in Sec. 67 differ from those in Sec. 66 only in the appearance of the barred symbols: iz"" T",~, X, '/'a, etc. Henceforth we shall refer to problems of these two types as plane elastostatic problems. In the formulation of these plane problems no restrictions on the connectivity of the region R was introduced. If the region R is multiply connected and finite, we shall supFIG. 51 pose that its boundary C consists of m + 1 simple closed contours Co, such that the exterior cont,

Tn'" U,lI

+ V.

The compatibility equation (66.8) now yields the equation

21' V'V. V'U =->'+21' If V is harmonie, then, as above, U is a biharmonic function. we are led to consider equations of the form

Otherwise,

in R.

V'U = F(xI, x.)

'10. General Solution of the Biharmonic Equation. The solution of the . fundamental biharmonic boundary-value problem can be made to depend on a certain general representation of the biharmonic function by means of two analytic functions of a complex variable. I We consider the biharmonic equation (70.1)

in R,

and if we let ViU "" PI(XI, x.), the function PI is, clearly, harmonic in R. Consequently' we can construct an analytic function F(z) "'" PI

of a complex variable z = XI

+ iP.

+ ix. by computing from PI the conjugate

• This representation was first obtained by E. ~ursat, Bulletin de la 8oci&e 11UJlM· 11UJlique de France, vol. 26 (1898), p. 236, who assumed that the biharmonic function is analytic. A derivation given here is due to N. 1. Muskhelishvili, Izvestilla (Bulletin) Akademiya Nauk SSSR (1919), pp. 663-686. The analyticity of the biharmonic function is not hypothesized here, and, indeed, it follows from the representation itself. • We uae the term harmonic function only for single-valued functions of class CS which satisfy Laplace's equation in the given region. Since the second derivatives of the biharmonic function U are related to stresses by (69.4), the function P. E V'U is necessarily single-valued. If p.(x., x,) is known, its conjugate P.(x., x,) is determined by integrating dP, - p •.• dx. + P". dx, = -p •. , dx. + P.,. dx" since Cauchy-Riemann equations demand that p,.• = -P •. , and p, .• = p....

J:'

Then

+ P ••• dx.) .Is independent of the path joining an arbitrary point M ,(x!., x:), with the point M (X" P.(x., x,) -

(-P."dx.

z.), II»' PI is harmonie. 'It follows that P.(Xl, x,) is determined to within an arbitrary constant C and, hence, F(z) .. P. + iP to within a pure imaginary constant Ci. If " x.), and hence F(z), is single-valued. In a the region R is simply connected, P,(x.,

multiplY connected region, F(z) is, in general, multipll'-valued, and we can confine our attention to some single-valued branch of F(z). The same considerations apply' to .,('1") ... ~~ JF(z) liz.

TWO-DIMENSIONAL ELAST08TATIC PROBLEMS

harmonic function p..

263

The function fO(Z), defined by

(70.2)

fO(Z) "" )4fF(z) dz

= PI

+ ip.,

is surely analytic, and therefore

+ l.op. - = -1 (P • + t'P) •.

'( ) op. fOZ=ox.

ox.

4

It follows from this, upon noting the Cauchy-Riemann equations = P'." PI.• = -P'.I, that

PI.I

PI.I PI .•

= p'.' = ~~p .. = -P2.l = -%p •.

Using these results and the fact that PI and p. are harmonic in R, we readily verify that

v'( U -

PIXI - pox.) "'" 0

in R.

Hence U has the structure, (70.3) where ql(XI, x.) is harmonic in R. Now if x(z) == ql + iq. is an analytic function of z whose real part is ql, the formula (70.3) can be written as U = m[ZfO(Z)

(70.4)

+ x(z)],

where z == XI - ix. and mdenotes the real part of the bracketed expression. Since fO(Z) and x(z) are analytic functions, it follows from (70.4) that U(XI, x,) is of class C" in R. The important representation (70.4) was first deduced by Goursat by different means. If we denote the conjugate complex values by bars, so that, for example, ~(z) == PI - ipo, then (70.4) can be written as (70.5)

2[1 = ZfO(Z)

+ z~ + x(z) + Xtz).

We shall make frequent use of this result in the sequel. 71. Formulas for Stresses and Displacements. The components ,."" of the stress tensor can be expressed in terms of the functions fO(Z) and x(z), introduced in Sec. 70, by substituting in the relations [69.4)

1"11 ...

U. n ,

1"11

= -U,lt,

1""

= U. ll,

from the Goursat formula, (70.5]

2[1 =

~(z)

+ z;W + x(z) + ifz).

lIlATREMATICAL THEORY OF ELASTICITY

To simplify calculations, we rewrite (69.4) as (71.1)

'1"11 {

+ ~'I"12 STU

'1"22 -

and compute first the find 1 that (71.2)

U,l

~U'12

= U,22 -

a

-i(U,l

7" iU,2)."

+ $U,12 a (U,l + $U,I),l, expression U,l + iU" from (70,5), = U,I1

+ iV,. =

I"(z)

We easily

+ Zl"'(z) + ~,

where we have set I/I(Z) a x'(z),

(71.3)

Calculating the derivatives of (71.2) with respect to Xl and x. and inserting the results in the right-hand members of (71.1) yield '1"11

+ iT12

'1"22 -

iTI2

=

I"'(z)

= I"'(z)

+ I"'(z) - ZI""(z) - I/I'(z), + I"'(z) + ZI"I/(z) + I/I'(z),

which can be written more compactly as' (71.4)

{

T11

+ Tn 1'11

'1" . . -

= 2[1"'(z)

+ 2iTn

=

+ 1"'(z)J "" 46{[I"'(z)], 2[zl""(z) + I/I'(z)J,

The formulas for displacements can be obtained by integrating the stress-strain relations (66.3), which we can write as

+

U, •• = 'Ar'J 21'Ul,l, = U,lI = 'Ar'J + 2I'u"., = - U,I' = I'(UI,2 + U',l).

T11'=

(71.5) {

1'22 1'12

Solving the first two of these for Ul,l and U,," we get 2",Ul,l

X + 2}o! • + 2(X + "') V U, X + 21' • U,2' + 2(X +-~ V U,

= - U,lI

2",U.,2 = -

and, recalling the definitions

V'U = PI = 4pI,I = 4p202 1

We omit the details of elemtmtary calculations making use of the formula

U . = aU at

.•

a. ilx,

+ au ~ iI' ax,

where 6 - Z, + iz. and if = z, - iz•. • These useful formulas were deduced finlt by G. V. Kolossoft in references given in Sec. 65. The derivation sketched above is due to N, I. Muskhelishvili. See, for example, See. 32 of his hook Some Basic Problems «f the Mathematical Theory of FJasHcity (1953).

TWO-DIMENSIONAL ELAST08TATIC PROBLEIIS

265

in Sec. 70, we obta.i.n 2"''''-1 = - U. ll

2"u.,! = - U. 22

2(X + 2,,) + X +" Pl.l, 2(X + 2,,) + X +" P.,10

The integration of these equations yields, 2(X + 2,,) + X +" PI + f(x,) , 2(X + 2,,) -U. 2 + X +" Pi + g(Xl),

- U. 1

(71.6)

where f(x.) and g(Xl) are, as yet, arbitrary functions. The third of Eqs. (71.5) serves to determine! and g. Since P,.' = -Pt." we easily find that f'(x,)

+ g'(x,)

=

0,

and hence !(x,) = ax, + fl, g(x,) = -ax, + 'Y,

where a, p, and'Y are constants. The forms of! and g indicate that they represent a rigid displacement and can thus be disregarded in the ana.lysis of deformation. If we set! = g = 0 in (71.6), recall that

f'(z) is &180 single-valued. Therefore

L ".

(b)

>fez) -

C.log (z - z.)

+ >f.(z),

k-1

where >f.(z) is analytic and single-valued in R. If we further suppose that the displacements u" are single-valued functions in R, then the increment acquired by 21'(ft, + iu.) in describing the contour C. is zero. Using this condition in (71.7), with 9' and >f in the forms (a) and (b), we find 21ri[("

+ I)A.z + ,.B. + 6.1

=

o.

Hence A. - 0, and C. = -xB.. The B. have a simple physical meaning explained in formulas (72.7) and (72.8). These follow from the calculation of the resultant force acting on each contour C. with the aid of formulas (71.1). The details of this argument will be found in Sees. 35, 36 of N. I. Muskhelishvili's Some Basic Problems of the Mathematic&1 Theory of Elasticity (1953).

If R is a finite multiply connected domain bounded by the exterior contour Cm+' and by m interior contours Ck (k = 1,2, . . . ,m) (Fig. 51), and if the displacements and stresses are single-valued functions throughout R, then 'P and !/t have the following structures:

(72.7)

!/t(z) = 2".(1 ~ x)

.

L

(Xlk) -

iX~k) log (z - z.)

+ ..yo(z),

k-l

where (Xik\ X~·) is the resultant vector of external forces applied to the contour C. and is an arbitrary point in the simply connected region R. bounded by C.. The functions 'Po(z) and !/to(z) are single-valued analytic functions in R. If R is an infinite region, bounded by several simple closed contours' C. (k = 1, 2, . . . , m), and if the stress components T'j are bounded in the neighborhood of the point at infinity, then' it is not difficult to prove that for sufficiently large Izl,

z.

, The region R in this case can be thought to be obtained from the region R of Fig. 51, by making the contour Cm +, expand to infinity. It corresponds to an infinite plate with m holes bounded by the C ..

TWO-DIMENSIONAL ELASTOSTATIC PROBLEMS

(72.8)

'P(Z I)

t/t(z )

Xl + iX, = - 211"(1 + ,,) log z =

K(XI - iX t ) 211"(1 + x) log z

. (B + 1.C)Z

+

269

+ rpo(z),

.

+ (B' + zC')z + t/to(z),

provided the origin of coordinates is taken outside R, that is, within one of the contours Ck. The Xl and X. are the components of the resultant vector of all external forces acting on the boundary C l + ... + Cm, so that m

Xl

+ iX.

L

=

(X\k)

+ iX~'»;

k-l

I"c(z) and fc(z) are single-valued analytic functions in R including the point at infinity.! The constants B, B', C' are related to the state of stress at infinity as follows:

(72.9)

2B - B' = TU( (0')

+ 4>(0')

~ ~4>'(O') + ",'(O')v(O')]

-

'" (0')

== N - iT

on 1.11 == 1,

where fI(.I) == ,y; (.I) (ai' (.I) ,

If we let Th == 1'.., 1';. = Til, 1';. == Tp~ in the formulas in the footnote on page 271 and recall formulas (71.4), we get the useful expressions,l Tpp

(75.12)

{

1'"" -

Tpp

+ T~"

== 2[4>(.1)

+ iW],

+ 2&", ==.plw~ Mn4>'(.I) + (aI'(.I)v(.I)] \.1)

16. An Elementary Method of Solution of the Basic Problems for Simply Connected Domains. The boundary conditions (75.7) and (75.8) have the form w(0') '77:\ 'T7:'\ (76.1~ a¥'l(O') ¥'l\O') ,yl\O') == H(tT), w \0')

+=

+

where IT ... ell is the value of .I on the boundary of the unit circle. In the first boundary-value problem a == 1 and HelT) "'" ft(iJ) i!.(iJ), and in the second problem a == -7t: and H(IT) == -2I'[gl(iJ) ig.(t1)1. If the region R is finite and simply connected, the functions ¥'l(f) and '{tl(r) can be represented in the power series

+ +

I

Note that eI'" _ ~ ...' (l"). pt;;7(ff

277

TWO-DIMENSIONAL ELASTOSTATIC PROBLEMS

.

e

(76.2)

lI'l(r)

=

l

U.re,

l

~l(t) =

k-O

bktk,

k-O

and it is natural to attempt to calculate the a. and b. by the method of undetermined coefficients. To this end we expand ,he right-haud member 'Of (76.1) in the i'ourier series (39.6) to obtain

.

(76.3)

H(a)

l

=

);=_

.

l

Cke"'~'"

Coer",

GO

and write the complex Fourier series for the known function (76.4)

The insertion from (76.2), (76.3), and (76.4) in (76.1) yields the equation 110

(76.5)

a

l

00

a.uk

l

+

1:=1

00

coer"

k=-110

l

GO

kU.u-H1

k-l

IlO

l

+

l

b.a-· =

k=O

C.cr",

k=~oo

if we take ¥'I(O) = ao = 0 and note that {f = e-i6 = a-I. On performing the indicated operations, which are surely legitimate if the involved series are absolutely convergent, we get GO

CIa

a

l

aoer"

+

1:-1

k-l

00

CIQ

CIO

l (L mamCmH-I) cr" + l (l ",-1

k-O

m-l

+

mamCm_;_I) a-'

.

