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1962-8

Stress Analysis of an Infinite Cylinder with anIrregular Shaped Cavity Using Complex VariableTechniquesGerald H. LindseyBrigham Young University - Provo

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£, %0<0O £

STRESS ANALYSIS OF AN INFINITE CYLINDER WITH AN IRREGULAR SHAPED CAVITY USING COMPLEX VARIABLE TECHNIQUES

A Thesis Presented to the Faculty of

the Department of Mechanical Engineering Brigham Young University

In Partial Fulfillment of the Requirements for the Degree

Master of Science

byGerald H. Lindsey

August 1962

This thesis, hy Gerald H. Lindsey, is accepted in its present form by the Department of Mechanical Engineering of Brigham Young University as satisfying the thesis requirements for the degree of Master of Science.

TABLE OF CONTENTS

CHAPTER PAGENOMENCLATURE.................................. v

I. INTRODUCTION .................................. 1Statement of the Problem............... 1Importance of the S t u d y .................... 1Scope of Application........... 2Limitations of the Solution................ 4

II. REVIEW OF THE LITERATURE..................... 5III. METHOD OF APPROACH........................... 3

Plane Strain Assumption................. 8Boundary Condition Difficulty .............. 9Complex Variable Approach .................. 10Completing the Solution.................... 10

IV. FORMULATION OF THE PROBLEM................... 12Elasticity Field Equations . . . . . . . . . . 12Equations of M o t i o n .................... .. . 12Stress Strain Equations ..................... 12Strain Displacement Equations ........ . . . 13Strain Compatibility ........................ 13Scope of Application........................ 14General Considerations in Defining theP r o b l e m .................................. 15

Plane Strain Assumption .................... 17

IllCHAPTER PAGE

Stress Function .......... . . . . . . . . . 19D*Alembert*s Principle . .................... 22Conversion of Body Forces to Boundary

Stresses . . . ............ . . . . . . . . 25V. THE PROBLEM IN THE COMPLEX PLANE...............' 30

The Stress Function in Terms of ComplexVariables . . . . . ............ . . . . . 30

Stresses in Terms of Complex Functions . . . . 36Displacements In Terms of Complex Functions. . 40Surface Tractions in Terms of ComplexFunctions . .............. 45

Moments in Terms of Complex Functions . . . . 47VI. THE PROBLEM IN THE TRANSFORMED PLANE........... 51

Arbitrariness of F and X ................... 51Dependence of the State of Stress on theElastic Constants ........................ 54

VII. GENERAL CURVILINEAR COORDINATES.............. 57Coordinate Transformation .................. 57Speoific Applications...................... 62

VIII. THE PROBLEM IN THE TRANSP0RM2D PLANE........... 63Boundary Conditions ........................ 63Transformation of the Complex Functions . . . 67Series Solutions for the Complex Functions . . 70

CHAPTERlv

PAGEIX. SUMMARY AND CONCLUSIONS...................... 74

BIBLIOGRAPHY.......... 77APPENDIX A. THE DYNAMICS PROBLEM .................. 81APPENDIX B. CONJUGATE FUNCTION NOTATION ............ 36

NOMENCLATURE

Hoags.

0 .................

E ..............G ..............

S . . . . . . . .

P q P Q P1 Q . . r © e . . . . . . s ...................t ..............U V w ...........u u u . . . .r e sX Y 2 ..........1 ? .....................X y Z ...........W Z . ..........m mw z .................

General Series Constant Constant Young’s Modulus Shear Modulus =*Gravitational AccelerationUnit Vectors • Cartesian CoordinatesDefined Analytic FunctionCylindrical CoordinatesContourTimeDisplacements in the X, Y, Z directionsDisplacements in the r, ©, z directionsBody Forces per unit volumeBoundary Stress ComponentsCartesian CoordinatesComplex VariablesConjugate Complex Variables

An

1 3 k

Greek

vi

Normal Stress » Cartesian Coordinates Shear Stress - Cartesian Coordinates Normal Stress • Cylindrical Coordinates Shear Stress - Cylindrical Coordinates Normal Strain - Cartesian Coordinates Normal Strain - Cylindrical Coordinates Shearing Strain - Cartesian Coordinates Shearing Strain - Cylindrical Coordinates Poisson*s Ratio Body Force Potential Function lame's Constant E-S

~ (U-iS) 0 -2i) Stress Potential FunctionDel OperatorMass Density

CHAPTER I

INTRODUCTION

The advancement of modern technology has brought with It a demand upon the engineer for more thorough analysis techniques. This has been precipitated by the advent of missiles and space-craft that do not permit regular inspec­tions to determine causes of failure, crack propagation, design weaknesses, etc. Thus, a more complete analytical work must be performed to facilitate a foreknowledge of reaction and performance of a particular design to its environment.

It was the purpose of this study to determine the stress distribution on the orojs-section of an infinite, right-circular oylinder with an irregular shaped cavity, which was subjected to a oonstant acceleration.

flaSa&gg.. &LContained within the wide range of elasticity pro­

blems are several applications of the solution of an infinite cylinder with an irregular cavity. Suoh cavities produoe stress concentrations that are of significant analy­tical importance because of the difficulty in obtaining experimental data from the actual body. Sternberg referred

Problemthe° LSMlSSlSfil

S M Zthe°L .im a x & m .

2to this problem in a recent publication. (4)

Stress concentrations in elastic bodies— -that is looal accumulations of stress— -arise from a variety of causes. The present remarks are confined to concentrations of stress which are either due to geometric disturbances, as exemplified by cavities, holes and notches, or owe their existence to material discontinuities, as illustrated by inelu- sions and reinforcements. A characteristic feature of such problems is that they Involvehighly

,£atur e A „ e.l.t.o...aa.i i S p j p l .g Q tip Oapproximation. This la especially true of three- dimensional stress-coneentratlon problems which are, in addition, not readily accessible by experimental means, at least in cases where the aggravation of the stress-field is produced by an internal cavity.

Thus, a direct analytical approach is required. As a specific example of its Importance in the construction of preBent-day solid propellant rockets, the propellant it­self becomes a component part that must be analyzed in the same manner as any other element of an engineering design. This means the determination of stresses, stress concentra­tions, displacements, failure levels, etc. This thesis is concerned with the calculation of stresses and displacements.

■SS.PJ2£.l.9X,iggli£l§ . aAny problem involving a cavity within a cylindrical

body that is long compared to its diameter or is constrained on the ends such that there is no longitudinal movement can be solved using the solution put forth in tills thesis. The solution employe a three-dimensional approach to a plane

of such problems la that they Involve highlylocalized effects in the form of steep stress gradl*

.ZgEX.nature, Ul-gulted,to,.arj,gradl-

are.entBnumerical methods of

3problem. Once again Sternberg comments on this general formulation of the problem. (4)

Finally, we discuss briefly some available three-dimensional studies of plane stress- concentration problems. The three-dimensional treatment of such problems presents great analytloaJL difficulties, as does the three-dimensional treatment of plane problems In general. In this context, we recall the significance of the two- dimensional solutions associated with the plane problem in elasticity theory. The plane-stress solution ordinarily violates the boundary condi­tions for the lateral boundary and supplies an approximation applicable to relatively thin plates.

thick Plate.Problems of this nature have come to the foreground

in the last fevr years, making solutions of the cylinder problem of interest to analysts in the field of applied mechanics and especially those associated with the missile Industry. During the period in Which a rocket is being fabricated, while it is being transported to a staging or launching site, and during the actual flight, the missile is subjected to environments that could produce large inter­nal stresses in the grain. In the construction of solid propellant grains there, of necessity, oust be slots out into the propellant which are of a symmetrical star shape. The comers of this unusual boundary are sources of stress concentration, which must be carefully analyzed. If the stresses become of sufficient magnitude to produce a failure

She plane-strain solution, on the other hand.

pjjBasL. AaggjL..aRa.. m , M l i l im .M sw jsL ss,.H i a s M . m&§LM, a psm&j&MSM

&ppr 0Xxsy5tx%, oxx valid 1 RU.3.USl.lr violates th e bouadagy conditions the

y i e l d s

In the grain, the propagation of the crack will open additional surface area of the propellant. Burning rate is proportional to exposed area, and if additional area is exposed, uneven burning results and the rocket malfunctions.

An additional facet of the problem is that solid propellants are classified as viscoelastic materials, mean­ing that they possess properties of elastic solids as well as certain properties of vlsoous fluids. However, when the problem is formulated through the application of D’Alembert's principle, the viscoelastic solution can be obtained from the elastic solution by invoking the proper correspondence principle, which relates viscoelastic material properties to elastic properties.