L

.

b.u-k =

k-O

L Coer",

k __ oo

and, on comparing the coefficients of like powers of a, we obtain: e

(76.6)

aUk

+

l l

mUmCm+k_1

= Ck,

(k = 1,2, . . . ),

",-1 e

(76.7)

6.

+

ma..c.,_;-1 = C_;,

(k = 0, 1, 2, ... ).

",-1

If the system of Eqs. (76.6) can be solved for the ak, the b. are determined at once from formula (76.7). In the first boundary-value problem the system (76.6) cannot be expected to yield a unique solution if the imaginary part of at is left unspecified, since the function ¥'t(r) is not determined uniquely unless the value of g[¥,~(O)/c.l(O)J "" g[at/",'(O)J is assigned. No such supplementary condition is needed for the second boundary-value problem, inasmuch as the condition !"1(0) = 0 completely determines both lI'l(t) and ¥tIm

278

:MATHEMArlCAL THEORY OF ELASTICITY

Further, to ensure the existence of solution of the first boundary-value problem for finite domains, the resultant force and the resultant moment of assigned stresses T .. (a) must vanish. It is not difficult to shawl that the vanishing of the resultant force implies that the function H(,,) is single-valued on the unit circle, while the moment condition imposes a restriction on the coefficients Ck in the representation (76.3). An important special case arises when w(!) is a polynomial of degree n, because, as we shall presently see, the determination of at's, in this case, reduces to the solution of the system of n linear equations in n unknowns. The practical importance of this becomes obvious if it is noted that the mapping function for a finite domain can be approximated with arbitrary accuracy by a polynomial. We note first that, when w(t) is a polynomial of degree n, the function w(,,)/;rr;;y has the representation' we,,)

w'(tT)

2: n

-- =

(76.8)

ckuk

+

k-O

~

2:

k C_kU- •

k-1

Consequently, on setting c. = 0 for k 2': n we get: 1

H the resultant force vanishes, then /1(8)

+ il,(8)

fe

= i

(T,

1.'..

+ 1 in

+ iT,) do

(T,

= O.

(76.6) and (76.7), But

+ iT,) do,

and hence the increment in I, + if. as the contour C is traversed is zero. This is another way of qaying that H(tf) = /1(r'J) + ij.(r'J) is single-valued on 1,1 - 1. The vanishing of the resultant moment requires that

fe (x,T. - x,T,) do

-

-

fe (x, d/,

+ x, d/t)

= O.

Integration by parts gives [X./,(8)

+ X".(8)]C

-

fe [/,(8) dx,

+ 1,(8) dx,]

= 0,

and since the function in the brackets is single valued, the bracketed term vanishes.

The integrs.l can be written as .

T t'T , tI

+ + 3" 2,,) pw 'R'" e' . d8

=

2>' 4(X

+ 3" pw"R'" + 2,,) e' .

It is clear from this that the problem of determining the stress distribution r~'J is identical with the uniform-pressure problem considered in (a) 2>' a b ove, wh ele we must set P = - 4(>.

+ 3" + 2,,) pw "R"•

ticity (1951), pp. 107-111, where this problem is solved by indirect means. This problem was originally treated by H. Hertz, Zeitschrift fur Mathematik und Physik, vol. 28 (1883), and later by J. H. Michell, Proceedings of the London Mathematical Society, vol. 32 (1900), pp. 35-61, vol. 34 (1902), pp. 134 142, who solved severa) similar problems by ingenious devices. A unified and systematic trep.tment of this category of problems was first given by G. V. Kolosoff and N. I. Muskhelishvili in Itvestiya Petrograd Electrotechnical Institute, vol. 12 (1915), pp. 39-55 (in Russian). 1 For different solutions of this problem see Love's Treatise, Sec. 102, and Timoshenko and Goodier's Theory of Elasticity, Sees. 30 and 119. The problem of the disk rotating about an axis normal to the disk at an arbitrary point of the disk was solved by Ya. K. I1'yn, Doklady Akademii Nauk SSSR, vol. 67 (1949), pp. 803--806 (in Russian). A solution of the problem of rotating disk with att!).ched concentrated m _ is outlined in Sec. 80 of Muskhelishvili's Some Ba.sic Problems of the Mathematic!).l Theory of Elp-sticity (1953).

MATHEMATlCAL THEORY OF ELASTICITY

It should be kept in mind that the va.lue of X in formulas (68.3), appropriate to the problem ofa rotating disk, is given by X = 21../1-/ (X + 2/1-), since we are dealing with the state of generalized plane stress. In the corresponding plane-deformation problem, that is,-in the problem of a rotating shaft, X is the Lame constant given by (23.5). 78. Solution of Problems for the Infinite Region Bounded by a Circle. If we consider the region Izl ~ R and map it on the unit circle in the r-plane by means of (78.1)

z = we!) =

R T

the functions 1I',(r) "" lI'[w(r)] and "',(!) "" ,p[w(r)] assume the forms [see (76.11)],

If

+ (B + iC) + 1I'°(r), ",SX, - iX.) log r + (B' + iC')!!_ + ",O(r). 211'(1 + x) .I

\O,(!) = ;:.\{; : ) log r

(78.2)

,p,(r)

=

-

We recall that X, + iX. is the resultant force acting on the circular boundary and the constantl! B, B', C, C' are related to the stresses and rotation at infinity by formulas (72.9). We shall assume that C "" 0 and take B "" ~4hl( 00) + TO.( 00 )], B' = ~2[T22( 00) - TU( 00 )], { c' = Ta( 00).

(78.3)

For the determination of the analytic functions 11'0(.1) and ",oCI) we thus have the boundary condition 1

(78.4)

11'0(0') - -11'0'(0') 0"

+ "'0(0')

= FO(O') ,

where 78. 5) (

BR F O() 0'= F() 0'- Xl + 211' iX, IogO'---;-

+ fa [;:"(1-;:) a -

BRu 2

J-

(B' - iC')Ra,

and F(u) '"" f,(O) + i}2(O), determined by the specified stress distribution on the circular boundary.

.

Setting

.yom

=

Lbk5\

A-O

in (78.4) a.nd writing in the right-hand member the Fourier series representation for the single-valued function,

287

TWO-DIMENSIONAL ELASTOSTATIC PROBLEMS

we obtain

l

l

..

(78.6)

..

a~ -

k-1 -

l

l

aD

kii.".-k-!

+

k-l

k-O

X, + iX. [ . + ~L. k1 (-k -~ 1I't q -

..

=

fj.".-k

uk)]

-

A"uk

k--" -

2BR -q-

-

(B'

t'C')Rq

-

k-l

Xl - iX.

2

+ 2... (1 + x) U- • The comparison of like powers of u then yields:

+ Xl + iX. - (B' - iC/)R 211' ' iX.1 k> 2 A + Xl + 211' k' -, A i(XI + iX.) 02 '

al = Al

ak=k

6

0=

(78.7)

61

= A_1

-

Xl

t

iX

• - 2BR,

iX.! b-2 -- A -2 _ Xl + 211' 2 -k A ( )b =

-k -

k - 2

+ X, -

ak_2 -

XI

iX.

+ x)' + iX 1 211' ii'

211'(1

2

k

~

3.

These formulas simplify considerably when Xl + iX. = 0, that is when the stresses on the boundary are self-equilibrating. We next specialize these results to several problems of technical interest. a. Uniform Internal Pres8ure. When constant pressure P acts on the boundary of the hole, T, = P cos fJ, T. = P sin fJ, and'

f.

+ if.

=

+i -i

t

(Tl

+ iT.) ds

f' Pe"R dfJ = -PRe".

Thus, F(u)

= -PRq-'.

1 The negative sign is introduced in the integral because the positive direction of integration along the circle is clockwise, inasmuch as the normal v is directed toward its cent-er.

288

MATHEMATICAL THEORY OF ELASTICITY

Clearly, Xl + iX, = 0 and, if we assume that the stresses at infinity vanish, B = B' = C' = O. Thus FO(O') "" F(".), and = ..

"1m

O(n,

I/I,(r) ,., I/IO(r).

Since all Fourier coefficients A k , with the exception

A_, = -PR, vanish in tne expansion for F(O'), we conclude from (78.7) that at b,

= 0, k = - P R,

~

1, bk

=

0,

k",cl.

Hence

1/11 = -PRj, and, therefore, Hz)

.. (z) '" 0,

=

PR2 Z

Using the formulas (77.6), we get, PR2 u,. = - , u, = 0, 2J.1r PR2 Tn' =

-T"

= - 7'

'Tr 8 =-

0,

and the problem is completely solved. b. Cuncentraied FQf'ce in tM Plane. The streB6 distribution produced by a concentrated force applied to a point in the plane can be obtained by analyzing the effect of the constant stress distribution

T2=~' 2'11'R

(78.8)

acting on the boundary of the circle JzJ = R. The resultant force produced by the stress distribution T, + iT, is, clearly, X, + iX 2. If we assume that the stresses at infinity vanish, FO(u) defined by (78.5) becomes, iX, I _!_ Xl - iX,. (78.9) F O(".) = F(U ) _ X, + 2'11' og U + u' 2'11'(1 + x) But

;~ J1 + ~n 80

that

+

'J'(T 'T)d 8 = = t I l .

+

i(Xl iX,) r'J F() ". = 211'

X, !!5

2 .X1 +iX 21r

-l

.. v,

+ iX, Iog "..

211'

Inserting this in (78.~j ..-e see that the right-hand member in (78.6) reduces to the single term X, - iX 2 -2 F O(" ) -_ 211-(1 + x) 0' •

TWO-DIMENSIONAL ELASTOSTATIC PROBLEMS

289

We thus conclude that a~

= 0, Xl

b

k

> 0,

+ iX.

• = 211"(1

k F- 2,

+ x)'

and hence ",O(r)

""O(r) = 0,

++iX.x) s' .

= X,

211"(1

Inserting these in (78.2), and recalling (78.1), we get

X,

(78.10)

+ iX. R + x) log ""i'

'" (z) = 211"(1

.1'(2) ..

-x(X , - iX.) 10 ~ 2... (1 + x) g z

=

The stress distribution in the region if we insert

(78.11)

Izl 2:

'( ) __ X, + iX.! '" z 2... (1 + x) z' ""(z)

= x(X , - iX,) 2... (1

+ x)

+ X, + iX. R'. 211"(1 + x) z' R is determined by (77.6),

"( ) _ X, 'I'

Z

-

+ iX.!_ + x) Z2'

2... (1

! _ X, + iX. R'. z

...(1

+ x)

z'

To obtain the stress distribution produced by the concentrated force X, + iX. applied at z = 0, we let R -> 0 and allow T, and T. in (78.8) to increase in such a way that the resultant force is always equal to X, + iX,. The resulting stress distribution is that given by formulas (77.6), where we use (78.11) with R = O. The result of simple calculations is, x TN"

(78.12)

"r" orr'

= -

1C

+ 3 X, cos (J + X, sin (J

+

1

x - I X I cos

+1

21fr (J

-,

+ X. sin (J 211'1'

'

=

X

=

x - I X I sin (J - X, cos X + 1 211'1'

(J

The solution recorded here corresponds to the state of plane strain. In dealing with the generalized plane stress, x in (78.12) must be replaced by ;( = (3 - u)/(1 + u), while X, and X, are reckoned per unit thickness of the plane. That is, X, = Xf/2h and X. = Xgj2h, where 2h is the thickness of the plate and X~ + iX~ is the concentrated force. c. Concentrated Moment in the Planl-. We consider next the effect of the stress distribution

M. T 1 = - 2... R' sm

(J,

T,

=

M 2... R' cos 8,

290

MATHEMATICAL THEORY 011' ELASTICITY

applied to the boundary 1:1 = R. This distribution is produced by the constant tangential stress T of magnitude Al/2TRt.

Binoe fl

+ if.