Limitations of the Solutionfhe true dynamics problem, involving time dependent

displacements and stresses, cannot be solved using this method. An approximation to this solution can be had by considering the body at a desired acceleration level, which does not vary with time. At present, methods are not avail­able to solve the vibrations problem because of the difficulty in expressing the field equations in terms of complex variables.

CHAPTER II

REVIEW OP THE LITERATURE

Volumes have been written on the field of elasticity and additional volumes have been written to document the historical development of the subject. However, as in all fields, significant contributions were made by people out­side the area working on other problems. Such was the case with E. Goursat, a geometrician, who before the turn of the century investigated the possibilities of expressing bihar­monic functions In terms of complex, analytic functions. (5) In 1909, M. Kolossoff made direct application of this technique to elasticity, where the biharmonlc equation appears in many of the mathematical relationships. He pub­lished his results in his Doctoral Dissertation at Dorpat in Russia. (10 and 14) Later his work was expanded by one of his students, H. I. Muskhellshvlll, also a Russian, and many of their Ideas are expounded by the latter writer in two books. (10 and 11)

In their published works are numerous solutions to two-dimensional problems of all varieties. It is a very exhaustive treatment of the subject, but it was not trans­lated until 1953. Thus, there are few applications of the method in English speaking journals. Much of the earlier work of these writers was overlooked in other parts of the

6world, so that Independent development of the use of complex variable methods for elasticity by subsequent writers con­tained some duplications. (6)

Methods involving the use of complex variables in the solution of plane elasticity problems appear in most of the textbooks on elasticity. (10, 13» 14, 15) In. these solutions, which normally apply to bodies of simple config­uration, beautiful mathematical derivations result} however, these solutions do not Include body forces. The same situation prevailed in the current publications where no solutions could be found in the engineering and physios Journals that applied the theory to problems Involving body forces. With a method developed by M. Biot, the problem presently under consideration is converted to a form that lends itself to the mathematics. (1)

Russian scientists have made considerable use of the complex variable approach as is indicated in the reviews of the publications in the field of applied mechanics.However, a few papers were found by American authors who used this approach to analyze apparent stress singularities. (8 and 16) In addition, J. S. Brock and associated authors used the same methods of Muskhelishvili to compute stresses around square holes in plates and beams. (2) The contri­butions by the English authors, which parallel some of the Russian work, is thoroughly reviewed by Green and Zerna, and

7they reference the original publications in which the work first appeared*

In summary, a survey of the literature yielded a few articles on the use of the complex variable theory as it Is applied to elasticity problems In general, but a very limited amount of discussion was found concerning problems involving body forces.

CHAPTER III

METHOD OP APPROACH

In elasticity, as In other branches of mathematical physics, It Is often difficult to find an explicit solution of a given three-dimensional problem, but its analog in two dimensions can frequently be solved. The theories of plane strain and generalized plane stress are well-known examples of two-dimensional theories. In these theories, as far as Isotropic bodies are concerned, the blharmonic stress function equation has a central place, and many solutions of this equation have been obtained with the help of real variable analysis. The presentation of two-dimensional elasticity* and the solution of special problems is, however, greatly simplified by the use of complex variable techniques. Moreover, the range of problems which can be solved is greatly extended.

To provide an overall view of the method of approach to the entire problem, a brief outline of each phase of the problem is given here so that the continuity of the solution is better followed. Later in the body of this paper each portion will be developed in detail.

In bodies which are long compared to their crossU msl Mtoa&BJkmwa&m.

9

sectional dimensions or are constrained such that the motion in the longitudinal direction Is small In comparison to the transverse direction, it can he assumed that each cross section Is subjected to the same stress environment. Thus, only a thin slice need he analyzed, where all strains and displacements are confined to this plane. Problems of this type can be handled nioely using the principles of plane strain. This basic assumption leads to the simplification of the field equations and a more readily available solution.

Sm flaez..In the solution of any elasticity problem, the

stresses and/or displacements at the boundaries must be used to determine the unknown constant in the solutions of the differential field equations. When the shape of the boundary is such that it cannot be defined using simple functions of the coordinates, the boundary conditions can­not be readily substituted into the solutions of the differential field equations. This is the situation in the problem under consideration. One solution Is to perform a conformal mapping on the actual configuration and transform It into a shape more easily described mathematically. Once this la done, the boundary conditions can be expressed in terms of simple equations related to the transformed boundary, and a legitimate solution to the stress field can be obtained.

Difficulty£ga£&&2&ggaaflax

10. ggaelfig.

In order for the transformation procedure to be employed, the stresses and displacements must be expressed in terms of complex functions, -which can be transformed and solved for, thus determining the stress and displacement field. The manner in which this is done is to express the stress in terms of two potential functions. One of these may be eliminated by removing the body forces from the pro­blem through the addition of a proper stress distribution at the boundary, fills leaves a new problem with no body forces in which one stress function completely describes the stress field.

By a classical method, the stress function is expressed in terms of complex variables, and it is only a matter of mathematical manipulation to express stresses and displacements as functions of the same complex variables.

Qompletihi-: the SolutionI'fhen the transformation function (which relates the

actual boundary in one complex plane to a simple boundary in another plane) is substituted into the stress and dis­placement expressions, a new problem is formulated. The boundary conditions are also related to the transformed complex functions and are used to evaluate them, fhe manner in which this is done is to assume general power series

gaaBlfig. ..Yaag^g,.ito?Egat^

11representations for the functions and evaluate the unknown constants from the boundary* conditions. Since the stresses and displacements in the original plane have been related to the now determined transformed functions, they are known also, and the stress and displacement fields are completely specified*

Thus, this thesis puts forth a solution to the pro­blem of calculating the stress field of a region with a complicated boundary, possessing body forces, by using complex functions coupled with conformal transformation theory. The solution encompasses bodies undergoing constant accelerations in one direction due to transverse loading, as w e n as the statics problem where the only force involved is the weight of the body. The second problem is only a special case of the first problem, where the acceleration is that due to gravity.

CHAPTER IV

FORMULATION OF THE PROBLEM

M W 2& & & L

The g e n e r a l f i e l d e q u a t io n s o f e l a s t i c i t y a r e

included here for completeness references#M l i M ? , 9 L M tX m -

c)<r» -- + + 2)1X1 + X

)r.,- -V c . * ybz

b'fxz O'Hjl 4- ^ 7b X ay -> Zjbt

Stress Strain Equations

and for clarity in later

II

't 2 ( l a )

- O ^V*- r * • a i j )

2)Lw/ H 2" ( l c )

‘**r <U - K^l * Gi)] (2a)e»'T (Hj - ii>C» + Gt)J (2b)

(ft - aSCct; v tr j (2c)ifXI) s TlujT ifxl - TxZ.

G tf'i'Z r ^4Z G

(2d)

Elasticity Field Eanations

Eamtions of Motloa

Strega ..gtsalii Muatlong.(Alternate Fora)13

CTx = A ( 6r Vfey) + (A+Z6 )£* (3a)

Gy = A ( 6* + £L) +• (A + 2G) £3 (3b)

<3~l : A(6 K + £y) + (A + £6) 6 z (3o)

1 *y - G &<.y T*i = 6 &a T\jz. - (b Vyz. (5d)

S tra in eisBlaqenent ;S9.uat.lpng

6 s ^ K ^

£ - ^ Aw, *z ' d z

(4a)

tfxs -d)u + c>Vc)*-) c)x.

(4b)

tfyz. =c>lT c) w^ <*y

(Ac)

tfxz -'c)u c)w c)z Ax (Ad)

Strain. compatibility Equations

6> I)* c) * 1A*-Ax A

(3a)

< 6y | c)1 *b Z.*- c) i}1

AL^yi AJ Az. (5b)

Strain 31 sglao enent. ^nations

s&sto.

ye. X u yd z l bz. b*

ye* b ~ 6> SI Jf- X.

by bz b \ cXy ^z.

c)*6ij 3 d>bLc)K *5 b % ()z.

ye* b b K’sz. xz.4 - ----------— -

b z 6>K Z

14

(5o)

(5d)

(5e)

(5f)

SagaaThe solution method to he derived in this thesis will

directly apply to any cavity with n axes of symmetry (n as 1,2,3*. •) where the greatest diameter of cavity is of the order of half the cylinder diameter. (This restriction Is imposed by the transformation used. It will apply to any cavity that the reader can conformally map to a simple shape.) A few typical geometries are shown here as examples.