-

if'

(TI

+ iT.) ds = ~ 2rR

f'

eil dB - -

Ali elf

2rR

'

Mi F(u) = - 2rR u- I • Thus, the only nonvanishing At in (78.7) is A_I = - Mi/2rR. stresses at infinity vanish, the system (78.7) yields, ak = 0,

k = 1, 2, .

Mi

b, = 2rR' inasmuch as Xl

+ iX

2

= O.

If the

., k"" 1,

bk = 0, Hence,

and 3.

= _

P~R,

Ie> 1, b: = 0,

291

TWO-DIMENSIONAL ELASTOSTATIC PROBLEMS

Thus

and, from formulas (78.2),

o~

0,

m

~

1,

transforms the region exterior to the ellipse (79.1) into a circle if we take a-b R = a + D, m = a + b' 2

It I ~

I,

It should be noted that, as the point t = eiD describes the circle 1 in the positive (counterclockwise) direction, the corresponding point z traces out the ellipse (79.1) in the clockwise direction, Accordingly, the parametric equations of the ellipse must be taken in the form:

It I =

x, = R(1

+ m) cos D,

x.

=

-R(1 - m) sin iJ.

If m = 0, the ellipse becomes a circle. When m = 1, the point in the z-plane traces out the segment of the Xl-axis, between Xl = 2R I>nd x, = -2R, twice, as the point t describes once the boundary It I = 1. Thus, in this case, the function (79.2) maps the z-plane, slit along the line joining the points (2R, 0) and (-2R, 0), onto ItI ~ 1. The solution of the first boundary-value problem for an infinite simply connected domain, as we saw in Sec. 76, can be reduced to the determination of two functions '1'0(1) and 1/10(1), analytic in the circle III < 1, which sati'lfy t.he boundary condition (79.3) , See See. 19.

~o(.,-)

+ wC"-)

",'(a)

rpOI(a)

+ 1/-0(a)

= FO(a).

TWQ-1>lMENSIONAL ELASTOSTATIC PROBLEMS

293

In our problem,

so that the coefficients en, in the expansion (76.4), vanish for all n ;:: O. It follows then from (76.6) that k;:: 1.

so that (79.5)

'l'0(S) =

l

.'l.r',

.=1 where (79.6)

with PO(o") given by (76.13). An integral representation for O(t) = PRt (2e 2ia 4

PRt

.!.O(I-) _ 'Y

Thus

p:r (2e

,

[2 _ 1 - e""'(r'

2(mt2 _ 1) m

-

2ia - m

",(I)

=

~(t)

= _ PR [e-2ia!

2

m),

-

r

+ m)].

+.p}

+ e2iar m

_ (1

+ m2)(e m

2ia

-

m) __ r_],

1 - mrs

from which the displacement and stresses can be computed without difficulty. 1 PROBLEMS

1. Compute the displacements and stresses in the problem treated in the illustration of Sec. 79a, for the case when m - o. I. Solve the problem of deformation of an infinite plate with an elliptical hole. when a constant tangential force acts on the boundary of the hole. 1 The solution of this problem was first obtained by C. E. Inglis, Transactions of 1M l'Mtitute of Naoal Artihitec18. London, vol. 55 (1913). pp. 219-230. The solution given

here is due to N. I. Muskhelishvili, [Zll68tiya (Btdletin) Akademii Nauk SSSR (1919). pp. 663-686. It is also contained ill See. 82&. pp. 337-339, of Muskhelishvili'. book, Some Basic Problems of the Mathematical Theory of Elasticity (1953).

MATHEMATICAL THl!IORY OJ' JllLASTICITY

SO. Problems for the Interior of an EUipse. Theoretically the method of solution illustrated in the foregoing can be used to solve problems for simply connected domains whenever the mapping function II == ",(1) is known. But the function ",(r), mapping the region interior to an ellipse onto a circle, is very complicated. However, as was shown by Musk.,_ helishvili,' it is possible to make an effective use of the function employed in the preceding section to solve the interior problem as well. l·pIane

FIG. 56

We consider (80.1)

z = ",(I) = R

(I + T).

and, upon setting I = pe;' and z = x,

R > 0,

m ~ 0,

+ ix., find

=

(p + ;-) cos t'J, R(p - ;)Sint'J.

=

Pl

x, = R X2

Thus, the circle of radius C, with the semiaxes

P

in the I-plane corresponds to an ellipse

(80.2)

provided that pf ::::: m. The circle of radius

p = p.

corresponds to another ellipse C. (Fig. 56),

N. I. Muskhelishvili, Prikl. Mat. Mekk., Akademiya Nauk SSSR, vol. 1 (1933), pp.5-12. See also, Some Basic Prohlems of the Mathematical Theory of Ela.sticity (1953), pp. 244-250. This problem was also treated by much more complieated means by O. Tedone, AUi della accademia delle 8cUnze di Torino, vol. 41 (1906), pp. 86-101, and T. Boggio, AUi del reale inatituto veneto di 8cUnze, lettere ed uete, vol. 60 (1901), pp. 591-6011. A solution of the problem, with the aid of integral equations, was also given by D. I. Sherman, Doldady Akatkmii Nauk SSSR, vol. 31 (1941), pp. 309-310. I

TWO-DIMENSIONAL ELASTOBTATIC PROBLEMS

297

and the elliptical ring bounded by C 1 and C2 is mapped conformally by (SO.I) on the a.nnulus formed by the circles P = P1 and P = P.. If PI is increased indefinitely, the function in (80.1) maps the region exterior to the ellipse C1 onto the region 'exterior to the circle P = Pl. For m = pI, the ellipse C 1 degenerates into a segment of the real axis. If we take m = 1, the mapping function

R>O

(80.3)

maps the entire z-plane, slit along the real axis between Xl =- -2ft and = 2R, onto the region III 2 1. As the point I = e'l) traverses the circle once, the corresponding point z traverses the slit twice, so that the points tT = e'/I, and tT = e-ill , correspond to one and the same point Po on the slit. The ring bounded by the circles P = Po > 1 and P = 1 then corresponds to the interior of the ellipse Co, cut along the real axis between the points (-2R, 0) and (2R, 0). If either the displacements or the stresses are specified on the boundary Co of the uncut ellipse, the functions ~,(I) and ",(t) are determined by the condition of the form Xl

(80.4)

"''1'1(1)

w(t) -:rr.:\ -;:-r.:'\ + wl(t) ~1\1) + >/11\1)

= H(I),

Since 'l'1(t) and 'h(t) are analytic in the ring 1 represented in Laurent's series as

.

(80.5)

'l'1(t)

=

L

k--

akr,

>/1,(1) =

< It I < po,

.

L M·. k--

110

they ca.n be

10

Moreover, the point Po on the cut corresponds to the points I = e'll • and I = rill. on 111 = 1, and the continuity of 'I'(z) and 1/I(z) requires that (80.6)

'l'1(tT) = '{>,(It) ,

1/I,(tT)

=

1/I,(It).

The condition (80.6) implies that the coefficients ak and b. in (80.5) are related by the formulas: (80.7)

k = 0, 1,2, . . . .

The further conditions connecting these coefficients, which enable one to determine the functions in (80.5), are obtained from the boundary condition (80.4) in the manner of Sec. 76. The reader interested in further calculational details will find them in the cited works of Muskhelishvili. 81. Basic Problems for Doubly Connected Domains. We shall see in this section that the method of solution outlined in Sec. 76 can be easily modified to yield an effective solution of the basic problems for the cir-

298

MATHEMATICAL THBOllT 0"' JlLASTlCJTY

cular ring. Although a doubly connected domain can be mapped conformally on a circular ring, a generalization of the formulas of Sec. 76 to doublY" connected domains usually leads ~o intractable systems of equations for the coefficients in the series representations of '1'(21) and '{t(z). The treatment of the first and second boundary-value problems for the circular ring is identical, and we confine our discussion to the first problem. Let the ring be formed by" a pair of concentric circles C.., a = 1, 2, of radii R", where R, < R.. To simplify" calculations, we shall suppose that the external stresses applied to each boundary- are such that the resultant force and moment vanish for each bound8.ry. In this event, the logarithmic terms do not appear in the representations (72.7), and the functions tp(z) and '{t(z) will be analytic in the ring R, < Izl < R •. Accordingly, we can write

LakZ", ~

(81.1)

'P(z) =

The coefficients (81.2)

tp(z)

where (81.3)

i

at

R,
0,

as we saw in Sec. 80, maps the region bounded by two confocal ellipses onto a circular ring of radii P = pa, ex = 1, 2. If the external stresses acting on the elliptical boundaries are such that the resultant force and moment acting on each boundary vanish, the functions holes is also considered in this paper. As an illustration of the "alternating method," the equilibrium of an eccentric ring is discussed in See. 88. The state of stress in a heavy semi-infinite sheet with one circular hole was investigated by R D. Mindlin, "Stress Distribution around a Tunnel," Proceedings of the American Soeiety of Civil Engineers, vol. 65 (1939), pp. 619....642. Stress distribution in a heavy semi-infinite sheet with two circulsr holes was studied in detail by D. I. Sherman, Prikl. Mat. Mekh., A1w.demiya Nauk SSSR, vol. 15 (1951), pp. 297-316, 751-761. An investigation of the stress concentration in a heavy semi-infinite sheet, near arch-shaped and trape~oidal openings stiffened by absolutely rigid rings, was made by I. S. Ham, DoptMdi Akademii Nauk Ukrmn'.koi RSR (1953), pp. 299-303.

302

MATHEMATICAL THEORY OF ELASTICITY

transfofms (81.9)

/t(r), analytic in the circle Irl < 1, which satisfy on its boundary l' a conditiOli of the type' (82.1)

"''I'(er)

w(IT) - -+~ 'I"(er) + >/t(er) w (er)

=

H(er),

where H(IT) == h,(,'}) + ih,(,'}) is a single-valued function having continuous derivatives with respect to i} satisfying Holder's condition.' The boundary condition (82.1) can be reduced to an integrodifferential equation for the determination of 'I'(l) and >/t(l) by a technique similar to that used in deriving Schwarz's formula in Sec. 42. 1 der If we multiply both members of (82.1) by 2ri er _ l' where Irl < 1, and integrate over 1', we get (82.2)

with

1 We omit the subscript 1 and the superscript 0 on", and f in the formulas (76.11), (76.12), (76.14) and in sll expressions of this snd the following three sections. I See Sec. 40.

MATHEMATICAL THEORY OF ELASTICITY

By Harnack's Theorem of Sec. 41, Eq. (82.2) is equivalent to Eq. (82.1). But for every F(r) continuous in IfI ~ 1 and analytic for IfI < 1,

_!_.1 F(O') du = F(f) ' 2 n _!_.1 F(u) dO' = F(O),

2n

~O'-f

so that (82.2) yields (82.3)

a-n n,

L K~l ~

+ Klr + Ko +

...,

m-~

where K" K n-

(84.6)

== (heft.,

I

+ 2a.c., == litc. + 2a.c. + ... + (n - l)an_lCn, = alcl + 2d.c. + . . . + nanc•.

=

iitCn_1

We do not write out the expressions for K~, m ~ 0, because, as will be seen presently, they are not required in the calculation of ",(r). It may be observed that Eqs. (84.6) contain only n coefficients c; in the principal part of the Laurent expansion (84.3). The determination of the principal part calls for quite elementary algebraic computations. If we now set r == 11 in (84.5) and insert the result in (82.3), we obtain

f

~ "'(0-) ",'(0-) du = 2," ~ ",/(U) U -::- r

i

.. -0

r-,

K ..

..

since

f . L K_,..r-

1 2ri.,

Thus (82.3) yields

1m

(84.7)

+ ~

a

+

1 we find

,,' (IT) W'W(IT) (IT) '-0

=

K(O) .".,-.

c

+

I

+ KjO)1T + K'o°) +

n-T'

K'!!;"IT- m ,

",-1

where K~O_:2 = alC n_2,

(84.13)

K~.'.3

= alCn _3

KiO)

=

aiel

+ 2a"cn-2,

+ 2a,c. + . . . + (n

- 2)a n-"cn_2,

Evaluating the. integral in (83.4), we get n-2

~ ( w(lT) - w(r) 1. In the first boundary-value problem a = I, and the imaginary part of a, can be set .,qual to zero. Equation (85.4) then yields

a,

+ a, '= 2a,

and from (85.3), with a = 1, K,

= Qi = _!__ 2

41ri

= C,

1 >

H(u) ,,'

.L

au.