Figure 1

A m O l c a t l o nofScope

15

•Figure 1 Continued

£ m s & L M K 2 £ S & 2 m

A typical problem is formulated as described inFigure 2. .y

z

Figure 2

As an example of an irregular cavity, the internal boundary of the cylinder is taken to be star-3liaped as shown in Figure lc and la a stress-free surface. The applied stress distribution oh the outer boundary will be some function of © between ©, and ©*, whose resultant is a reaction force equal to the weight of the body. The resultant resists the vertical body forces acting on each portion of the material in the cylinder. (W . jVlA ) examples are shown in Figure 3.

Two typical

General Considerations in Defining the Problem

16

fills completely defines a boundary value of the second kind, typical of the type to be analyzed.

She stress distribution at the boundary becomes quiteImportant in this problem because of the comparative sizeof the stressed boundary to the diameter of the body. Theprinciple involved here was put forth in 1855 by SaintVenant in ills memoir on torsion, and Is stated asfollows* (13)

If some distribution of forces acting on a portion of the surface of a body Is replaced by a different distribution of forces acting on the same portion of the body, then the effects of the two different distributions on the parts of the body far removed from the region of application of the forces are essentially the same, provided that the two distri­butions of forces are statically equivalent.

Previously, It has been demonstrated that a rule of thumbfor this principle is that the stress distributiondiminishes exponentially, and the differences produoed bythe two distributions is less than one per oent at a pointIn the body a distance away from the applied stresses equalto the distance over which the stresses are distributed. (12)

Figure 3A Constant Figure 33 Sinusoidal

17Thus, when the largest dimension of the body is approxi­mately equal to the largest dimension of the area over which the stress Is applied, Saint Venant's Principle states that the distribution must be close to the actual distribution to produce good results. To ascertain the reaction of the body to various distributions, several different forms should be assumed in order to determine sensitivity to various distributions.

Because the problem being solved is a cylinder in which motion along the axis is restrained, plane strain con­ditions can be assumed. The theory of plane strain assumes the followingi

U = u ( x , y ) (6a)

V i / 'f o c j) (6b)

w = o (6c)

Zero displacement in the longitudinal direction and the remaining displacements are not functions of z.

X =X (* .y ) (6d)

Plane Strain Assuantion

18

Y - - Y M (6e)

Z = O (6f)

No body forces in the longitudinal direction and the remain** tag body forces are not functions of z. These assumptions affect all of the field equations and reduce their complexity* Returning to equations (4) it can be seen that several of the expressions are identically zero.

(7a)

(7b)

(7o)

This leaves only the strains In the XT plane

(7d)

(7e)

These results lead to considerable simplification of the compatibility equations, (5)

c)6Hd z c) L

5 O

19

(3)

because u and v are directly related to and (Equation ?d) which are not functions of z . Thus, equations (5b) and (5c) are identically zero, since the remaining terns in the equation are already zero. For the same reason equations (5d), (5e) and (5f) are identically zero, because

which results again from the fact that u and v are not functions of z (7e). The only compatibility equation remaining is (5a).

f <)x. e>y (10)

To formulate a stress function, the above compatibil­ity equation (10) must be converted to a stress compatibility equation. This is done by returning to the stress strain equation (2) as they are simplified by the plane strain conditions.

6x - j)( (Ty dt) (11a)

Stress Ftmctlon

20

Ly - Cy - l ( x ♦ Cl) (11b)

— cl - i)(c; *<rs) - o (lie)

= = ° (lid)GG 6

Differentiating and substituting into equation (10), a compatibility equation in terms of stresses Is forthcoming.

In this expression a baslo relationship between Young*a Modulus and Shear Modulus has been introduced (see nomenclature). Yo eliminate G*. from the equation so that only stresses in the XT plane are involved, equation (11c) Is solved for Ct .

Cl = i!) (Cl + G J (13)

Making this substitution into equation (12),

21

4- c> - 6v) +

( l - -51) (Tcj - ii(.' + >5>) 61y_d)x.L

= z(<+>s) (i4)c)x b^

It is now appropriate to define a stress function /. by the equations

G*. z /$ + i V 'T - _ c)Vs " STS (15)

where is related to the body forces. Substituting (15) into (14)

bfo . 5 VO - ^ 1)b ^ b L)'

I t +<)lj* ^x2 *

b*1 "5x7 - 6(i-*-6) y^_ + a vb ^ 1 bx.'buf (16)

Simplifying

' n + y ^x.2 c>51 (i-^) + - £ 4 (> - y = o6>X.4 b t fb i f o b f (17)

22Using the V operator and rearranging, the equations reduce to

+- Vl/3 - o (18)

This places restrictions upon ^ and f t .

As was mentioned previously, conformal mapping is part of the method of solution to this problem? however, when a dynamic analysis is attempted, the acceleration terms of the equations of motion (Ho. 1) produce difficulties in expressing the stress function in terms of complex vari­ables. This will be shown clearly in later steps of the solution. As a result of this limitation, D*Alerabert*s Principle will be employed to express the acceleration terms of equations (1) as inertia forces and include them in the body force terms. The term "body forces” refers to those forces which are distributed over the volume of the body as is a gravitational force. These forces are distinguished from those that are distributed over the surface of a body such as a hydrostatic pressure.

Prom (7) it is seen that

t y i - O z /u)Z.G

(19a)

J£S2bB23>£3D*Alembert*3

o

23

>%z. (19b)

With the restriction that the acceleration is in the Y direction only and is oonotant, (1) tinder the plane strain restrictions reduce to

2>(Tx

X T

'dt,±±

cifn.

XY

f Iti7

p >JLi bM-

pf 2F

- o

r K

: O

(20a)

(20b)

( 2 0 c )

where X I3 zero and Y I3 the weight, '..lien the equili­brium equations (1) are derived, the body forces are designated as forces per unit volume; therefore, Y =^3 represents the weight per unit volume which is also a constant in the negative Y direction. Waking these adjust­ments on (20),

6>flxbx.

b ?b x

*L)

b U

6><K,

X = O (21a)

- y - o (21b)

whereX = oy - < n z y ° *

24

By substituting the stress function relationships of (15) Into (21),

AA<)x. ^ x ^ q 2 ^L(2

y * , y * + ^5x^3 T lj~

X -- o

7 --o

This results in a definition of

(22a)

( 2 2 b )

i A ; O (25a)

Y - - f - r(23b)

Integrating and noting from equation (23a) that ^ is not a function of x.

= / ^ (24)

where the integration constant has been defined to be zero. Since p a . is a constant for any stress state of the body,

25

(25)

And It follows from the definition of y*

= O (2 6 )

Hence is a harmonic function, and from (18) it must follow that ^ is a biharmonic function.

This Is exactly the same equation that governs the condition for zero body forces. The question now becomes one of what is happening at the boundary under such conditions.

Considering the relationship between the boundary forces and the internal stresses from an element on the sur­face where ds represents the free boundary,

W o (27)

Y

Figure 5

Conversion of Body Forces to Boundary Stresses

26

x as -- (h dy + Th, dx. (28)

Y d s -- (^d\ + dy (29)

Let 1 and ra be the direction cosines of the normal to the boundary with the X and Y directions. % and f are the com­ponents of the force applied per unit length of the boundary.

Cos C 05 /3 - cl x.<ds rr\

X - 6* |) +■ 1 xy (T\ (30)

Y = & +■ 64 on (31)

Substituting in the definitions of stress (15)

X- W

If we now define some new terms

(32)

(33)

x'- X - PS (34)

27

Y x =. Y - m & (35)

crx = ^ -- c - $

< - - 0 ‘ ^ ^v .. y y

(36a)

(36b)

(36c)

Rewriting the boundary conditions of equations (32) and (33),

r = CO ■ + T/t, m (37a)V' = CO + Cm (37b)

Hence, the problem has now been redefined where the defini* tlon of the stress function of (36) insures that equilibrium is satisfied and from the compatibility equation (10) V Y = O must also be satisfied. Thus, the problem has been reduced from one with body forces to one with no body forces but with a different load applied at the boundary equal to

X - P £ and Y - ™ p

This is actually seen to be the original boundary force X and f plus a hydrostatic pressure equal to p . Suppose

28a stressed boundary with components as shown In Figure 6, the resultant force per unit length Is

R. + ( ~Wf ' - p

R = /°ai.j i^aSin 0

Figure 6

This stress distribution must be superimposed on the actual boundary stress to produce the same Internal stresses as In the original problem. However, when the primed stresses (T/, CTy and 'hy are solved for, (h , (Ty and T.y can be determined

using (36). The altered problem now appears in Figure 7. (Refer to Figure 3a) Henceforth, all primed stresses strains and displacements refers to the new problem of

Before SuperpositionFigure 7b

Altered Problem

The shape of the inner boundary has been simplified for purposes of illustration. Since there was no assumption in the derivation of (30 and 31) concerning the shape of the boundary, this approach ia not limited to simple config­urations. It can exist on a cylinder of any shape providedall of the conditions previously discussed are satisfied

CHAPTER V

THE PROBLEM IN THE COMPLEX PLANE

.t s s B s ms £ , ? , g&clfi&g&

When the stress function equation (18) is satisfied, both compatibility and equilibrium have been satisfied and all that remains to completely specify the stress field of the body is to satisfy the boundary conditions. This is a difficult task einoe the boundaries themselves are very difficult to describe using either cartesian or cylindrical coordinates. We resort to a conformal mapping technique to obtain simple boundaries to be used in the boundary condi­tion equation. Therefore, it is now appropriate to show how the problem can be extended to the complex plane where stresses, displacements, boundary forces and moments, as well as the stress function are expressed in terms of analy­tic functions of complex variables.

In 1898, E. Goursat published his method for representing the biharmonic function by means of two analytic functions of a complex variable. (5) This is done by defining a new function p,

W V* W

p V'JC --V*p - o

7 7 7 7

(38)

(39)

The Stress Function In Terms of Complex Variables

Prom (36)p - (u' + CTy' = <£. + (Kj - X/3

p : (U * Cf; -£/^y

(40)

(41)

p must be a single-valued function, since the stresses are single-valued at every point In the body. Thus W must be single-valued also.

Equation (38) states that p Is a harmonic function by definition.

A new function Is now formulated,

where z is now a complex variable and q is the conjugate function of p. This means that q Is harmonic, that f(z) is analytic and that p and q are related through the Cauohy- Kiemann equations (3). Using the Oauohy-Rlemann relation­ships, whioh are given here for convenience, it can be seen that q can be determined within a constant.

ju) = p + LC| (42)

bp _ < q bP _ by (43)

Supposing that p is known to a function of X and I, q can be found by integration

3 = \ S(*,Lj) dy + k(x.) + C

32

(45)

Prom the second equation of (43),

Since g(x, y) and p(x, y) are known functions, (46) can he used to calculate h*(x) which in turn can he used to deter- mine h(x). Thus, from equation (45), q is determined within a constant. When q is substituted hack into (42), f (z) has an unknown imaginary constant in it which results from the unknown constant in q. The evaluation of the constants will be deferred until later in the derivation when they are accounted for by assuming a general power series for F(z).

As a matter of convenience in a later portion of the mathematics a ne w funotion is defined.

F ( l ) - — 1 Ul )c \L . p * L Q (47)J<X

Remembering that f(z) is analytic, F(z) is analytic from a theorem of complex variables which states, "The Integral of an analytic function Is therefore an analytic function of its upper limit, provided the path of integrations is con-

33fined to a simply connected domain throughout which the integrand is analytic.” (3) This can be accomplished by performing the integration over a contour that makes the region simply connected.

Figure 8

Making use of (47) and evaluating two different forms of the derivatives of F(z)

Equation (48) results from the fundamental theorem of inte­gral calculus for complex variables. AI30 with complex functions there are different forms of the derivative, depending upon the path selected, as the derivative is evaluated in the limit process. (3)

dF (43)

dF r ap , • b ad z < K T * (49)

and

dF_ _ _ • 2)P + c)Q_di (50)

Equating the real parts of (48) to (49 and 50) respectively,

2>P p ¥ 4 6>lj 4 (51)

Adding these two equations and simplifying,

^P o>Q. *3+ 2 t v = P (52)

Goursat recognized that equation (52) could be written in another manner by the following algebraic manipulation

v'-UP-yQ) = *(-}£* 4 ^ ) * + +(53)

Remembering that ? and Q are harmonic, which means that

V*P -- V*Q = o

Thus

V l(xP + yQ.) + 2 c>aP

(54)

34

But

p = v2-

35

And therefore

7 l (xP + y Q) - V*/

This can be rewritten by transposing + yQ.)

V L(y- xp - yQ) = o

This defines a new harmonic function which shall be designated

(55)

(56)

(57)

^ - x P - y Q , p{ (53)

Solving for £

^ = xP + yCX + P, (59)

This is the desired expression, for each of the terms of (39) can be expressed as functions of the complex variable z. Noting that

ZF(l) - (x-ly)(P + LQ.) - (xP + yQ) + l(*Q. - y P) (60)and

If Cl) ( x + i - y ) ( P - i a ) ( x P + y Q ) + i-(yp - XQ) (61)

36where

Z = K - uj F (z.) -- P - lQ.

thus

£ I FU) + z f (l) X- P +• Q_ (62)

which Is an expression for the first two terms on the right side of (58)* To include the last term, a new complex function is needed and will be defined as X (z.) a P( + i Q,, where Is the conjugate function of 3?-. Thus

P, (63)

where

x[i) -- p, - ia, (64)

Combining (62) and (64), mb have expressed the stress function in terms of two complex functions.

I F U ) ZF (i) + XU) + ZU) (65)

Stresses In Terms of Como lei: FunctionsAs was stated previously, the only step remaining to

37

completely determine the stress field through the potentialfunction £ , is to satisfy the physical conditions at the boundary. The external restraints on the body will be expressed in terms of stress in this particular case because stresses are known directly at each of the boundaries. We therefore use (65) to express the stresses in terms of complex functions F(z), X(z).

bib y ^F(l) iz.T T ~ "5T

L . v r w + fcl) + i IfS- A L +.b lS T b z b \

p(7) <>L + + c> X' ' d)x T I 51 z IT

(66)

HoweverL - X ■* ty I = X - iy

z.

. aFCz) . f '/zn "IT- 4z_

^ f (Z) _ aF(z)) z a z --F'(Z) <67>

Mz) _ dx(z-) __ ^ b 7 ~ 4 7

5x(z) = ax® . £'g) < z az

5x7 F'(L) + F(z.) + Z F '( i) + F(ZJ + X'(z) + til)j_

2

Similarly

TIT" ZF'(z) ~ F(i) -ZFU) + F(I) + X'U) " X'(*) (68)

Adding (6?) and (63), simplifies the relationship to

i f +l4f- = Fti> + ^ ' ( D * Z ' a ) (69)

Differentiating (69) first with respect to x and then with respect to y.

F'(z) * Z F"(z) * F'(z) + X''(Z.)

F'(z) - ZF-(z) + F ' ( i ) -r(i)

(70)

(71)

By multiplying (71) by and then subtracting from (70), we obtain a relationship for stress using (36),

^ + 2>V6)XZ (f LJ1 X + (Ty' = F'GO + F'(,I) C (72)

where C is some real function.If (71) Is multiplied by and is added to (70) another relationship for stress is formulated.

b vTx1 T q 1" + ZL -- Z ZF"(z) + X''(z) (73)

38

In terms of stress39

ov-tfi'-nr.; - 2 z r @ * (74)

To reduce the number of conjugates In (74), the conjugates of both sides are equated, remembering that the conjugate of the sun and product is the sum and product of the con­jugates. (3)

( a V - C ) = 2 z r ( L ) ♦ x ’(r) = D (75)

Where D and E are functions of s. Therefore, collecting the equations, the stresses can be aolved for using simultaneousequations If F(s) and X(s) are known.

61' + (Ty ' - C (76)

( ( ^ ' - ( T f ) + i(zr;) = D 4-1E (77)

Equating real and Imaginary parts

a;7 +• (n3' - c (78)

CTy' - 6 7 = 0 1i/a E£ (79)

From (78) and (79) 61 , (T;/ and f l y can be solved for.

equations i f J?(z) and X(s) are known.

Equating real and Imaginary parts

40

The stress-strain relationships for plane strain in the new problem are

. (Tx/ ' -6 (<Jy ■+■ (i~L )

£y ' T 07 - n3(cfx' + <rt')

0 ; r <n' • i(<r/ .