312

lIlATHEMA TICAL THEORY 01' ELASTICITY

The substitution in (85.1) then yields .p(r), for the first boundaryvalue problem, in the form (85.5)

(0 .p

=

..!_, (

J.... ( H(fT) duo

H(fT) du _.1_. ( H(tT) du _ r 4ri}y fT' 2ri

211'1 }~IT -

j..,

fT

The corresponding function tIter) from formula (84.11) is determined by!

tIter) =

(85.6)

~ iITH~~ du -

.ply) + ~r i HtT~) duo

In the second boundary-value problem ex = - x, and at is completely determined by (85.4). We leave it to the reader to write out the appropriate solutions in the form analogous to (85.5) and (85.6). Instead of using formulas of Sec. 84, it is frequently simpler to determine the function 'P(r) directly from Eq. (82.3). We illustrate this by solving the problem of Sec. 79 for the infinite region bounded by an ellipse. Inasmuch as the boundary condition (79.3) is identical in structure with (82.1), the integrodifferential equation for .p0(!) is (85.7)

'P0m

r

+ ,L ~ ,.,UI(tT~ dfT + "'0(0) ~ ...~. mn }~(J) (fT)

IT -

•

_..

If we insert "'(IT) ",'(tT)

1

rFO(fT~ duo

}ytT - •

+ ma-'

= tT(m

- fT')

from (79.4) in (85.7) we get (85.8) 'P

0(1)

r

1 ma-~ 'Pi)I(tT) dq 2ri}ytT(m - tT')fT - r

+ _!_.

+

+ "'0(0)

=

_!_.J,

FO(IT) duo 211'1 ytT - r

Since,

lui ;:::

1,

the value of the integral in (85.8) is zero.' Thus, ,.,O(r) =

But ,.,°(0)

=

~i:::-~du -

"'°(0).

0, so tha.t

• Since a, - el" X, - tt•. Note (79.4), and reea.U that the expansion for .... ("') contains no positive po_ ol •• I

TWO-DIMEN810NAL ELAST08TATlC PROBLEMS

313

and, hence, (85.9) The calculation of (85.10)

.;O(t) =

_!_,

2ri

1

FO(u) du

"yU -

t

+ r(r' + m) 'l'0'(t) 1 - mr t

was carried out in detail in Sec. 79. For m = 0 these formulas yield the solution of the first boundaryvalue problem for the region exterior to the circle III = 1. The reader may find it instructive to solve these problems by determining the function 'I'~(t) from the integral equation (83.5) and by following the argument of Sec. 83. For either of the mapping functions considered in this section the integral in (83.5) vanishes, so that

'I'~(t)

=

.! A'(t).

'"

The substitution in (83.4) then yields at once "'1'0(1) = A(t)

+ p,

where p is a constant. This constant and the constant k in (83.3) can be easily determined by making use of (83.6) and recalling that '1'(0) = o. 86. Further Developments. Multiply Connected Domains. The methods of solution of plane problems considered thus far depend vitally on the knowledge of the mapping function. Since only simply connected domains can be mapped conformally on a circle in a one-to-one manner, the considerations of Secs. 82 and 83 do not apply to multiply connected domains. However, there is a simple connection between the mapping function W(I) and Green's function for the domain.! Thus the integral equation (83.5) can always be written in the form whose kernel is expressed in terms of Green's function. Since Green's function can be constructed for multiply connected domains, this at once suggests a generalization of the integral equation. One such generalization has been made by Mikhlin, who reduced the basic problems of plane elasticity in multiply connected domains to the solution of certain Fredholm integral equations whose kernels depend on Green's functions.' Although Mikhlin's equations serve admirably to establish the existence of solutions in multiply connected domains, they possess the disadvan1 H Olle writes th& mapping function in the fonn r - /(z) and makes the point z - '. of tbe region R correspond to the center of the circle Irl - 1, tben Green's functioa G(p, P,) - (1/2.0) log (l/ll(z)l), with tbe pole P. at the point I .. • A connected account of this work is contained in a monograph by S. G. Mikhlin. I!Iltitled Integral Equations (1949) (in Russian).

314

MATHEMATICAL THEORY OF ELASTICITY

tage of being dependent on the solution of an auxiliary Dirichlet's problem for Green's function. It is clearly desirable to formulate the relevant equations so that they depend only on the assigned boundary values. The fs.ct that this can be done was demonstrated by Lauricella l in a rather involved paper concerned with the integration of the equilibrium equations for the clamped elastic plate. This particular problem, as we have already observed in Sec. 69, is closely related to the first boundaryvalue problem in plane elasticity. The Lauricella equations have been rediscovered by Sherman, 2 who deduced them in a very simple way and made use of them in solving the standard boundary-value problems, and certain important new types, in plane elasticity. A detailed account of Sherman's work would consume more space than we have at our disposal, and we give only a sketch of the essential ideas. We recall that in a finite simply connected domain ",,(z) and I/I(z) are analytic in the interior, and on the boundary C they satisfy the condition, (86.1)

a",,(t)

+ t,,/ (t) + I/I(t)

= J(t)

on C.

Sherman seeks to represent ",,(z) and I/I(z) by the following integrals of Cauchy's type:' ",,(z) = _!_. ( W(8) ds

2" Jes - z

(86.2) !{t(z)

r 2"}es-z

= ~

'

W(S) ds _

__!_ ( AW' (8) ds, 211'z}e s -z

where w(s) is an unknown density function whose derivative satisfies Holder's condition on C. . The choice of w(s) is restricted by the boundary condition (86.1). We proceed to determine the nature of this restriction by substituting from (86.2) in (86.1). We first note that I

1

"" (z) = 2-' 11'1.

1

w(s)

ds , c (--)2 s- z

I G. La.uriooll&, Acta MatMmatica, vol. 32 (1909), pp. 201-256. • D. I. Sherman, Doklady Akademii Nauk SSSR, vol. 27 (1940), pp. 911-913, vol. 28 (1940), pp. 25-32, vol. 32 (1941), pp. 314-315; Prikl. Mat. Mekh., Akademiya Nauk SSSR, vol. 7 (1943), pp. 301-309, 413-420, vol. 17 (1953), pp. ~92. Equstions .,.bQSe sppesrsnce is strikingly simiIa.r to the Sherman-Lsurioolls equstions have also been deduced by N. I. Muskhel.ishvili, Doklady Akademii Nauk SSSR, vol. 3 (1934), pp. 7 and 73. However, the content of Muskhelishvili's equstions is quite different, and they appear to be less lIIII!OOptible of extensions to the new types of eIa.tJtoststie pI'Oblems. • These forms are suggested by the known solution of the equilibrium probleln lor the aemi-infiuite plane, See &Iso formuls (82.4}. Integrsls of Csuehy'll type wertintroduced in See. f().

315

TWo-DIMENSIONAL ELASTOSTATIC PROBLEMS

which is not an integral of the Cauchy type. parts we get '(z) =

(86.3)

'"

~

But on integration by

r W'(S) ds.

2n}os-z

II .ve now let z in (86.2) and (86.3) approach an arbitrary point t of C. we get from Plemelj's formulas (40.7), r, including the point at infinity.

FIG. 60

The functions .,.(O)(z), I/I(O)(z) will be determined in the region so that w(O)(t)

+ t/;(O)(t)

=

-pt

Izl < R,

on Co.

These functions, clearly, will not satisfy the conditions (88.1) on the boundary C I • We next obtain the solution '1'(1), >/;(1) in the region Iz - al > r, corresponding to the zero stresses at infinity, such that (cp(l)(t)

+ tcp(l)'(t) + I/I(I)(t)

=

-l~[",' r, from the boundary conditions, tp(l)(t)

+ ~ + ",U)(t)

'"' -L(,,(O), ",(O)}

'"' pea 1 I

Ie.

+ t).

TheIle follow from Eq. (85.S) and (85.6) upon setting, - Rr. TheIle follow from (85.9) and (85.10) upon setting m - 0 and , .. ,./. where

• - . -0. ~ Ia verifyinc

these ca1cuIationl, note that I - B' It.

TWO-DIMENSIONAL ELAST08TATIC PBOBLlD¥S

SettingFl

..

pea

+ t) in (88.6) and integrating. we find, ",(1)(0) = 0, ",(I) (0) = + pa,

ri

80

that

(88.8)

",(I)(z)

r 2p ",(I)(z) '" - -

= 0,

z-a

+ pa'

Iz - al > r.

We next form I

L(",(l) ",(1) = J!!__ l' Ii - a

and determine ",(·)(z), "'("(z) for L(",('), ",(2»

Izl < R

Ie. =

+ pa'

from the boundary condition,

-L(rp(l), ",(1»lc.

= -

pr' r=a - pa.

Making use of the formulas (88.5) with Fo(t) given by the right-hand member of the expression just found, we obtain,

+ az) 2R'(R' - az)' pr2a ",(2)(Z) = -pa + (2)( ) '" _

(88.9)

'"

pr2z (R2

z

(R2 - az)2

(2R2 - az).

This process can be continued to obtain the approximating functions of higher orders. The series (88.2) constructed in this manner converge, but clearly the rapidity of convergence will depend on the magnitudes of the parameters a, r, and R. As noted earlier, this problem can be solved more simply in bipolar coordinates. l b. Concentric Ring under Concentrated Forces. Let the ring bounded by concentric circles Co and Cl of radii Rand r, respectively, R > r, be acted on by the concentrated forces P at z = ± Ri. The functions ",(z), "'(z) are determined in the region r < Izi < R from the boundary conditions: (88.10)

",(t)

+ t7m + ~ = const = J(t)

J(t) = 0,

for t = Re",

=P,

for t = Re",

on C1 , on Co,

1 See, for example, Ya. S. Uflyand, Bipolar Coordinates in the Theory of Elasticity (1950), pp. 204-210 (in RWl8ian). • See Sec. 77c·

326

MATHEMATICAL THEORY OP ELASTICITY

We again seek a solution in the form (88.2), where '1'(2.) , >/I r. AI> our first approximation '1'(0) (z), I/I(O)(z) we take the known solution, dedy.ced in Sec. 77, for the solid circle of radius R, under the action of concentrated forces. It is, '1'(0)(:)

=

!2 (log: - ~R + ~), 2r

.1.(0)(:) = Pi " 2.".

(10

z + ~R z - ~R g Z + ,R

R

+ _!!!__,_ _ ~). z - ,R z + $R

The subsequent approximations are determined from the boundary conditions (88.4), with the aid of formulas (88.5) and (88.6). Although the process indicated here leads to convergent series (88.2), the convergence is slow. However, because of the special character of loading, it proves possible to deduce the general expressions for '1'(1,,), t(l,,) and sum the dominant terms in the resulting series. Narodetzky' obtained in this manner an approximate solution, valid to any specified degree of accuracy. Variants of the Schwarz method have been used by Mikhlin and Sherman to solve certain integral equations furnishing solutions of the first elastostatic boundary-value problem for the semi-infinite plate with an elliptical hole. 2 89. Concluding Remarks. The principal object of this chapter has been to introduce the reader to certain powerful general methods of solution of the two-dimensional problems in elasticity. These methods have recently been extended to plane problems in anisotropic elastic media and modified to include the problems of transverse deflection of thin plates and several categories of contact problems s in elasticity. Among the more comprehensive contributions of this type are: 4 S. G. Lekhnitzky, Anisotropic Plates (1947). 1. N. Vekua, New Methods of Solution of Elliptio Equations (1948). 1. Ya. Shtaerman, The Contact Problem of Elasticity (1949). S. G. Lekhnitzky, Theory of Elasticity of an Anisotropic Elastic Body (1950). . 1 M. Z. Narodetzky, IZ1Je$tiya Akademii Nauk SSSR, Technical Serie8, No.1 (11148), pp. 7-18 (in RUllI!ian) . • S. G. Mikhlin, Trudy Seisnwlogical Institute, Academy of Science of the USSR, No. 391 (11134) (in RllBSian). D. 1. Sherman, Trudy &iBmological InstitutB, Acade>/ty of S~ of the USSR, Nos. 53 and 54 (1935) (in RUIlSian). • The contact problems are treated in Chap. 13 of N. I. Muskhelishvili's Singular Integral Equations (11153), as well as in his monograph Some Basic Problema of the Mathematical Theory of Elasticity (1953). • With the exception of the book by Green and Zerna all these monographs are in the RUIlSian language.

TWO-DIMENSIONAL ELASTOSTATIC PROBLEMS

327

G. N. Savin, Concentration of Stresses around Openings (1951). A. E. Green and W. Zerna, Theoretical Elasticity (1954). Savin's book contains solutions of numerous special problems on the stress concentration near openings in stretched isotropic elastic plates. I A survey of the recent work on the theory of plates, published in the USSR, is contained in a paper by G. Dzhanelidze, Prikl. Mat. Mekh., Akademiya Nauk SSSR, vol. 12 (1948), pp. 109--128. An English translation of this paper, prepared by the American Mathematical Society, Translation 6 (1950), is available. References contained in this translation should be supplemented by the following papers dealing with the deflection of thin elastic plates whose boundaries are simply supported, clamped, or partly clamped and partly simply supported. All these papers' appeared in vols. 14 to 17 of the Russian journal Applied Mathematics and Mechanics (Prikl. Mat. Mekh., Akademiya Nauk SSSR): Z. 1. Havilov, vol. 14 (1950), pp. 405-414; M. M. Friedman, vol. 14 (1950), pp. 429--432, vol. 15 (1951), pp. 258-260, vol. 16 (1952), pp. 429--436; G. F"Mandzhavidze, vol. 15 (1951), pp. 279--296; V. K. Prokopov, vol. 14 (1950), pp. 527-536, vol. 16 (1952), pp. 45-56; A. 1. Kalandiya, vol. 16 (1952), pp. 271-282, vol. 17 (1953), pp. 293-310, 692-704; G. A. Greenberg, N. N. Lebedev, and Y. S. Uflyand, vol. 17 (1953), pp. 73-86; G. A. Greenberg, vol. 17 (1953), pp. 211-228. 1 These may be supplemented by J. R. M. Radok's paper concerned with the problems of plane elasticity for reinforced boundaries, Journal of Applied Mechanic., vol. 22 (1955), and by Eugene Levin's doctoral dissertation entitled "Reinforced Openings in Plane Structural Members," University of California, Los Angeles (1955). See also I. S. Hara's paper cited in Sec. 81, and I. O. Abramovich, Doklady Akademii Nauk SSSR (NS), vol. 104 (1955), pp. 372-375. , The following papers on the deflection of thin elastic plates were published while this book was in press: V. A. Likhachev, Prikl. Mat. Mekh., Akademiya Nauk SSSR, vol. 19 (1955), pp. 255-256; O. M. Sapondzhyan, Izve8tiya Akademii Nauk ArmyanskdL SSR, Phy•. Mat. Nauki, No.5 (1954), pp. 19-43, No.6 (1955), pp. 27-34; D. I. Sherman, Doklady Akademii Nauk SSSR, vol. 10 (1955), pp. 62:Hl26.

CHAPTER

6

THREE-DIMENSIONAL PROBLEMS

90. General Solutions. The key to effective treatment of the twodimensional boundary-value problems, discussed in Chap. 5, is in the special representation of solutions of appropriate field equations with the aid of certain arbitrary functions. Although several attempts have been made to construct analogous "general solutions" of the three-dUnensional field equations of elasticity, such solutions have not been exploited in a systematic way. The so-called general solutions are but particular forms of solutions of the field equations involving arbitrary functions of special types. Thus one can construct a solution of Navier's equations, containing srbitrary harmonic functions that enter in particular combinations with certain known functions. The choice of known functions and the form of solution are determined, in part, by the differential equations and, in part, by the topology of the region. Another" general solution" of Navier's equations can be constructed with the aid of the biharmonic functions, and there is no a priori reason why one form of general solution should be readily transformable into another. The criterion of the generality of a given form of solution lies in the possibility of determining the arbitrary functions 80 that the boundary conditions are fulfilled. Thus, in dealing with the two-dimensional elastostatic problems in simply connected domains, the general solution of the homogeneous Navier's equations was obtained in the form 1 (00.1) where lP(z) and 1/I(z) are single-valued analytic functions. This solutioll is general in the sense that the unknown functions II' and", can be determined, essentially uniquely, when suitable boundary conditions are Unposed. If one relaxes restrictions on the connectivity of the region, or on the behavior of displacements on the boundary, the representation (OO.I) may cease to be valid. An equivalent form of the general solution involving four arbitrary plane harmonic functions can be deduced from (00.1) by setting, 1 See.

71. 328

TBllE1!l-DDlENSIONAL PROBLEMS

t "" " ,

(OO.5~

so that (90.3) can be written in the form V2(j.lU;

+ w,,) = o.

Hence (90.6)

j.lU;

+ >t"

= '11"

where 4>, is an arbitrary harmonic vector. j.lU;,;

It follows from (90.6) that

+ V'>t = 4>",

and, on noting (90.5), we get _ A + j.I '" V2.T, ,.. - A + 2j.1'*"'"

(90.7)

A particular integral of this equation is ~ : : ;j.I x,.4>" and hence' the ~en­ eral solution can be written in 'the form (90.8)

where 4>0 is an arbitrary harmonic function, Referring to (90,6), we see that the displacement vector U; can be represented in the form (90.9)

j.lU;

= 4>, - 4>0,; -

+21'I' (X,.4>,.)", '12 >..>.. +

involving four arbitrary harmonic functions, This formula can be cast in the form whose structure i~ identical with the representation (90.2) of the displacements in plane elasticity. On carrying out the indicated differentiation in (90.9) and simplifying, we find, p.U;

>.. + 3j.1 = 2(>" + 2j.1) 4>, -

1 >.. + JI. A + 2j.1 xlIIi,' - 4>0.,.

'2

But >.. + 3j.1 3 - 4a 2(>" + 2j.1) = 4(1 - a)'

>.. +2j.1 - - = 2(1 - 0')

>"+,u

and hence

3 - 4a 1 ,uU; "" 4(1 _ 0') 4>, - 4(1 _ 0') Xlpi,' 1

Note that V'(z,..) - 24>"" since 4>, is harmonic.

+G.l,

THB.l!lE-DIMENSIONAL PROBLEMS

331

and .if we define, tp; "" 4\/[2(1 - u)], and recall from (71.8) that x = 3 - 40-, we get

(90.10)

21lu;

=

X"" -

"'0.'.

Xj0

+ X;if>i,

(i = 1, 2, 3),

where the 4>'s are harmonic. 1 Any two of the functions 0 and

' + II)

(92.5)

2"D

+7

=

o.

The solution of (92.5) and (92.4) yields P D = - 41r(>' + II)' 41r" and the substitution of these values in (92.3) gives,

C=~,

P

(92.6)

I

Xa:I:..

7

'U ..

= 41r"

'Us

= ~~ 41r1' r'

P

- 41r(>' +

x.

p)

rex. + r)'

(O!

= 1,2),

+ P(>. + 21') !. 41rp(>. + II) r

It is worth noting ' that at a great distance from the origin the displacements vanish as 1/r, and hence the stresses vanish as l/r2. In this connection it should also be observed that the concept of the concentrated load is a mathematical abstraction resulting from specific assumptions concerning the behavior of continuous distributiOlis of loads when a definite limiting process is followed. It is not surprising, therefore, that different limiting processes might yield singular solutions different from (92.6). A decision about the practical validity of any given singqlar solution should rest on physical rather than mathematical grounds. The definition of the concentrated load in the instance of curved surfaces obviously involves an even greater degree of arbitrariness. Because of the usefulness of the solution in the,form (92.6) it is natural to use it as a criterion for an acceptable definition of the concentrated load acting on a curved surface. s The solutions (92.6) can be generalized, in an obvious way, to yield the displacements produced in an infinite region Xa ~ 0 by suitably restricted continuous distributions of normal loads. 1 See remarks in See. 74 regarding the behavior of displacements and streMea in the two-dimensional ease and their bearing on the uniqueness of solution. • See in this connection: E. Sternberg and F. Rosenthal, "The Elaatic Sphere under Concentrated Loade," Juumal t1/ Applied Mechanics, vol. 19, No.4 (1952), pp. 413-421. E. Sternberg and R. A. Eubanks, "On the Singularity at a Concentrated Load Applied to a Curved Surface," A Technical Report to ONR, Department of Mechaniea, mina JDstitute of Technology (1953). A. Huber, "The Elaatic Sphere under Concentrated Torques," QuarlerJw of A~ lIt111wtNJtica, vol. 13 (1965), P.iI. IllH02.

341

THREE-DIMENSIONAL PROBLEMS

p(~, 'II) be

If we let

the distributed normal load acting at the point

eE, 'II) of the xtxrplane, the resultant force P on an element of area du is P = peE, 'II) duo Inserting this in (92.6) and integrating over the XlXr plane, we get,

I

'U a

(92.7)

(7 p(~, 'II)rl d~ dTJ _

_

x,xa

-

4...", ))

Xa

4".(X

(7 pet, 'II)+dt dfl,

+ "'»))

-~

r(r

x.)

-~

i[

r

+

where r2 = (Xl - t)' (X. - 'II)' + X:. The evaluation of the double integrals in (92.7) presents serious computational difficulties except in those cases where simplifying assumptions are made about the nature of the load pet, '1) and the shape of the regiol). over which the load is distributed. If the load is axially symmetric about the x.-axis, it is possible to deduce tractable expressions for the displacements by the method of Hankel transforms. l A solution of the problem of deformation of the semi-infinite elastic half space by the concentrated force acting in the interior of the solid was given by Mindlin.' Mindlin's solution specializes to that of Boussinesq when the force is assumed to act on the boundary of the solid. 93. The Problem of Boussinesq. As an illustration of the use of general integrals of N avier's equations (93.1)

",V'Uj

+ (X + ",).,J.; = 0,

we construct a solution of the second boundary-value problem for the semi-infinite region x. > 0 bounded by the plane x. = 0. 3 'I. M. Sneddon, Fourier Transforms (1951), pp. 468-486. This book contains a treatment of several problems concerned with the deformfltion of semi-infinite elastic media. ' The torsion of an elastic half-space by shearing forces distributed over a circular area was considered by N. A. Rostovoev, Prikl. Mat. M ekh., Akademiya N auk SSSR, vol. 19 (1955), pp. 55-60. I R. D. Mindlin, Compte. rendUB hebdomadair.. de8 seances de I' acadhnie de8 8ciences, Paris, vol. 201 (1935), pp. 536-537; Ph1l8ie8, vol. 7 (1936), pp. 195-202. An exposi~on of Mindlin's work is contained in H. M. Westergaard's monograph Theory of Elasticity and Plasticity (1952), pp. 142-148. • The first, second, and certain types of mixed boundary-value problems of elasticity for the semi-infinite region bounded by a plane are associated with the names of J. BoU8llinesq and V. Cerruti. These authors solved a number of special problems with the aid of potential theory. A resume of earlier work is contained in Chap. 10 of !.eve'. Treatise. Love [philHophicBl Tr0nsacti0n8 0/ the RolJal Societll (London) (A), VIIL 228 (1929), p. 377J applied the Bouasineeq method to study the defonnation of the eemi-infinite spaee by pressures di8tnouted over a circle and • rectangle. An account of recent developments in related problems utilizing the Fourier and related transforms, is contained in I. N. Sneddon'. book Fourier Transforms (1951), pp. 450--

342

MATBJlMATlCAL THEORY OF ELASTICITY

Since the displacements resented in the form

14J

(93.2)

are biharmonic functions, they can be rep-

UJ ... 'Pi

+ XI"'.i,

where the 'Pj and " are harmonic functions. These functions, as noted in Sec. 90, are not independent since the 14 satisfy Navier's equations. Indeed, on substituting (93.2) in (93.1) we easily find that

[(). + 31')"'•• + (). + p.)'PJo.k].i = o. Hence, on disregarding the nonessential constant, we get (93.3)

).+1'

>/t•• = -

).

+ 31' 'Pl.l.

The functions 'Pi and", must be chosen so that, on the boundary x, = 0, the displacements 14j assume specified values/i(xl, x.) and vanish at infinity in a suitable manner. Setting XI = 0 in (93.2), we see that the harmonic functions 'Pi are required to satisfy the boundary conditions, (93.4) The determination of the 'Pi has thus been reduced to the familiar problem in potential theory, and there are several methods available for constructing these functions. Perhaps the simplest of these is a method based on the Fourier integral representation of harmonic functions. If we suppose that (93.5)

'Pi(Xl,

X2,

XI)

=

.. II gi( -.

01,

fJ)e 7 • m (

343

The substitution from (93.7) in (93.5) thel1 yields the desired functions 0 is axially symmetric, the problem can be treated more effectively by the integral equations and Hankel transform methods. Problems of the indentation of a semi-infinite space by a rigid punch of circular and elliptical cross sections are in this category,' The method of Fourier integrals can also be applied to solve the corresponding first boundary-value problem, but since such calculations present no points of novelty, we do not include them here.' The possibility of reducing the elastostatic problem for the semi-infinite space to the simpler problem in potential theory hinged on the special form (93.2) of the general solution of Eqs. (93.1). In applying this method to problems involving spherical boundaries, it is natural to take solutions in the form U; = h"aik (1928), voL 6.

344

MATHEMATICAL THEORY OF ELASTICITY

mentary mea,ns. Such is the problem of deformation of a spherical shell by uniform internal and external prt:ssures. Let the internal and external radii of the shell be a, and at, respectively, and let the interior pressure be PI and the exterior pressure p •. We take the center of the shell to be at the origin and consider the system (94.1) (i = 1,2,3), P.V'Ui + ()I. + p.)"., = 0, with appropriate boundary conditions. Since the deformation of the shell is symmetric with respect to the origin, we take displacements in the form .

Ui = ",(r)x"

(94.2)

where r = XiX, and '" depends only on r. in (94.1) yields the equation 2

d'", +! d", dr'rdr

=

The substitution from (94.2)

0 '

whose general solution is (94.3)

The stress-strain relations 'T'i

=

)l.uuo'i

+ I'(Ui.i + 1.4,;),

upon using (94.2), give (94.4)

where iJ = 3A 1• given by

'Tij

= Mo,;

+ 2p. ["'0;; + ~ "" (r)xiX'].

The stress Tr in the radial direction

P,

= xilr is

and, on inserting in this formula from (94.4) and (94.3), we find (94.5)

Tr = (3)1.

+ 2p.)AI -

4p.A.

--,:s'

Also, the stress T, acting on the planar element with the unit normal to p; can be easily computed from

n; orthogonal

T, = 'T,j1t.1I;, = (M

But n;n; = 1, and noX, = 0, (94.6)

+ 2p.",)n.-n. + 2p.r ",'(r)(n;x;)'. 80

that

T, .., AD ==

+ 2!i 0 for To > 0, the hoop stress is a monotone increasing function, 80 that the largest value of T, is on the outer surface of the shell. The extreme value of T. occurs when 1 r' = 3aiai/(ai + ala. + a:). 101. Two-dimensional Thermoelastic Problems. The state of stress induced in a long cylindrical body by the distribution of temperature, which does not vary along the length of the cylinder, can be determined by solving the familiar problem in plane elasticity. For if we take the xraxis along the length of the cylinder and assume that the temperature T(X1, x.) is independent of the x.-coordinate, then the stress components T,.; will not depend on x,. These stress components can be balanced by the application of suitable longitudinal forces and bending couples applied to the ends of the cylinder, so that its cross sections remain plane. If, now, the solution of the plane-deformation problem is superimposed on the solution of the simple problems of tension and pure bending, the result will represent a valid solution of the original problem not too near the ends of the cylinder. We thus need consider only the plane-deformation problem wherein Ua

= 0,

(a = 1,2).

Since e'j = e13 = en = ea.

Yz (14,j + Uj,') ,

(i, j = 1, 2, 3),

= 0, and we see from (99.5) that' 7'13 = 7'23 = 0, 7'.~ = Xu",o.~

(101.1) {

1'33

+ Il(U.,~ + u#,.)

- kT0a#,

= Xu", - kT.

From the second of these equations we find that 7'••

U"r

+ 2kT + Il) ,

= 2(A

so that (101.2)

T ..

=

X 2(X

+ Il)

kll 7'0. -

X

+ Il T.

Thus, the longitUdinal stress component 1'33 is completely determined by To. = 1'11 + 7'22 and T, For the determinatiolJ of the 7'.~ we have the equilibrium equations, (101.3)

7' a#,~

= 0,

which are identically satisfied if we take (101.4) 7'11 = W. n , 7"2 = - W,n, Tt! = W,n, where W(X1, x.) is the stress function. 1 A discussion of numerical results of engineering interest and further references to such results are contained in Chap. 14 of Timoshenko and Goodier's Elasticity. • Hereafter we denote the constant {J ~ (3).. + 2p)'" introduced in Sec, 99, by k to avoid possible oonfllsion with Greek indices having the values 1 and 2.

365

THREE-DIMENSIONAL PROBLEMS

On recalling. the compatibility equation

+ e".ll

ell."

2e U •lt

=

and making use of (9904) and (lOlA), we find that W satisfies the equation (101.5)

V"V'W

+ cV'T =

0,

where I C

21'k = ---. A 21'

+

If we set

w=

(101.6)

U- V,

where V is a solution of the Poisson equation (101.7)

V'V

=

cT,

we find from (101.5) that U is biharmonic, so that (101.8)

V'U

o.

=

Thus the problem can be phrased entirely in terms of the biharmonic function U and some particular integral' of (101.7). We can take such an integral in the form (101.9)

V(XI, x.) =

i; III T(x~, x.) log r ax;,~,

where r' = (XI - XI)' + (x, - x~)' and R is the cross section of the cylinder. When the tractions T a are specified on the boundary C of the cross section R, we have (10LlO)

and since, from (lOlA) and (101.6), 1'tl

=

U. ll

-

V. u ,

1'11

= U .•2

V ...,

-

1'12

=

V. lS

-

U. lI ,

we find that l (101.11)

dU.• _ T ( ) (Ii' 18

+ Ts' dV. t

dU. I = -T.(8)

ds

+ dV. ds

1,

where 8 is the arc parameter measured. along C. 'In terms of E, .., and the coefticieat of Jinear expansion a, C - Eat(l - ..). The harmuic function entering in the pneraisoiution of (101.7) can be absorbed in the general ~ntatioll of W inasmueh .. the aenerai solution (70.4) of the biharmonic equation contains an arbitrary harmonic function. • Ct. (69.8). I

866

MATHEMATICAL THEoRY OF ELASTICITY

Accordingly, the boundary conditiOIlll for the biharmonic function U can be written in the form (101.12)

U,l

+ iU"

=

i/.'..

"" 11(8)

(T,

+ iT,) lis + V,l + iV,s + coIlllt

+ il,(s) + const

on C.

This is precisely the boundary-value problem we have considered in detail in Chap. 5. When the flow of heat is steady, V'T

=

0,

and it follows from (101.5) that W is biharmonic. In this ease we can take V ... 0, and we conclude from the foregoing that the state 01 stress induced in a cylinder by the steady heat flow i8 identical with that present in the same cylinder at constant temperature (that is, with T = 0) under the same sur/ace loading. Reference here is made only to the stress components 'T'-IJ (a, 13 = 1, 2). The stress 'T'33 necessary to maintain the state of plane

deformation is given by the formula (101.2). The straiIlll and, of course, the displacements do depend on T(Xl, x.), and they can be computed from (101.1) once the 'T'a~ are determined. As a corollary to the italicized statement just above, we can state that, when the cross section of the cylinder is simply connected and the cylinder is free of external loads, the steady heat flow produces no stresses r-IJ. These remarkable results, pertaining to the steady heat flow in cylinders, were established by Muskhelishvili,l who was also respoIlllible for an interesting physical interpretation of the discontinuous, or multiplevalued, displacements that arise in the study of the thermoelastic problems in multiply connected domaiIlll. A comprehensive treatment of the two-dimensional thermoelastic problems, based on methods developed in Chap. 5, is contained in a dissertation "Thermal Stresses in Long Cylindrical Bodies," University of Wisconsin (1939), by Gatewood.' As an illustration Gatewood considers the deformation of a composite circular cylinder with a concentric circular core when the temperature T(r) is a function of the radius r. He also solves the problem for a composite circular cylinder with an eccentric circular core when the temperature T is cOIllltant. It is easy to show that, 1 N. I. Muskhelishvili, Bulletin oj 1M Electrotechnical Imtitule, Petrograd, vol. 13 (1916), pp. 23--37; Atti tW.l4 occademia nazionale tki Lincti, Rendictmti, ClaM. di 8cieMe ji3iche, matemaliehe • natural.., Rom.. aero 5, vol. 31 (1922), pp. 548-551. A detailed discussion is also _tained in Muskheliahvili'. monograph Some Ba.aie Problema of the Mathematiea.l Theory of EJaatieity (1963), pp. 157-165. • See also B. E. Gatewood, p~ M~, aer. 7, vol. 32 (1941), pp 282 301.

THREE-lHMENSIONAL PROBLEMS

367

when the cylinder is of radius a and the temperature T(r) is a function of the radius alone, then 1 T....

== .; a

roe

=

10f" rT(r) dr - ~r 10[' T(r) dr,

~ 10fa rT(r) dr + -;. t r 10

a rre = O.

rT(r) dr,

Thermal stresses in a circular ring when the temperature T is a function of both the radius and polar angle were calculated by Lebedev. 2 The contents of this section can be modified in obvious ways to apply to the two-dimensional problems involving thin plates. The transverse deflections of thin elastic plates, under fairly general distributions of temperature, have been considered by 3 Galerkin, Nadai, Marguerre, Sokolnikoff and Sokolnikoff, and Pell. 102. Vibration of Elastic Solids. We have formulated the basic dynamical problems of elasticity and discussed the existence and uniqueness of their solutions in Chap. 3. Analytical difficulties attending the determination of explicit solutions of such problems are so great that the available explicit solutions are concerned with special types of vibration in spheres anJ. cylindrical rods, and with a few types of propagation of elastic waves in unbounded media. In this section we indicate one mode of attack on the problem of vibration of bounded elastic media, and in the remaining sections of this chapter we discuss some important aspects of wave propagation in the infinite and semi-infinite solids. In the study of free small vibrations of coupled dynamical systems with a finite number of degrees of freedom, it is shown that the most general motion about the equilibrium configuration is compounded of a finite number of certain special modes of vibration, known as the normal modes. The number of such modes is equal to the number of degrees of freedom. Each particle in the system executing a given mode moves with simple harmonic motion, the period and the phase of which are the 1 This problem, and the corresponding problem for the hollow cylinder, can also be BOlved hy an elementary method of Sec. 100. See also Timoshenko and Goodier's Theory of Elastieity (1951), pp. 408-416. • N. Lebedev, Prikl. Mat. MeM., Akademiya Nauk 8SSR, vol. 3 (1936), pp. 76-84. • B. G. Galerkin, Ingenieurbaulen und Baumeclw.nik, Leningrad (1924), pp. 131-148. A. Nadai, Elastische Platten (1925), pp. 264-268. K. Ma.rguerre, ZeitBchriJtfiJ.r angewandle MafJJematik und Mechanik, vol. 15 (1935), PI>. 369-372; Ingenitur Arcli.U1, vol. 8 (1937), pp. 216-228. I. 8. Sokolnikolf and E. 8. Sokolnikoff, TramactionB of tk American Matliem4tical Societg, vol. 45 (J939), pp. 235-255. W. H. Pell, Quarterly oj Applied MafJJematica, vol. 4 (1946), pp. 27-44.

368

MATHEMATICAL THEORY OF ELASTICITY

same for each particle. Thus, the general motion of the system of n degrees of freedom can be represented by a linear combination of n simple harmonic motions with n distinct frequencies. These frequencies are determined by solving the secular equation which is completely determined by the quadratic forms representing kinetic and potential energies of the system.' When the system is continuous, the corresponding secular equation has infinitely many real roots and hence infinitely many characteristic Juneti(}1l.8 representing normal modes of vibration. These characteristic functions are solutions of the differential equations of motion with appropriate boundary conditions. Thus, in dealing with small free vibrations of an elastic solid, the characteristic functions, satisfy the equations in r,

(102.1)

and the homogeneous boundary conditions (102.2)

Ti/Pj

=

0

for all t,

on 2:,

that correspond to the absence of external surface forces. Since normal oscillations are simple harmonic, it is natural to Beek particular solutions of this system in the form (102.3)

Ui(X, t)

=

U:(Xl, X" x.) cos (wt

+ E).

On inserting this in (102.1), we find that the functions u: satisfy the equations (102.4)

with ~' = uti' The solution of Eqs. (102.4) satisfying the boundary conditions (102.2) is known to exist only for a denumerable set Wk (k = 1, 2, . . . ) of values of 101, all of which are real. Thus, the characteristic functions are (102.5) ulk)(x, t) = u: Ck ) cos (Wkt + e), (k = 1,2, . . .), where the u:(t, are the appropriate solutions of (102.4). One then concludes that every oscillation of the body can be represented in the series

.

1);;

=

L Akul

t

),

k-I

where the Ak are suitable constants, whenever the ulk ) are orthogonal with respect to the region under consideration. 2 1 See, for example, E. T. Whittaker, Analytiea.I Dynamics, Chap. 7, or H. Goldetein, Quaieal Mecha1!ics, Chap. 10. • The mode of 8Olution described here is precisely that UIIed in aoIviq the probllml of lIQl&Il traoaYer8e 'Vibrations of an elastic string by the Fourier method. The A. are determined from condition. ebaracterisinc the initial diaturbance.

369

THREE-DIMENSIONAL PROBLEMS

It is tolerably clear that the determination of charaeteristic functions, even for such simple regions as spheres and cylinders, is accompanied by very laborious computations. As a simple example we consider the determination of these functions for the problem of oscillation of a sphere, every particle of which executes a vibration in the radial direction. We take

(102.6) where r' =

u, = zd(r) cos (wt X;Z;.

+ .)

Then

u: = zd(r).

(102.7)

On substitution from (102.7) in (102.4), we find that the function fer) is required to satisfy the equation,

f"

(102.8)

+ ~ f' + kif = 0,

where

k2=~.

(102.9)

X

+ 21'

The solution of (102.8), which does not become infinite for r = 0, is

f =

(102.10)

~ (kr cos kr

- sin kr),

where C is an arbitrary constant. The component of displacement u. in the radial direction is.

z,

u.=u,r = rf(r)

where we have recalled (102.6). direction is

But from (102.6)

u,.•

= (31

cos (",t

+ .),

The component r .. of stress in the radial

+ rj') cos (",t + .),

so that (102.11)

r" = [(3),

+ 2p)f + (X + 2p)rf'J cos (wt + .).

Since the condition (102.2), in our case, is r .. = 0 for r = a, where a is the radius of the sphere, we firid, on setting r = a and on using (102.10) in (102.11), that

(). + ~)((2

- k'al ) sin ka - 2ka cos

kaJ + 2A(ka cos ka

- sin ka) = O.

This is our bequeaey equation in which k is related to '" by (102.9). The

370

MATHEMATICAL

Ta~RY

OF EL.\STlClTY

values of the six lowest roote of thi8 transcendental equation for ~ = p" which corresponds to IT = ~, are recorded' in Sec. 196, p. 285, of Love's Treatise. It ill clear from this simple example that the determiuation of characteristic functions in the more general case ill extremely difficult.' 103. Wave Propagation in Infinite Regions. If a region ill so large that the effects of the boundaries can be dillregarded, it ill possible to represent the disturbance as a sum of two waves propagated with velocities that depend only on the density and elastic constante of the medium. Indeed, the displacement vector u can be represented as a sum of two vectors, one of which is solenoidal and the other irrotational. This leads to a consideration of two special types of disturbance for one of which div u = 0 and for the other curl u = O. N ow if div u = 0, u;" = " = 0 and the general equations (103.1)

,.~'u;

+ (X + ,.)"., =

pu.,

reduce to (103.2)

These are familiar wave equations of the form 02U;

Tt2

(103.3)

=

ctv2u"

where the velocity c of propagated waves ill

c=

(103.4)

~

This is the case of an equivoluminal wave propagation, since div u = 0 for waves moving with this velocity. On the other hand, when the motion is irrotational, curl u = 0 and the vector identity curl curl u = V div u - V'u, enables one to write (103.1) in the form (103.5)

(X

+ 2,.)V2u, =

,nl;.

These equations show that the velocity of propagation of irrotational waves is (103.6) 1 These l'eIIIlltB are due to H. Lamb, Pf'tlClll!diR{13 of tIte Ltnulon M~ ~Y. vol. 13 (1882), but the problem of radial VIOlations of the solid sphere was first diacuued by PoiaIIon in 1828. A lIOlution of the correaponding problem for the hollow sphere is given in Sec. 198 of Love's T.reat.iae. • The reader mq lind it iDatmctive to - u a JIIIODfI&I'aPh by H. .KoWr:7. svWaves in Solids (19$3), which _taiaa a IIWDIBIII3' of _ t eontriblltieD11 flo problema of longitudiDal, s..u.oimJal. and 8enraI 'fibration of eylindera.

THREE-DIMENSIONAL PROBLEMS

371

We thWl IlOO that the disturbance in an infinite medium can be described with the aid of two special types of waves; one of these travels with the equWoluminal velocity Cl = Vii/P, and the other with the irrotational lIelociJ;g c, = V(~ 2p)!p. Clearly, c, > Ct. When the initial disturbance is symmetric about the origin, the displacement is a function of rand t only and Eq. (103.3) can be written as

+

a!(ru)

---;)iI =

C'

as

ars (rIu).

On recalling the D'Alembert solution of the wave equation, we have 11

1

= j:i F(r - ct)

+ J.i1 G(r + ct),

where F and G are arbitrary functions. This represents two trains of spherical waves, one diverging from the source r = 0 with the velocity C and the other moving toward the source with ihe same velocity. At a great distance from the source, spherical waves become nearly plane. This suggests that in an infinite medium plane waves can travel with one or the other of the velocities found above. A direct verification of this is simple. When the waves are plane, (103.7)

ct),

tIi = Fi(X;,,; -

where the Fi are arbitrary functions and the Vi are the direction cosines of the normal to the plane of the wave. If we insert (103.7) in (103.1) and eliminate the Fi from the resulting equations, we get the equation for c in the form (pc' - p)(pe' - ~ - 2p) = 0, which shows that plane waves travel with the equivoluminal and irrotational velocities. PROBLEMS 1. ShOW that. when the displacement vector u is written in the form u = v ..

+ curl ,,",

then Eqs. (103.1) are satisfied if

II'"

/It' -

C:VS...

Ct '"'

.Ji+ 2,. "V~--. p

and Cl-=

~. p

I. Referring to Prob. 1, IIhow that a clua of particular solutions of (103.1) oan be pmerated by taking

.. _ 04......+4"1. ",_~_-..oI,

when Va - 0 aad a, aad a. are independent of :.:..

372

MATHEMATICAL THEORY OF ELASTICITY

8. Referring to Prob. 1, show that when the gradient and curl are exprellll!d in cylindrical coordinates (r, '. #), then. in axially symmetric problems,

u, - 0, Md

a",

(i'",

I

iJr

r iJr

a'", 1 a'", at' =--, cl at' at,y lat,y

~+--+t

at,y

lay,

I

-+- --of+= --. ar r or rt az c~ ott J

l

Deduce two sets of particular solutions of these equations by taking of - G(r}.' 0, b > O.

Bcce-h::-HC••l-"l),

The period of the waves in (104.8) is 2.../p, and the wave length is 2.../8; hence the velocity of propagation is (104.9)

=

C.

1!.. 8

To determine the constants in (104.8), we insert from (104.8) in (104.4) and (104.5) and find that at { b2 -

(104.10)

S2

8

2

+ k2

=

+ h' =

0, 0,

where

k'

(104.11)

Further, since

Va

= PP', I'

and w. represent equivoluminal and irrotational waves, W •• ~ = W~ •••

Hence and we can take the vectors in (104.3) to be

(104.12)

v, = Aac"'" sin (8X, v. = A8e--' cos (axl WI = Bse--l>z· sin (ax, w. = Bbe--l>z· cos (ax, -

pt), pt), pt), pt),

where A and B are real constants. Further restrictions on the choice of constantsjn (104.12) are provided by the boundary conditions (104.2). Since ", = 0, II. = -1, Eqs.

cr.

1 Prob. 2, Bee. 103. In accordance with the wrual practice, the desired vectors v. and w. are determined. by the real parta of expreesion.a in (104.8).

374

MATHEMATICAL 'l'HllIOBY OJ' ELASTICITY

(104.2) demand that

1'22

1'lt

=;

x, = 0 and thus, from .Hooke's

= 0 on

law, (104.13)

+ 2"ul,l = + = 0,

{ Xu..... U1.2

0,

U2.1

for x.

~

O.

The substitution from (104.12) in these equations then yields two linear equations 2"a8A + [2"b' + X(b 2 - 8')]B = 0, { (a' + 8!)A + 28bB = 0,

(104.14)

which have nonvarushing solutions for A and B if, and only if, [X(b 2

(104.15)

8')

-

+ 2I'b'](a' + 8')

- 41''tWs'

= O.

On eliminating a and b from (104.15) with the aid of (104.10), we deduce the equ~tion,

(2 - 82k,)4

(104.16)

(1 - 82h 2

= 16

)

(

1 -

k')

82 .

But

~ =~, 8'

C{

as follows from (104.11), (104.9), (104.7), and (104.6). Hence Eq. (104.16) can be cast in the form, (2 - ",)4 = 16(1 - ",,,,)(1 - "'),

(104.17)

where (104.18)

'"

cf ci

== c' ..!, cf

I'

x=-=--'

X + 2"

On simplifying we get, ",' - 8",'

(104.19)

If we take X = 1

1',

+ (24

then x =

- 16",)", - 16(1 - x) = O.

~

and (104.19) becomes

3",1 - 24",'

+ 56", -

or ('" - 4)(3",' - 12w

32 = 0,

+ 8)

= O.

The roots of this equation are "'1

= 4,

"" =

2 + 2/0,

"'3

= 2 -

2/0.

It is easy to check that the first two of these do not give positive values Thus the only suitable root is '" = 2 - 2 Vi, and hence, from (104.18),

to a and b in (104.10).

Ca

=

Cl

V;;;

= 0.9194

~~.

375

'J.'IHtEE-DI1IENSIONAL PROBL1DI8

In the limiting case of incompressible body (" Eq. (104.19) assumes the form (oil -

&>2

+ 24w -

~

0,

(f

== ~, x = 0),

16 = 0,

and one finds, as above, that the velocity of the surface wave is

c.

= 0.9553

~.

In either case, C3 is slightly less than the velocity of the equivoluminal wave. Having determined C" one can compute a and b with the aid of Eqs. (104.10) in terms of s and write out the corresponding expressions for the ua • The rate of attenuation with depth depends on a and b, and it is easy to see that the waves of higher frequency are attenuated more rapidly than those of low frequency. Since Ca is independent of wavelength, there is no dispersion. Waves roughly similar in appearance to Rayleigh's waves are often recorded by seismographs. However, seismographic records of distant earthquakes indicate a dispersion, which is to be expected since the earth is not a homogeneous medium. If we consider forced vibrations of the semi-infinite solid by normal forces P cos (Bl:l - pt) distributed along the xa-axis, we must replace the boundary conditions (104.13) by 'Tn = 'T12

}..u....,

= !'(U1,2

+ 2!'u2,2

+ U2.l)

= =

P cos

pt),

(SXl -

o.

Equations corresponding to (104.14) in this case are 2,.asA

+ [2~! + }..(b2 - s2)]B = (a! + s2)A + 2sbB =

P, O.

These can be solved for A and B. The corresponding expressions for u.., with x. = 0, give the displacements on the free surface of the solid under the action of forces P cos (Bl: - pt) distributed along the xa-axis. These can be generalized l in a familiar way with the aid of Fourier integrals to provide formulas for the displacements on the free surface due to the forced vibrations of a more general sort. The analysis of resulting formulas shows that the steady-atate disturbance of tpe free surface consists of three waves one of which moves with velocity Cl, the other with velocity C., and the third with the Rayleigh velocity Ca. A similar analysis has been carried out by Lamb for the semi-infinite 1 H. Lamb, PhiloaophictJl T~ of the Royal 8ociet1l (London) (A), vol. 203

01 the Royal8oci.t1l (I..ondon) (A), vol. 93 (1917), p. 114. See also S. Timoshenko, Philo8ophical MGglWine, vol. 43 (1922), p. 125, and J. H. 1_, ProeNdingB of the Royal8oci.ty (I..ondon) (A), vol. 102 (1923), p. 554. (1904); Proceedings

376

MATHEMATICAL THEORY OF ELASTICITY

half space a portion of whose boundary is subjected to the axially symmetric forced vibrations. This problem w~ treated by a different method by Smirnoff, Soboleff, and Narychkina. 1 1 V. Smirnoff and S. Soboleff, "Sur une methode nouvelle dans Ie probleme plan dee Vibrations 6lastiquee," Trudy, &is~ lnatitute of the Academy of Scimcei of the USSR, No. 24 (1932); No. 29 (1933). S. Soboleff, Matematichuki Sbornik, vol. 40 (1933). E. Narychkina., "Sur lee vibrations d'un demi-espace aux conditions initial"" arbitraires,"·Trudy, 8eisnwlogicallnatitute of the Academ!l of Scienceo of the USSR. No. 45 (1934); No. 90 (1940). See also closely related papers by: Ya. A. Mindlin, Dokladll Akademii Na"" SSSR, vol. 15 (1937), vol. 24 (1939), vol. 26 (1940), vol. 27 (1940), vol. 42 (1944); Prild. Mat. Mekh., Akademiya Nault 88SR, vol. 10 (1946), pp. 229-240.

CHAPTER

7

VARIATIONAL METHODS

1015. Introduction. The determination of the state of stress in the preceding chapters was made to depend on a solution of certain boundary-value problems involving partial differential equations. A different approach, exploiting certain broad minimum principles that characterize the equilibrium states of elastic bodies, is developed in this chapter. This approach is based on the use of direct methods in the calculus of variations, first proposed by Lord Rayleigh and W. Ritz and extended by R. Courant, K. Friedrichs, B. G. Galerkin, L. V. Kantorovich, S. G. Mikhlin, E. Trefftz, and others. We shall see that it is possible to construct certain integrals, related to the work performed on an elastic body in the process of deformation, and to show that these integrals attain their minimum values when the distribution of stress in the body corresponds to the equilibrium state. This in effect reduces the problem of stress determination to certain standard problems in the calculus of variations that can be solved by direct methods. The concluding sections of this chapter contain a brief treatment of several numerical methods of solution of problems in elasticity. Some of these are suggested by the variational techniques but do not explicitly depend on them. 106. Variational Problems and Euler's Equations. We shall be concerned with the calculation of the extreme values of functions defined by certain integrals whose integrands contain one or several functions assuming the roles of arguments. As an example, consider the integral (106.1)

[(y) =

1(II" ... F(x,

y, yf) dx,

where F(x, y, yf) is a known real function F of the real arguments x, y and yf "'" dy/dx. The value of the integral (106.1) depends on the choice of y =0 y(x), hence the notation [(y). We shall use the term functioru'Jl to describe functions defined by integrals whose arguments themselves are functions. To Illake the symbol I(II) meaningful, it is clearly necessary to impose some restrictions on the choice of the argument y(x) and on the pre377

378

MA.'l'HEHATIClAL -TRIiORT 011' JllLASTIClTY

scribed function F appearing in the integrand of (106.1). We shall suppose that the admissible arguments y(x) belong to a class 0' and 8B8Ume at the end points of the interval (Xo, Xl) the specified values yo and Y1. Thus, y(xo) = yo, { y(X1) = Y1,

(106.2)

where yo and Y1 are prescribed in advance. The entire set IY(x)} of admissible arguments Vex) can thus be viewed as a family of smooth curves passing through (xo, Yo) and (Xl, Yl). As regards F(x, y, y'), we shall suppose that it is of class 0 2 for all values of y' in some specified region of the xy-plane containing the curves I y(x) }. For a given curve y = g(x) of the Bet I y(x)}, the integral (106.1) yields a definite numerical value I (ii), and we pose a problem of determining that particular curve Vex) in the competing set which makes the integral (H)6.1) a minimum. If Vex) minimizes this integral, then every function ii(x) in the neighborhood of y(x) can be represented in the fom).

+ E1I(X),

fi = Vex)

(106.3)

where E is a small real parameter.

We shall call the difference,

Vex) - vex) = E'7(X), the variation of Vex) and write, oy

==

E71(X).

We note that the function Vex) is determined by (106.3) with E = o. Moreover, every function in the set {y(x) I satisfies the end conditions (106.2) and thus ,,(xo) = '7(Xl) = O.

(106.4)

Since y(x) minimizes (106.1), I(y)

~

I(y),

+ .,,)

~

I(y).

or (106.5)

I(y

The left-hand member in the inequality (106.5) is a continuously differentiable function of E, and therefore a necessary condition that 1I(x) minimize (106.1) is

I_.0 "".

O.

+ ..,) ... f: F(x, y + .."

1/'

dI(y + ..,) dE

(106.6)

But. l(y

+ ..,') dx,

379

VARIATIVNAL METHODS

and on differentiating under the integral cign we obta.in (106.7)

dI{lI: e7/)

I

e-O

= /.""

('qFv

:h

+ Fv"{) dx = O.

AI!

is customary, the Bubscripts on F indicate partial derivatives of F with respect to the arguments denoted by the SUbscripts. Integrating the second term in the integral (106.7) by parts, we get

%0 /.""

Fv'f/ ' dx = Fv'f/

::to '"''

-

/.z....

f/ _,:_' dx, u;c.; dF

and since f/(xo) = f/(x.) = 0, we can write (106.7) in the form (106.8)

.. , ( dF) "" F. - ~ ,,(x) dx =- O. /.

Thus the integral (106.8) vanishes for every flex) of class C' satisfying the condition (106.4), and we conclude thatl F - dF" = 0

(106.9)

v

dx

'

is a necessary condition that the integral (106.1) be minimized by y = vex). Equation (106.9) is the Euler equatioo associated with the variational problem I(y) = min. On expanding it we get the secondorder ordinary differential equation (106.10)

d'y F v'.' dx 2

,dy + ]i.,. dx + F". -

Fv = 0

for the determination of y(x). We have assumed that the minimizing function is contained in the admissible set Iy(x) \. However, the condition (106.6) is merely a nece8sary condition for the extremum of I(y). Ordinarily, it is important to verify that the solution of the Euler equation indeed satisfies the inequality (106.5) and thus minimizes the integral. If suitable restrictions are placed on the function F(x, y, y'), the appropriate solution of (106.9) will in fact minimize the functional I(y). Thus, if F(x, y, y')

=

p(X)(y')'

+ q(x)y' + 2f(x)y,

(106.10) yields a self-adjoint second-order differential equation (106.11)

:x (w') - qy -

f .. 0,

1 The proof of this lemma, due to Lagrange, is contained in many books. See, for example, R. Courant and D. Hilbert, Methoden der Mathematischen Physik, vol. 1, cr 1. S. Sokolnikoff, Te1l8Or Analysis, pp. 154-155.

MATBEMATICAL TBEORY OJ' ELASTICITY

whose solution satisfying the end conditions 1/(Xo) = 1/0, 1/(Xl) = 1/1 actually minimizes the integral 1(1/) = j",. r'" [P1/'2

(106.12)

whenever! p(x)

> 0 and q(x)

(106.13)

1(1/) =

+ q'/l' + 2/1/1 dx; 11'2

~ -

- - - pm'. in (Xo, Xl).

Xo A functional of the form (106.12) arises in the one-dimensional elastastatic problems concerned with the study of deflection of bars and strings. Similar calculations performed on the functional Xl -

(z, F(x, 1/, '/I', 1/", ••. ,'/I(R» dx

jz.

yield the Euler equation (106.14)

F~ -

d

dxF,;

~ + dx2F,;'

-

...

• +- (-I)" dxRF~R)

= 0,

when certain obvious restrictions on the continuity and differentiability of F and '/I(x) are imposed. We consider next the problem of minimizing the double integral (lO6.15)

leu) =

g

F(x, '/I, u, u z ,

~)

dx d'/l

on the set lu(x, 1/) I of functions of class C', where each u(x, '/I) in the set takes on the boundary C of the region R specified continuous values U = . [11 + (6u,).,J[11 + (01tj),i]

+ p[e,; + ~~(IlUi),; + ~2(llui).,J[e;j + Yz(01t'),i + 72(01t;)"j - 2A D'

- p.e,;eij.

Upon expanding this, we get (107.7)

.6.W = M(oU;)"

+ 2",e,i(ou,),j + P,

where (107.8)

P

A == '2l(ou;U2

r + 4:p. \(&Ui),; + (llu;),,1 2 ~ O.

We note that P = 0 only in the trivial case when

oe;;

= 72[(OU;),j

+ (OUj)"J

ADo,;

+ 2",e'h

Since Tij

=

= O.

Eq. (107.7) can be rewritten in the form AW = (MOi; + 2p.e;;)(Oui),j = T;;(Ilu;),; + P.

+P

VARIATIONAL METHODS

Accordingly. the increment l::.U in strain energy U is, (107.9)

+ J. P dT

l::.U = J. l::.W dT = J. T,;(8u;)J dT =

J.

=

J 'E T"V' '11 ou; du J.,. T-""1·'

(T,;

J.

au;).; dT -

r

T'j.;

+Q otl; dT + Q au, dT

,

where (107.10)

Q""

J. P dT ~ O.

I

But if the body is in equilibrium, T'M 'Tij'IJi

= -F,

= T,

in T, on ~,

and, therefore, (107.11)

l::.U =

J:E T, ou, dO" + J. F, ou, dT + Q.

Recalling the definition of potential energy V, we get, l::.V=aU- J:ET,ou,dO"- J.F,ou,dT

and, inserting in this from (107.11), we have finally l::.V = Q. Since Q ~ 0, we have proved the following theorem: THEOREM OF MINIMUM POTENTIAL ENERGY: Of all displacement/! satillfying the given boundary conditiOn/! those which satisfy the equilibrium equatiOnl! make the potential energy an absolute minimum. In applications one is usually concerned with the converse of this th~­ orem. We now prove that the converse theorem is true. Let us suppose that, by some means, we have obtained a set of functions u; + fiu; of class C 3 which satisfy assigned boundary conditions and such that (107.12)

r

l::.V = aU - J % T·o,,·dO" - JF·ou·dT >0 • ti.Q\\.

We assume with Saint-Venant that the only nonvanishing stresses are ,.•• and ,.." and recall' that the displacement components in the cross section are (110.1)

u

='

v

-azy,

=

azx.

In the formula for complementary energy, (108.10]

V* =

u-

J

Z.

Till; d0-

'

the surface integral must be evaluated only over the ends of the cylinder, since the external forces are known over its law.ral surface, We recall that

u= w=

~1"'JlIJ ... (Tue..

Using the stress-strain relations

we find

1. W d,.,

+ T,..e.,,).

401

VAlUATIONAL lOlTIIODS

and hence

i,.l

u=

(T:c

+ T:.) dT.

To compute the surface integral in (108.10), we make use of (110.1) and get, for the end z = 0,

fa T."U; dtr since u = " = 0, for z = O. On the end z = I, we have

fa T."U; dtr

=

V

(T ••U

= 0,

+ T'IIIi) dx dy

ff (-alYT•• + alxT..,) dx dy.

=

E

Thus, (110.2)

V· = ;,.

U+ (T:.

T:.) dx dy - al

[f

(XT.., - YT •• ) dx dy.

In this case the admissible stresses satisfy the equilibrium equation in R,

(110.3)

and the boundary condition (110.4)

T ••

cos (x, v)

+

T ..

on C.

cos (y, v) = 0

Equation (110.3) will clearly be satisfied if we introduce the stress function vex, y), such that

av

(110.5)

T ..

=-p,C; in the set (112.4). These restrictions pertain to a special character of approximation of every admissible function y(x) by functions of the set (112.4). DEJl'INITlON: Let Vex), of class O' in (xo, x,) 8atisfy the end conditions 1/(xo) == 1/0, y(x.) = y.. If, for every E > 0, there exist8 in the family (112.4) a function y:(x) = y .. (x, at, a~, . . . ,a:) such that Iv! - 111 < E and Iv:' - 11'1 < Efor all x in (xo, x.), then the set of junctiofi.s (112.4) i88aid

to be relatively complete. We prove next that when the Bet (112.4) is relatively complete the sequence 19,,1, defined by (112.5), is such that lim I(Y .. ) = I(y*) = m. Indeed, when the set (112.4) is relatively complete, there exists a function 1/:(x, a~, a~, . . . , a::) that approximates arbitrarily closely both the exact solution y·(x) of the problem I(y) = min and the derivative of 1/* (x). That is,

Iy: - y*1