(30a)

(SOb)

(30c)

Substituting (80c) back into (80a) and (80b)

£ C(»-V) - ( T ^ lIt4) (81a)

(81b)

It is now necessary to express the strains 3m terms of displacements and the stresses In terns of the stress function (4) and (15).

c>-u!~JT~ E (l*4l) Y /

T *Fx3( 1 +■ i)

h v '^ T

(I-*)

(82a)

IE (82b)

Displacements In Terms of Complex functions

Referring to (51) and (55)

ip , 4®. -- -l pdm b Lj 4 'D ^ + £ £

P b K*

¥e obtain from (82a)

bvb b k

b uC)K

bu.'bx

_l_E ( p - ^ ) O - y ) - - $ £ K ' » * ) ] (83)

pO-'i1) (84)

(35)

In like manner from (82b)

b i f ' I by' E

bv' ib ILJ LW _ j_bij ~ E

P O - ' W 0 + 4)

40 ( ' - ^ - 0 O * ' J)

(86)

(37)

(88)

Integrating,

U -- 4 P ( l - i 5 ‘) + 9,(9) * c, (89)

41

42

4Q. (l - {**) b / O * ) + 9*00 + C-L (90)

To evaluate the unknown functions and constants, we turn to the shear stress-strain equations. (2)

*4 - Tm = 2-0 + j) r “ 'G E

Or using (4) and (15)»

b u ' , 2)v' 2_(i + 6)<kj S = E

Appropriately differentiating (89) and (90) and adding, a relationship is found from (91).

g.'M + H M - - - 2 ( u i ) £ £ <92)

Prom the Cauchy-Riemann equations

= ^Q-< x

Thus (92) simplifies to

<3.'0) 4- SJOO s o (93)

d>LJ (91)

which means that for this equation to be true for all x and

y, Si * (y) and Spf( ) must bo equal to a constant

9f(y) - - e ; w -- c

43

(94)Integrating

9. CSJ) Cy + Cgt(x.) = -cx + Cj

(95)(96)

Prom the strain-displacement equation (4), It is seen that these displacements •will not Induce any strain but represent the rigid-body displacements and have no bearing on the stress field. Thus,

<£*' --

-4 '

chi' __!_!>)C>K E T k

i~5Tj~ e

<W , bir' _ I t)

4P(l ~ 6 l) +- o (97)

+ o (98)+ c + (99)- c

Returning to the displacement equations of (89) and (90), we have as final expressions for displacements, where g^(y) and g2(x) have been dropped

11 ■-¥ 4p(l - -61) - - ^ 0 4 *0 (100)

(97)

(98)(99)

(101)4Q(l - ll)

The constant terms have also been omitted from these expressions, since they, too, only represent rigid body displacements and have no bearing on the stress field.

How to express these displacements as functions of the complex variables, (101) is multiplied by J,, and added to (100).

U'+ iv' 1 + iSE 4( P + iQ)0 - 3 ' ( - ^ 4 L 4^) (102)

Referring to the relationships already derived in (38) and recalling that F(zjb P+ l Q

U -h i t 1 6 FU)(5-46) - IF'(l) - Z'(i) (103)

Collecting the expressions for displacements and stresses (72, 74 and 102) into one group for convenience, we have the equations into which the boundary conditions are to be introduced.

<n/ s f 'U) + r'U) (104a)

44

45

<ry'- cr/ + Z l?^' - ^ Z F " U ) + * “(z) (104b)

/ • » U H I T 12G FU)(3-4 4) “ Z.F'(l) - X'(I) (104c)

?&5aa Q ^ o o m i m -.The resultant foroe on the grain boundary, of -which

a portion is characterized in Figure 9, will be calculated. Let S ds and f ds represent the s and y components of the force acting on the element of arc ds of the boundary. Relating these components of the resultant to the normal and shear forces acting on the element, the following is founds

Figure 9

X/ - 6~J C05 °C + Uy (105)

Y ' ~ CTI) S lw <x. Cos (106)

Surface Tractions in Terns of OonDlcz Functions

Prom PI {jure 9

AALels

46

Cos = S v.rv ^ - cU els

This relationship can be substantiated in each of the other three quadrants. If the potential functions are now sub-stituted for the stresses in (105) and (106),

veiqels

e)2" eix. e)x A s

b i' 6><A A 4 , A / b/\ A xX = ^4 1 As> "Jx. V V

This equation expresses the chain rule

Similarly

— / A_f 'b/ \el 5 V )

A - ( I t \el S \ 7 x T J

(107)

(108)

(109)

(n o )

F and P can be calculated by integrating the resultantJstress components over the boundary.

F* (111)

47

Ay ' Is - -

L <dsl W L h< L ( ^ L ) c\ b -_ (112)

(113)

Factoring out a ~i, a similar expression to (69) results.

This is the boundary condition equation for all boundaries, and the actual applied stresses at the surfaces of the body must be expressed in terms of (115), or equations (104a, 104b).

Aft, ,.TFurthermore, boundary conditions can be imposed which

involve the Moments due to the applied forces. Summing moments of the components of the resultant force at the boundary around the origin, (Referance Figure 9), counter­clockwise positive,

Hence

F„+iRj - -I t-(z.) + i f ' L l ) + £'(Z) (115)

fthere and 3? are forces per unit length If AB Is unity.

Momenta in Terns of Coirrolex Functions

48

M - (x. Y' - y x') <ds''A. (X16)

Substituting in (109) and (110)

M -- -

m --- U^i -bx J•6

. bu j

(117)

(118)

Integrating each term by parts

rbx Iu ( ¥ ) -

• 54 h < uOX (119)

Performing the same operation on the second term of (118)

-& . \ . \ / 6 rl- I v ( ^ ) - - - y y - +

A4 © aby •) (120)

Returning to (118) the expression for M becomes

M -- ~ U b</* i r + • J n r

"I E> J b-Hdu (121)

L ** ^-A “'A

The integrand of (121) is merely the definition of the total differential

M = - x +Si ■t B>

K- (122)

49Expressing x and y In terms of z.

= z U 2] y =21Z -z

Prom (6?) and (68)

W i I F'(z) + F(z) + ZF'(Z) + F(z) + X'(z) +XTz)

y zz ■ i^ir r 1 lf‘(z)- fll)- Z- F'(z) + F(z) + %'(z) ' X'(2

z ^ z>

t(Z-£)

(123)

(124)

Simplifying M becomesM - ~ J \ l i Viz ) + zF(z) + Z X'(z) + £ F(Z-) + z l F'(z)

7 U 'W +ZF(z) + zF(Z) X(z.) + X(ZJ (125)

Noticing that each term in (125) has a conjugate, M can be expressed as

M -- Re. Re

-(zzF'Cz.) + £ F(z) + Z X-'(z))

ZF(z) . XU)(126)

Simplifying,

M = Re XU) - 11 F'U) - Z X'U) (127)

2he algebra in later manipulations can be simplified con' slderabiy, if we introduce the followings

<f>(z) -- F'(z) A (z) -- t'Li)

The equations then appear as

<C + (h,' = 4Re^(z) = Z 0(z) + #(J)

Ch ~ cr; + 2l?^ , z L 0Xz) + A'(zJ

U + -£g LF(z)X -Z (z) -AU)

F* + 1 Ft) = - l FCz) + Z 0Cz) + A(z)

WhereX =

(128a)

(128b)

(128e)

(128d)

With these basic equations we now Investigate the general •form of the solution.

50

CHAPTER VI

GENERAL FORM OP THE SOLUTION

M J j m & .%.The general for® of F and X for multiply connected

regions can now be derived.

C + < - 4 Re jzf(z)

The real part of 0(.z.) must be single valued beoause of the single valued nature of the stresses, but the Imaginary part of may not be. In making one closed circuit inthe interior of the body, its value in general changes by

L ^ » where K denotes the kth contour.Let

6>n. = 2.rrA*cand define

Figure 10

t \i> -- m A*. I03 Cz. -zO (129)

where Z K is a point Interior to a "hole*, normally the center.

Arbitrarinese of P and X

Log ( Z - Z*.) increases 2.m every revolution, thus after one revolution

52

and

^*(z) r 0(z) + 2mA* - 5.

4> *(z) is single-valued.

0(0 = 2. A*. log (z-z*)

A*, lo g (Z -Z * ) - 2 m At

+- (130)

where k represents the number of contours. The real part of <f>(z) is single-valued here as we require.

F(z) = ( (z)c\z + CJz.

-- j>(Z-ZjA* log (z-zj -fz-zj +t--i

0 * ( z ) A z . + C

(131)

0*(2) dz. may be multiple valued around a contour, soz.

doing the same as above,

A z ) d z = C*. l og (z-Zw.)r t=l

single valued function of Z •

Combining terms

m -- Z ]L At log (.Z-Z*) * 5. & loq(Z-Z*) + F * U ) (132a)

Similarly, from (128b)53

A(z) - Z ^'og(z-zA v A* U)K.M (132b)

To make displacements single-valued

z f c C u ' + i v O = y ifU ) - i?(X) - A Cl) (133)

Putting in for F, and A at o and Zvr and subtracting, we find the difference to be

A 26 (u' * LIT') 2 m " (X+') Avc Z- +- X V *Thus,

A* = o X^tc +• - O (134)

and X may be found with one more equation. It is found .by considering the resultant force on the contours of the "holes”.

F** v l FIk. = -l- $ (z.) + Z0'(Z) + t'iX) (135)

Substituting in we find

Fx. +■ l Fix. ~ — M K. ~ $K. (136)

Prom equation (134)

- K i d

Thus,54

F**. +• i Fg - 2 IT Kit + X i t

Fk» 4- L F2 tt ( I + X)

v' _ yC ( F*< + i F^*)2 rr (i + X)

(137a)

(137b)

The final form for the analytic functions turns out to be,

F(z) , - FHk) \ ocjCz - zO +• F*U) (138a)

= 7n - L Fy«) 1 09 Cz.-zt) 4 A*U) (138b)

where F*^ and represent resultant force vectors on or around the contours.

M m S L ,S$g£?s ■In a simply connected body the stress state depends

only upon the external loading and the shape of the body.It does not depend upon material properties. However, when the body is multiply connected the solution Is different.

In the previous seotion It was found that for single­valued displacements

3^ A + ^ k- - O

Popondenoe q£ tfoe, State of Stress on the Elastic Constants

55If we assume that a problem has been solved for a body with material properties expressed by X . To determine if the same solution applies to a body with properties , wewrite

DC Mx. + k_ - o

Subtracting the two equations ae a simple means of relatingX. to X_

Putting in the appropriate expression fori(X- X') +

2 tr ( l + x)- o

In order for the same solution to apply for }(. differentI

from X. t the resultant forces on the contours must all vanish. (Fx^-F^-o) Otherwise, the stress state of the body depends upon material properties. By using the method outlined in the thesis, the elimination of the body forces puts resultant forces on the boundaries and makes the solu­tion dependent upon material properties.

Although the analytic functions have been expressed in their general fora, before they can be evaluated coor­dinate transformation must be introduced. Before pursuing the solution further we will divert our attention to the

56

equation of transformation to general curvilinear coordin­ates.

CHAPTER VTI

GENERAL CURVILINEAR COORDINATES

% '- f. ' LJ) >t = F.C^LjJ

These equations represent coordinate transformations from cartesian coordinates to general curvilinear coordinates £ , ^ . As F, (x.tjJ is allowed to take on various con*

stant values, a family of curves is generated, which are coordinate curves in the new system. The same 13 true for

For example, consider polar coordinates

£, - r Vxl+ ljz ' rt = e = arctan JJ_xThe values of x and y which make r constant trace out a curve in the x,y plane which is a coordinate lino. In tills case. It Is a circle for (, and a ray for ^ .

The same would be true for elliptical coordinates, where £, a constant would be oonfocal ellipses in the x,y plane and ^ = constant would be hyperbolas. In other words, we select the coordinate system for a problem that best fits the boundary of the body whether it be circles, ellipses, straight lines, etc.

Often times the boundary Is not a simple shape, and it becomes a matter of formulating a new type of coordinate

Coordinate Transformation

system that fits the body. Our problem becomes one of find- in C a transformation relating the new coordinate system to something more standard lilce cartesian. Rewriting the transformation equations

I R 0.L3) *1 = F*(>Uj)where £, s= constant describes the boundary of the body.These tiro equations can be combined into one using complex variables

Z - K < 0 (139)

where

I - * + L £ * 1 \

Confining ourselves to the class of transformations that are conformal, we know that angles are preserved through the transformations, and orthogonal systems transform to ortho­gonal systems.

Figure 11

53

£> = Constant

Co n s t a n t

59Using the standard transformation laws for stresses and displacements (which can be derived from tensorial consider­ations) we obtain the followings

11 G I C o s + CTIj Si-rCx. + T * y S la. (140a)

11 C / + C h j ' C o s 2^ - T*,j Sl^v 2 < < (140b)

2 ? « . s « - c ; ' ) Sikv + z t *:3 C o s z<<. (140c)

From these we find

t- (Trg - 6~jL 4- (j~y (14la)

( d V - G V + £ Lr/ij) e ^ (141b)

To obtain an expression for £ we notice the following:

cix.I AM

(140a)

(140b)

(140c)

Figure 12

60However t

dx = d a a

H 1

(142a)

(142b)

For simplicity we will consider a variation in K only, l.e., we are moving along a line of constant •

d y _ d £,T a m -

In general, then,

dy - J S in.

dx dx

dx

(143)

- J Cos °c

where J is a real number.

dz dx d yAt i t + L i t (144)

Since Z = -f

A 4 L 4 ) - i Vr)H 1 d < H ' H j (145)

Thus equating (144) and (145)

no <S>XJ ( C o 5 < + L Sm°0 = J e LoC (146)

Taking the conjugate of both 3ldes

V { 0 --J e'u

This provides a compact expression for C Zl^ in terms of the transforation equation.

2i*. no n o (147)

It must he remembered that the transfomation equation is referred to a cartesian system only, and this expression for must be related to rectangular cartesian coordin­ates. Using standard coordinate transformations for vectors, we find the displacements

11^ - U.' Cos< + iT'S un<< (148a)

= If Cos®^ ~ ll/ S li\ oC. (148b)

which gives

U^+ L-W- r ( u '+ Llf') e ~ LoC (148c)

61

Similarly, the resultant tractions on the surface

+ L r (h<. +- l F^) e . 'L'*‘ (149)

Specific ApplicationsIn the problem of the star-shaped cavity for

Instance, we transform to "star" coordinates} or If the cavity Is diamond-shaped, we transform to “diamond" coordin­ates. Kantorovich and Krylov derive the following function that transforms cavities of id anes of symmetry Into circles. (9)

Z -- +- B < v'* + C < ‘- ^ + ... (150)

where the coefficients are real. Needless to say, this is a very powerful tool that facilitates the solution of many difficult problems.

he now proceed to the final stages of the problem by formulating the equations in which the boundary conditions may be used to define the analytic functions F(z) and X ( z ) •

62

CHAPTER VIII

THE PROBLEM IN THE TRANSFORMED PLANE

BflaSteL ConditionsFrom Figure 7, It is seen that the boundary stresses

on the outer periphery are sinusoidal stresses superimposed on a constant radial stress, which only acts over a portion of the boundary. This makes the boundary stress a step function of © and will be expressed as a Fourier Series.To do this, © is redefined to have its origin at the bottom of the body so that the constant radial stress is symmetri­cal with respect to the origin. This involves a phase shift of 90° in the representation of the sinusoidal stress resulting from the removal of the body forces.

^ now becomes- / ^ CJ Cos ^ (151)

Figure 13 Redefined Angular Coordinate

64which applies at all boundaries. In order to express these stress boundary conditions as a function of 6 , we must eliminate 0^ from equations (141). This normal stress has no meaning on a coordinate line £_ = constant. Subtracting (141b) from (141a)

— ' - -p7 1^ ' L ^ " ~Z

This relationship can be used for the stress conditions at the Inner and outer boundaries. Using equations (128) the stresses in terms of the new variables can be expressed in terms of the analytic functions F(z) and i t t 7-).

(JV- 4>U) +£(z) "(Z <ph) 4 A(z))ezu (153)

( d V + <n>') - ( c r : / - ( T / t Z t T x y ) e Zi< (152)

The actual boundary conditions are then expressed as follows:

( C f ' - L 'U'l ) inner - ^ r ^ 05> & (154a)boundary

outerboundary

Coi i

i, 2-n -k.(154b)

outer _ _ M r Cos X 4- C (154c) boundary 'Ztt i

where C = constant radial stress (see Figure 7). All of the

65terms except for the constant tern of the boundary conditionare already in the form of a Fourier expansion. The step function is expanded in terns of complex Fourier Series and the remaining sinusoidal elements of the boundary stresses are superimposed. Considering only the constant stress portion on the outer boundary

G« = hh = 1 (155)

where C is defined asn

o(156)

For this problem¥,

+ Yrr (o)e-ln»dK

J L s uhTr

The entire expansion becomes

(157)

Superimposing this function upon the sinusoidal, one of the boundary conditions are now expressed as

(158a)(O f'- lT/O- r - P & r C03ub.

- L Oo.b. r "/?<r(r + A^)ei,x55 (158b)

To be able to transform these boundary conditions, they must be expressed in terms of z

y~ Cos 8 - j - ZlL~ Z (159a)

In.*(zz)

=\a z. (159b)

The final form of the boundary conditions in terms of z only

( V - n d , b . =

( o Y - ^ d U . --

,££LZu (z.-Z)

i. b.

(Z - Z) + 2.(-^r binA'.ho '

(160a)

(160b) lo.b.

which can be transformed to the < plane by substituting the transformatlon equation Z n 4(0.

The general method of completing the solution is to equate the boundary conditions of 160 to 153.

iT^Z-z) 4- U. \-

?(z) - e2L< It'Ll) + A(z)..b.

66

67

r

<fi(i) + 0(z) - Zd'(z) + A(z)

To evaluate the unknown functions by means of these boundary conditions, we must transform to the C plane. We now derive relationships that will be useful In performing the tran sformat1on.

Transformation of the Complex FunctionsWith the aid of a conformal mapping function

Z = -f- (£) we may express the functions F(z), JZ (z)»$ U ) in terms of the variable d* as follows:

(161a)

(161b)

(161c)

Differentiating implicitly,

(T'(Z) r _ Cl F,(dQ cl d _ jr'^j ddZ. cl< dz ' dz (162)

60The mapping function 13 normally not easily adapted tofind - d £ .

d L

_ J _ _ 1A z 6z_ d 4«)

d<And (162) becomes

F'(z)

\

4'«J

(163)

Similarly,

rcz)f «)

0'U) * : « )

VIA)(164)

Upon Insertion of these expressions into the formulas for stresses and displacements, the boundaries have been trans­formed from the complicated shape of the cavity to a unit circle. The outer boundary has remained a circle so that the location of the applied stress can be designated by a radius. The inner boundary will be designated R = in thetransform plane, and R - R will designate the outer2boundary.

(Tr iTR -- + &C<) - k.’MIL'Cff)

A'CO 4r « )

A d dno (165)

69where Z a k(w) represents the transformation from general curvilinear coordinates to cartesian coordinates.Z. =s -f(O is from general curvilinear to polar. The

relationship between the two is found to be the following!

z - 4 ( 0

< = 9 (vv)

z = 4 6 W ] k(w)

■f'W J 4 cl vV

The transformation from the ^ to the ¥ plane is

< =

k'M-- fY <Jew © ) - . | ^ ) e i

If we let w =5 ul-v

k'(w) ' +Y4)

in terms of <K(w) __ -f'CO kiw) \ X i ) Z

The stresses in terms of coordinates now become,

U<)<t.(4) * A , OS) (166)

70Sejflea Solutions for the .Qomplex •^notions

Referring to equation (132a) and (132b) it will be recalled that F*(z.) and X*(z) were single-valued. In an annulus any analytic function can be represented by a Laurent's Series. The general form of the solution appears below.

F' ^ ;2tXk) § r ( ^ + 1F'^ l o ° (■*-<*) * £ K L < f - £ . Y (167a)

/ YY\ **A.«) = 7 t — jt Z ( ^ - i -<?.)' (167b)

< K lvX.) id-, n-.v>

Since the resulting Laurent*3 expansion is uniformly conver­gent in the annulus when differlentated term for term it follows that,

l 1- £ ^ , ( 4 - ^ . ) " " (168)

The expressions for stress in the general curvilinear coor­dinates are obtained by substituting expressions (167b) and (163) Into (166), where is taken at the origin. Since there is only one cavity, we will take at the originalso

71r ' • 'T/6, - 1

!?3_ P C 00Hew +• L VH Zn (_i +X) XK.1

Hit<

n A ^ " +r\- *«

2tr(l + X.)Rli« C Fi4 it

*-1 <- T n A » ^ +

£f(£) 2tt(|+X)

m P - ; F • + (169)It-1

n A , r -Ir\- - *>

X vrv Fxk ~ I h)«. +2rrC>4X) frr

n.-o

The unlcnown constants A*, and b* are determined from the two boundary conditions of equation (160) , where at the boundary

i t U u . -- U O - V J . ) J L.b. (170a)

( y ~ y ^ i . b. A 4 ^Zi m - m

Z (TFT S m n*0

(170b)

r\--«o1(4)U 4 )

0.6.

72If equation (169) la written In the complex trigonometric form ( 4 = d e L ) and evaluated at R= = 1 and R a R^,It Is possible to find the undetermined constants A . A ,n nBn and B . Performing these operations with equation (169) and substituting it into (170) and (171),

-2n(i + x)e 1 4 ( e L*) - X F J e 2. ^

4'(e'0

s An. e + (171a)

- f H e ^ fiA^e - 5.el 2. { ( e * ) - { ( ? “)

2Tf(l -v-X.)IF*

R«.4(R.eu) \ + x lo‘)(ete*)e“fn ^ ) /

«0nA p-gU.-o^ + „ A, e r e - 11" * ' - (171b)

r\. -«o

4 0 ^ ) er'e^’0 + etc u#

-4^21*30

+ K C er- . ..>/ 4 tee*)ntr -5LA *6'. 4‘te<g)

73where F . = 0 in keeping with the general problem being2Tconsidered. By setting the coefficients of each power of

G e q u a l to zero separately, a set of simultaneous equations result which can be solved for the four general undetermined coefficients. An electronic computer will be required to evaluate the number of constants necessary for the desired accuracy of approximation for the series of

R (£) i A , (4) and ( £ ) .

Having found expressions for F , ( £ ) , A 6 A and j z U O , the primed stresses can be calculated from

(166). However, as will be recalled in the early formula* tions of the problem, these are not the true stresses. The stresses can be found by (36) which relates the actual stresses to the primed stresses. Physically, the expression for represents a hydrostatic pressure which mustbe added to the primed stress to calculate the actual stressdistribution

CHAPTER IX

SUMMARY AND CONCLUSIONS

In summary, the method derived herein provides a method of solution for bodies with complicated boundaries. The stress field can be determined for a desired inertia force, which is a good formulation of the steady accelera­tion problem. The amount of tedious and voluminous algebra that ensues depends upon the degree of complexity of the boundary and the stress distribution Imposed upon it. When the boundaries are quite irregular, the conformal transfor­mation function Is usually of a series form. In addition.If the stress distribution is of a nature that it must be expressed In teri.13 of a series expansion of some type, the computations become very involved.

In general, if the boundaries are simple and the distribution of stress Irregular, this method can be employed with relative ease by deleting the step of trans­forming the boundaries to another plane. Similarly, if the boundary is complicated, but the distribution of stress is simple, the solution Is quite readily obtained. However, If both the boundary and the distribution must be expressed In terms of series, there results a substantial amount of numerical work, in which a number of simultaneous equations must be solved. The size of the matrix of equations depends

75upon the number of terms used in the various series. Tills is to be expected in a problem of thi3 nature, and raerely necessitates the use of an electronic computer.

Another consideration i3 the alteration of the actual cavity of the body due to the approximation made by the mapping function. Obviously the better the transformation function duplicates the actual boundary, the better the results '.rill be. In some cavities, it would be conceivable that the entire shape could be duplicated, but for a compli­cated shape, this would probably complicate the computations to a degree that they would be too cumbersome to handle. If such should be the oa3e, the problem could be worked by considering a portion of the cavity in one problem where the transformation accurately maps this fraction of the boundary letting the remainder of the configuration be what it may. After determining the stresses around tills area, the remain­der of the problem Is solved with a transformation that accurately maps other critical portions. In this manner the stresses around all of the crucial areas could be found by breaking the problem into parts where each area is consi­dered as a problem In and of itself.

By reflecting upon the entire procedure, it is found that the actual problem Is not transformed to another complex plane, worked out and then transformed back.Stresses are not transformed to other stresses, neither are

76displacements transformed to other displacements. Rather, the stresses are related to general functions that are con­formally transformed and subsequently determined from transformed boundary conditions. After stresses are obtained, displacements can be calculated through the gen­eral field equations of elasticity.

B I B L I O G R A P H Y

BIBLIOGRAPHY

1. Biot, M. A. ‘'Distributed Gravity and Temperature LoadingIn Txfo-Dimen3ional Elasticity Replaced by Boundary Pressures and Dislocations," Jourxial of Applied Mechanics. 1935# p. A-41.

2. Brock, J. 3. "Analytical Determination of Stress AroundSquare Holes with Round Corners," David i). Taylor Model Basle Report 1149. November , 1957, p. 29.

3. Churchill, R. 7. Complex Variables and Applications.New York: McGraw-Hill Book"Company, Inc., I960.

4. Goodier, J. N., and Hoff, N* J., Editors. StructuralMechanics: Proceedings of the First Symposium onNaval Structural Mechanics.

5. Goursat, E. Bulletin de la soc!1 ete* laathefaatlque deFrance. . ol. 2$ TTo9g ), p. 23^.

6. Green, A. E., and Zema, ¥. Theoretical Elasticity.London: Clarendon Press, Oxford, 1954.

7.

8.

Heller, 3. R. Jr., Brock, J. S., and Bart, R. "TheStresses Around a Rectangular Opening with Rounded Comers in a Uniformly Loaded Plate,*1 Proceedings of the Third P. S. National Congress■on Applied MgcEaHias. June, 195^ 5. 3^-36^.'

Huth, J. H. "The Complex Variable Approach to Stress Singularities," Journal Applied Heohanlos 75. 1953, p. 561.

9. Kantorovich, L. V., and Krylov, V. I. ApproxjjaateMethods of Higher Analysis. New York:T inverseience Publishers,'Third Edition, 1958.

10. Muskhelishvili, N . I. Some Baslo Problems of theMathematical Theory~*o? Elastl'cTty. P. Eoor'dhoff Ltd., Groningen-Holland, 1953.

11. Muskheltshvill, N. I. Some Integral Equations. P.Noordhoff Ltd., Groningen-fiollab.b, 1953.

12. Sechler, E. E. Elasticity in Engineering.John Wiley & Sons, Inc., 1952.

New York:

7913. Sokolnikoff, I. 3, Mathematical Theory of Elasticity.

Hew York: McGraw-Hill Boo'k Company, Inc . , i960.14. Timoshenko, S., and Goodier, J. N. Theory of Elasticity.

Hew York? McGraw-Hill Book Company, Inc.» i'Jol.'15. Wang, C. Aonlied Elasticity. Hew York} McGraw-Hill

Book Company, Inc., 1^53.16. Williams, M. L. "Stress Singularities Resulting from

Various Boundary Conditions In Angular Corners of Plates in Extension," Joumal Applied Mechanics,24. 1952, p. 526.

17. Wilson, H. B., Jr. "Conformal Transformation of a SolidPropellant Grain with a Star-Shaped Internal Perforation onto an Annulus," American Rocket Society. August, I960, p. 7o0.

18. Wylie, C. R., Jr. Advance Engineering llathematlcs.Hew York} McGraw-nill Boole' Company,r tnc., "I'$51.

A P P E N D I X

APPENDIX A

THE DYNAMICS PROBLEM

Since stresses are related to strains, which are In turn related to displacements, stresses can be solved for in terms of displacements. (See 3)

<Tx

cr- =

K i - •£>) ^ u.

x b\± + M

61 -- A b ’LLA*

A

+ + Abi

I - 6) b\r A b wiS bij bl

bv~ , A( 1+&) b»J& ~Jz

(la)

(lb)

(lc)

(ld)

(le)

(lf)

By appropriately differentiating and substituting into (1), now expressions for the equations of motion are obtained in terms of displacements only.

/ c> V cfir blir

82

(2b)

^2Z ij I_ y^i

y J z / f ° ~ J i I (2c)

In vector fora equations 2 become

GV'u. + (G*/) V ( V ■ u) = (3)

whereu - i l l * j V" + k W

Prom the Helmholtz Theorem of Vector Analysis, which statesthat any vector field can be expressed as the sun of thediveryance of a scaler quantity plus the curl of a vectorquantity and, since the displacements of a body form avector field, the displacement vector can be written as

U. -- W + V * H (4)

^ e r e H -- LH* - ] Hy 4- k Hz<j> - Scalar

This defines three functions of position In terras of four unknown functions; one for the scaler potential and three for the components of the vector potential. However, anadditional requirement of the Helmholtz Theorem is that

V • H = o

83

(5)

which eliminates one of the redundant functions*Continuing by substituting (4) into (3)

G V ^ W + V ^ h ) + (G*4) v ( V - W + V'V^H) = ■*- v 'tfl)(6)

Prom the vector identity

V ZA = V(tf-A) - (7)

+ V*- h) - SJ (V- \7^ + V*V x H) ~ V* (V x V<*>) + (8)

- V* *7 * (V* h)

From the derivation of Helmholtz’s Theorem,

7 * = o (9a)

V • V* H = o (9b)

Simplifying (6), noting in cartesian coordinates 7-V^=j y 2<

G v(7*0) ~ V* v *(V*H) + (G^)V(V(Z*J - - ^ ( v ^ 7 x h ) (a.0)

Considering V* H as a single vector A and using the identity of (7),

V*V'x(v*n)= y(y>y*H) - v z(yxR)

84

(11)

Using (9b) and (11), (10) simplifies to

(A+2G)V(vV) + G V 2(V«H) =at (12)

Combining like terms,

V (a *2g)v V V* 6 \7ZH -r m * o (13)

For this equation to be satisfied, the vectors represented by each terra of tills expression must either both be zero or they must be equal and opposite. Since they are not in gen­eral equal and opposite, both must be zero, and thus what is inside of the brackets must be zero.

w c,z

V*H - c 11 T t 7

whereA + 2.6

(14)

(15)

These are wave equations, and to attack a dynamics problem sinusoidal inputs would be assumed for and H .

85

$ - e Luot (16)

14 z i4 e(17)

Equations (14) and (15) become

(IS)

V 2H ‘- -C*u3*R (19)

The time dependency has been eliminated, but there remains a variable on the ri$it side of the equation and at present there are no known methods of expressing this equation in terms of general complex functions. Functions, other than the bUiarmonic, can be formulated in a manner similar to the one formulated by Goursat. (5)

A method of handling the Poisson’s Equation with complex variables presents an avenue of work that would be a significant contribution to the field of elasticity.

APPENDIX B

CONJUGATE FUNCTION NOTATION

I t i s a p p r o p r i a t e t h a t some e x p la n a t io n be made a t

t h i s p o in t c o n c e rn in g th e u se o f th e f u n c t i o n a l n o t a t i o n

in v o lv in g th e c o n ju g a te . To d e m o n s tra te t h i s , assum e a

g e n e r a l p o ly n o m ia l i n z ,

■fU) - A z v 6zz + Cz3 + . • •

forming the conjugate of both 3ides,-f(z) : Az +- fez* + Cz5 - A z + fez1 + Cz3

•f(z) - Az + Bzl Cz3T hus, i t can be se en t h a t f ( z ) i s a f u n c t io n o f Z and

t h e r e f o r e ,

1 Mzl Z

I t i s n o t ic e d t h a t

f(z) = A z + fez1 +- Cz 3

A n o th e r d e f i n i t i o n c o n s i s t e n t w ith th e p r e c e d in g ,

f(z) : Az + fez1 + Cz*T h e r e fo r e , l e t

Hl) -- Hi)t o I n d i c a t e t h a t -fCz)is a f u n c t io n o f Z •

.ABSTRACT

An analysis was made of an infinite right-circular cylinder with an irregular shaped cavity to determine the stress distribution throughout the body. The body is assumed to be in a state of constant acceleration, which Includes, as a special case, the statics problem where the forces are gravitational.

The constant acceleration problem was solved by mak­ing the assumption of plane strain and employing I^Alembert’s principle, where the acceleration terms of the equations of motion are expressed as body forces in the general elasticity equations. The body forces are included In the solutions by expressing the stresses In terms of two potential functions: (1) The Alry-Stress Function and(2) A Body-Force Function. By adopting a principle devel­oped by M. Biot, the body forces are removed from the problem by altering the stresses at the boundary. The resulting stress function is represented in the complex plane by complex functions using the methods of Goursat and Muslchelishvili.

Due to the complexity of the boundary of the cavity, the configuration is conformally mapped onto a circle where the boundary conditions can be conveniently handled. The solution is completed by assuming series representations for the analytic complex functions in the transformed plane, and

The analytic complex functions In the transformed plane, and the unknovra. constants are evaluated using the boundary conditions.

A sample solution for a typical problem Is obtained in general functional notation to illustrate the manner in which the method can be applied and to provide a pattern for future applications to similar problems.

Approved: