Complex variable solutions of elastic tunneling problems

TU Delft

Interfacultaire werkgroep

Gebruik van de Ondergrondse

Ruimte (GOR)

Cob

Pos.^bus 5048 ^ ^ 2600 GA DELFT L O JS

COMPLEX VARIABLE SOLUTIONS

OF ELASTIC TUNNELING

PROBLEMS

A. Verruijt

Technische Universiteit Delft Faculteit CiTG

Bibliotheek Civiele Techniek Ste vinweg 1

2628 CN Delft

COMPLEX VARIABLE SOLUTIONS

OF ELASTIC TUNNELING

PROBLEMS

Arnold Verruijt August 1996

B'bliothesk Facuuoit der Civiele Techniek

(Bezoekadres Stevinweg 1) Postbus 5048 ?600GA DELFT

1. Introduction

In this report it is investigated whether certain problems of stresses and de-formations caused by deformation of a tunnel in an elastic half plane can be solved by the complex variable method, as described by Muskhelishvili (1953). The geometry of the problems is that of a half plane with a circular cavity, see figure 1.1. The boundary conditions axe that the upper boundary of the half

+2^

•- X

Ö

Figure 1.1. Half plane with circular cavity.

plane is free of stress, and that the boundary of the cavity undergoes a certain prescribed displacement, for instance a uniform radial displacement (the ground loss problem) or an ovalisation.

In the chapters 2 - 6 of this report the complex variable method for the solution of elasticity problems is recapitulated, and some simple examples are elaborated. These include problems for a continuous half plane and problems for a circular ring. By combining the techniques used in these chapters the actual problem of the half plcuie with a circular cavity can be solved, starting in chapter 7. Chapter 7 describes some properties of the conformal transformation. Chapter 8 contains the main derivations of the complex equations appearing in the boundary conditions. In this chapter the consequences of the stress-free boundary at the ground surface are investigated, and the basic equations are given for the case that the stresses aie prescribed at the boundary of the circulsu-cavity. In chapter 9 the problem for the Céise of a prescribed displacement at the boundary of the circular cavity is solved. This solution is elaborated in chapter

10. A computer program to validate the solution is decsribed in chapter 11. It should be noted that in the classical treatises of Muskhelishvili (1953) and Sokolnikoff (1956) on the complex variable method in elasticity, the problems studied here are briefly mentioned, but it is stated that "diflaculties" arise in the solution of these problems, and it is suggested to use another method of

solution, such as the method using bipolar coordinates. It is the purpose of this

report to show that these "difficulties" can be surmounted.

In this report two elementary problems are considered in detail. These are

the problem of a halfplane with a circular cavity loaded by a uniform radial

stress, and the problem in which a uniform radial displacement is imposed on

the cavity boundary (this is usually called the ground loss problem). In a later

report it is planned to consider Mindlin's problem of a circular cavity in an

elastic half plane loaded by gravity.

The results of the ccilculations are shown in graphical form in chapters 10

cind 13, which may be of particular interest for tunnel engineering. The results

are also available in the form of a diskette containing two MS-DOS programs

(TUNNELl and TUNNEL2) which will show numerical or graphical results on

the screen.

R e f e r e n c e s

H. Bateman, Tables of Integral Transforms, 2 vols., McGraw-Hill, New York, 1954.

E.G. Coker and L.N.G. Filon, Phoioelasticity, University Press. Cambridge, 1931.

L.N.G. Filon, On a quadrature formula for trigonometric integrals, Proc. Roy.

Soc. Edinburgh, 4 5 , 356-366, 1966.

A.E- Green and W. Zerna, Theoretical Elasticity, Clarendon Press, Oxford,

1954-W. Gröbner and N. Hofreiter, Integraltafel, Springer, Wien, 1961.

G.B. Jeffery, Plane stress and plane strain in bipolar coordinates. Trans. Royal

Soc, series A, 2 2 1 , 265-293, 1920.

G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and

Engi-neers, 2nd ed., McGraw-Hill, New York, 1968.

E. Melan, Der Spannungszustand der durch eine Einzelkraft im Innern bean-spruchten Halbscheibe, ZAMM, 12, 343-346, 1932.

R.D. Mindlin, Stress distribution around a tunnel, Trans. ASCE, 1117-1153, 1940.

N.I. Muskhelishvih, Some Basic Problems of the Mathematical Theory of

Elas-ticity, translated from the Russian by J.R.M. Radok, Noordhoff,

Gronin-gen, 1953.

C. Sagaseta, Analysis of undrained deformation due to ground loss,

Géotech-mque 37, 301-320, 1987.

I.N. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951.

I.S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd ed., McGraw-Hill, New York, 1956.

S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, 2nd ed., McGraw-Hill, New York, 1951.

A. Verruijt, Stresses due to Gravity in an Elastic Half-plane with Notches or

Mounds, Ph.D. Thesis Delft, Van Soest, Amsterdam, 1969.

A. Verruijt and J.R. Booker, Surface settlements due to ground loss and oval-isation of a tunnel, Géotechnique, to be published, 1996.

2. Basic equations

In this chapter the basic equations of plane strain elasticity theory are presented,

using the complex variable approach (MuskheUshvih, 1953).

2.1 Plane strain elasticity

Consider a homogeneous linear elastic material, deforming under plcine strain

conditions. In the absence of body forces the equations of equilibrium are

^ + ^ = 0. (2.2)

The stresses can be expressed in the displacements by Hooke's law,

. „ = 2 p — + A ( — + ^ ) , (2.3)

. , , = 2 M ^ + A ( ^ + ^ ) . (2.4)

(Txy — 0"yx

,dux dUy. , . , .

= "(37^ IT)- (=^-»>

Substitution of eqs. (2.3) - (2.5) into the equations of equilibrium (2.1) and

(2.2) gives

, V V + (A + ri|(^ + ^ ) = 0 , (2.6)

. V ^ ^ + (A + , ) | ( ^ + ^ ) = 0. (2.7)

These are the equations of equilibrium in terms of the displacements.

It follows from (2.6) and (2.7) that

Furthermore it follows from (2.3) and (2.4) that

< r „ + . . „ , = 2 ( A + M ) ( ^ + ^ ) . (2.9)

Thus it follows that

2.2 Airy's function

It follows from (2.1) that there must exist a single-valued function B{x, y) such that

c . „ = — , 0-,. = - — . (2.11) Similarly, it follows from (2.2) that there must exist a single-valued function

A{x,y) such that

dA dA .^^^,

' ' ' ^ " " • ^ ' ' ' ' ' " ^ - ^ ^ - ^ ^ ^

Because a^y = o'yx i^ follows that

S=^-This means that there must exist a single-valued function U such that

The stresses can be expressed in the function U, Airy's stress function, by the relations

d'U d'U d^U . _ , .

''"-~d^' ^ y ^ - - ^ ^ ' ^ ^ ^ = 0 ^ - ^^-^^^

With (2.10) it follows that Airy's function must be biharmonic,

V^V^U = 0. (2.16)

In the next section a general form of the solution will be derived, in terms of complex functions.

2.3 T h e Goursat solution

In order to solve eq. (2.16) we write

V^U = P. (2.17)

Because U is biharmonic the function P must be harmonic,

V 2 p = 0. (2.18) The general solution of eq. (2.18) in terms of an amalytic function is

where ƒ is an analytic function of the complex variable 2 = x -f iy. We will

write

Q = Im{/(z)}, (2.20)

so that

fiz) = P-^iQ. (2.21)

Because ƒ (2) is analytic it follows that the functions P and Q satisfy the

Cauchy-Riemann conditions,

dP^^dQ dP^ _dQ

dx ~ dy' dy ~ dx' ^ • )

A function 4>{z) is introduced as the integral of f{z), apart from a factor 4, so

that

dz ~ ^'

= i / ( ^ ) - (2.23)

The function <f>{z) is also an analytic function of 2. If we write

(i){z)=p-biq, (2.24)

it follows that

Thus, using the Cauchy-Riemann conditions for p and q,

We now consider the function

F=U~ ^z4>iz) - \z4>{z), (2.27)

or

F=U -\{x- iy){p + iq) - \{x + iy){p - iq) = U -xp-yq. (2.28)

Taking the Laplaciaii of this expression gives

V^F = V^U - x V V - yV^q - 2 ^ - 2^. (2.29)

Because p and q are the real and imaginary pzirts of an analytic function their

Laplacian is zero. Thus, using (2.26),

V-F = \/^U -P. (2.30)

V^F = Q. (2.31)

This means that we may write

F = Ke{x{z)} = iixiz) + x(^)}, (2.32)

where xi^) is another analytic function of z. The imaginsiry part of xi^) will

be denoted by G, so that

xiz) = F-hiG. _ (2.33)

From (2.27) and (2.32) it follows that

2U = z<f>{z) + zé{z) + x{z) + Xiz), (2.34)

or

U = Re{zó{z)-\-xiz)}. (2.35)

This is the general solution of the biharmonic equation, first given by Goursat.

In the next sections the stresses and the displacements will be expressed into

the two functions <f>{z) and

xi^)-2.4 Stresses

The stresses are expressed in the second derivatives of Airy's function U. First

the first order derivative of U will be determined. The starting point is equation

(2.28), in the form

U = F + xp + yq. (2.36)

Here F, p and q are harmonic functions, and p and q are complex conjugates.

Partial differentiation gives

dU .dU dF .dF . .dp .dp. . ,dq .dq.,^^^,

j^^'^=j^^'i^-''^''^'^d^^%^^'y(d-y-'iy^'-''^

The first two terms in the right hand side of eq. (2.37) can be expressed in the

function xi^) by noting that

dx dF .dG dF .dF , ^

dz dx dx dx dy

so that

dF dF

where

dx(z)

^ ( ^ ) = ^ - (2-40)

The third and fourth terms in the right hand side of eq. (2.37) together just

form the function é{z), see (2.24).

The last terms in the right hand side of eq. (2.37) can be expressed in the

function éiz) by noting that

dé dp .dq dp .dp

dz ox ox ox oy

or

^ = |i + i|i = |ï_.f. (2.42)

dz dy dx oy dy

From these equations it follows that

dp .dp. . fdq .dq

. * ' ( . ) = . ( ^ + e ^ ) + : v ( ^ - i ^ ) . (2.43)

All this means that eq. (2.37) can be written as

dU .dU

^ + i ^ = <?i(z) + z<t>'{z) + ^{z). (2.44)

In order to obtain expressions for the second order derivatives oiU, the quantity

dU/dx -f- idU/dy is differentiated with respect to x and y. First differentiation

of eq. (2.37) with respect to x gives

dHJ_ .d^U _ d^ .d^F .dp .dq_. .dp .dp.

dx^ dydx dx^ dydx dx dx dx dy

.d^p . d^p . . . d^q .d^q. ,^ , ,

Secondly, differentiation of eq. (2.37) with respect to y gives, after multiphcation

by - z ,

d^U .a^U ^d^F .d^F dq .dy ^ dq .dq

dy"^ dxdy dy^ dxdy dy dy dy dx

In these equations the first two terms can be expressed into the second derivative

of X, that is the first derivative of ^, by noting that

d^X -^

^ = — = 312- - '^Z^. = - ^ 3 - - ^ ^ T ^ ' (2-47)

so that

j ^ + iirsr. = *'(-). (2.48)

dip d^F . d^F

dz dx"^ ' dxdy

d^F

d^F

dy^

.d^F

'dxdy'

and

I^-'JIE^- -''^'^- (2.49)

The terms 3-6 in eqs. (2.45) and (2.46) add up to 2dp/dx, respectively 2dq/dy

because dp/dy = —dq/dx, and using (2.41) and (2.42) these can be written as

dp dq

2^ = 2-^ = é'{z) + é'{z). (2.50)

Finally the last terms in eqs. (2.45) and (2.46) can be expressed into the second

derivative of é by noting that it follows from differentiation of (2.41) or (2.42)

that

fé^d^_ .d^ ^ ÜL + i ^ (2 51)

dz^ dx^ dxdy dxdy dx"^'

or

^ = _ ^ _ 2 _ ^ = _ ^ _ i ^ (2 52)

dz"^ dy"^ dxdy dxdy dy"^

From these equations it follows that

and

cf^V •d^'p. . . d^q .d^q.

Substitution of all these results into (2.45) and (2.46) gives, finally,

)2rr a2

d^U . d^'U

dx^ dydx

and

d^U . d^U

dy"^ dxdy

+ i^-^ = zé"{z) + é'{z) -f- é'{z) + rP'{z), (2.55)

= -zé"{z) + é'{z) + é'{z) - rp'{z), (2.56)

It follows, finally, by using the expressions for the stresses in terms of Airy's

function, and by adding and subtracting the two equations (2.55) and (2.56),

that

(Txz + ^yy = 2{<?i'(2) + é'{z)}, (2.57)

<7yy - ^xr + 2iaxy = 2{zé"{z) + ip'{z)]. (2.58)

2.5 D i s p l a c e m e n t s

In order to express the displacement components into the complex functions é and i) we start with the basic equations expressing the stresses into the dis-placements, see (2.3) and (2.4),

^ dux , / dux duy.

a „ = 2 ; . - ^ + A ( ^ + ^ ) . (2.59)

„ duy , / dux duy. . . .

- , » = 2 ^ ^ + A ( ^ + ^ ) . (2.60) Addition of these two equations gives

<.„ + , . „ = 2 ( A + ^ ) ( ^ + ^ ) . (2.61) With (2.15) and (2.17) we have

axx+(Tyy=V^U = P, (2.62)

so that we may write

^ dux d^U X ^

''-d^ = w- 2(xT;r/^ ^'-''^

^ duy d^U X ^

Because V'^U — P these equations may also be written as

dx dx^ 2{X -\- n) ^ '

„ duy d^U X-\-2fx ^

According to (2.26) the quantity P may also be written as 4dp/dx or Adq/dy. This gives

dux _ mj_ 2{X + 2y)_dp

^^^-'T^^ x^a Yx' ^^-^^^

^duy_ d^U 2(A + 2fs) dq

The two equations have now been obtained in a form in which in the first equa-tion all terms contain a partial derivative with respect to x, and in the second equation all terms contain a partial derivative with respect to y. Integration gives

2,„. = - f ^ + 2 ( A ^ ^ ^

OX A -h //

2.«, = - f + ^ ( ^ , + ,(.). (2.70)

where at this stage /(y) éind p(x) are arbitrary functions. In order to further

determine these functions we use the expressions for the shear stress üxy

Dif-ferentiation of (2.69) with respect to y, of (2.70) with respect to x, and addition

of the results gives

2 , ( ^ + ^ ) = _ 2 | ^ + a(A±M(|p + | i ) + ^ + ^ . (2.n)

^ dy dx ' dxdy X-\-n ^dy dx' dy dx

Because p and q are complex conjugates it follows that dp/dy + dq/dx = 0.

Furthermore it follows from (2.5) and (2.15) that

2''(t-^) = ^^- = -='S- (-^'

Comparison with (2.71) shows that

df da

•j- + -T- = 0. (2.73

dy dx

The first term is a function of y only, and the second term is a function of x

only. This means that the only possibihty is that

df da

where £ is an arbitrary constant, and the factor —2^1 has been included for future

convenience.

It follows from (2.74) that

f = 2ix{a-ey), g = 2fi{b-\-2tx£), (2.75)

where a and b axe two more arbitrary constants.

Substitution of (2.75) into (2.69) and (2.70) gives

2fxux = - ^ + ^ ^ I T T ^ ^ + 2M(a - sy), (2.76)

2fiuy = ~ + ? ( A ± ^ ^ ^ 2/z(6 + EX), (2.77)

Oy A -'T fi

The last terms in these expressions represent a rigid body displacement, of

magnitude a in x-direction, b in y-direction, and a rotation over an angle s. If

this is omitted, on the understanding that when necessary such a rigid body

displacement can always be added to the displacement field, we may write

2 M ( . . + i . . ) = - ( f + i f ) + ^ ^ ( p + iq). (2.78)

The functions p-\-iq together form éiz), see (2.24). With (2.44) it finally follows

that

2fi{ux + iuy) = Ké{z) - zé'{z) - ^(z), (2.79)

where

K = 4 ^ ^ = 3 - 4i/. (2.80)

Equation (2.79) was also derived first by Kolosov-Muskhelishvili.

It may be noted that for plane stress conditions the same equations apply, except that in t h a t case the coeflacient K has a different value,

5A-K6^ 3 - i / , „ „ , ,

" = 3AT2;r = ÏTZ- (2-81)

2.6 B o u n d a r y c o n d i t i o n s

The solution of a certain problem is determined by the boundary conditions. These may refer t o the displacements, or to the surface tractions. In the first case the boundary condition can easily be interpreted in terms of the quantity Wx + iwj,, as given by eq. (2.79). In the second case it is sometimes conve-nient to specify the boundary conditions in terms of o-„ — icxy or in terms of

<Tyy-\-iayx, whlch can immediately be expressed in the stress functions é{z) and

•é{z) through the relations (2.57) and (2.58). This is especially convenient for

horizontal and vertical boundaries. For straight boundaries under a constant angle it may be convenient to use the transformation formulas

Ux' + +iuy> = {ux + iuy)exp{-i9), (2.82)

Cx'x' + <''y'y' = O'rx "t" ^yy > (2.83)

ay'y' — Cx'x' + 2i(Tx'y' = {<Tyy — (Txx + 2zo-3:j,)exp(2fö), (2.84)

where B is the angle over which the axes x and y must be rotated to coincide with x' and y'.

In the more general case of a curved boundary, see figure 2.1, it is more con-venient t o derive a formula in terms of the integral of the surface tractions. Let the boundary condition be that the surface tractions tx and ty are prescribed, as a function of a coordinate s along the boundary (such that the material is to the left). We may write

tx = Cxx cos a -f (Tyx sin a, (2.85) ty = Cxy cos o; -f- o-yj, sin a. (2.86)

Figure 2.1. Boundary condition.

Because along the boundsu-y dy = ds cos a and dx = —dssina, and the stresses

can be related to Airy's stress function U through the equations (2.15) it follows

that

_d^dy d^U dx _ d .dU.

dy"^ ds dxdy ds ds dy

_ d^dx d^U dy _ _d .dU.

(2.87)

(2.88)

^ dx^ ds dxdy ds ds ^ dx

These two equations can be combined into a complex equation

If the boundary traction is integrated along the boundary, and this integral is

defined as

F = Fi-\- iF2 = i I {tx + ity)ds, (2.90)

where SQ is some arbitrary initial point on the boundary, then we may write

(2.91)

where C is some arbitrary constant of integration. With (2.44) this gives, finally,

Fi + iF2 -\-C= é{z) + zé^ + ï^. (2.92)

This means that the integral of the surface tractions defines the combination of

functions in the right hand side.

^ .^ ^ dU .dU

F i + 2 F 2 + C = ^ - + Z - ^ ,

2.7 R e c a p i t u l a t i o n

The formulas can be recapitulated as follows. The solution can be expressed by two analytic functions é{z) and ip{z). The stresses are related to these functions by the equations

axx + ffyy = 2{é'{z) + é'iz)}, (2.93)

^yy - (Txx + 2i<T^j, = 2{zé"(z) + Xp'(z)}. (2.94)

2fxiux + iuy) = Ké{z) - zé'{z) - xP{z), (2.95)

where for plane strain

K = ^ = 3 - 4f/. (2.96)

A-H/i and for plane stress

5A -I- 6AZ 3 - 1/

' = ÜTT, = TTZ- (2-9^)

The integral of the surface tractions, integrated along the boundary, is related to the analytic functions by the equation

Fi -f- zFz + C = é{z) + zé'{z) -f- tP{z). (2.98)

The techniques to determine the complex functions éiz) and rpiz) from the boundary conditions will be demonstrated in the next chapters.

The two basic problems of the mathematical theory of plane strain elasticity are that along the entire boundary either the surface tractions or the displace-ments are given. In the first case the function F is given along the boundary, cind the functions éiz) and Tp{z) must be determined from (2.98). In the second case the function ^{ux + iuy is given along the boundary, and the functions éiz) and tpiz) must be determined from (2.95). It may be noted t h a t these equations are very similar (they differ only through the factor K), SO t h a t the methods of solution may also be very similar.

It may also be noted that the addition of an sirbitrary constant value to the two functions éiz) and ipiz) does not affect the stresses, but leads to an addi-tional homogeneous displacement. This may represent an arbitrary rigid body displacement of the field as a whole. In the case of a simply connected region, with a single boundary, with the surface tractions defined at the boundéiry (this is the first boundary value problem), the displcurements are determined up to an arbitrary constant. The constant C in (2.98) then may be taken as zero, without loss of generality, and provided t h a t it is remembered that a rigid body displacement can be added to the displacement field. In the case of a multiply connected region, when there are several disjoint boundary segments, the inte-gration constant C may be taken equal to zero along one of the boundaries, but must be left as an unknown value on the remaining boundaries.

3. Solution of boundary value problems

In this chapter the general technique for the solution of boundary value prob-lems for simply connected regions, in particular regions that can be mapped conformally onto a circle (such as a half plane) are discussed. In later chapters the theory will be applied to multiply connected regions, with circular bound-aries (a ring) and to problems for the half plane with a circular hole. Many of the solutions have been presented also by MuskheUshvili (1953) and Sokolnikoff (1956).

3.1 Conformal m a p p i n g onto t h e unit circle

Suppose that we wish to solve a problem for an elastic body inside the region

R in the complex 2-plane. Let there be a conformal transformation of R onto

the unit circle j in the ^-plane, denoted by

2 = u;(C). (3.1) We now write

éiz) = éiu^i(:)) = MO, (3.2) rP{z) = xPiu;iO) = MO, (3.3)

where the symbol * indicates that the form of the function é- is different from that of the function é- The derivative of é is

3.2 Surface traction b o u n d a r y conditions

If the points on the boundary in the C-plane are denoted by cr = exp(zö), the boundary condition for a problem with given surface tractions can be written as follows, starting from eq. (2.98),

F{<7) + C = é*i<T)+ ^i<T)t^ + Ma), (3.5)

uj'ic)

or, omitting the symbols *,

UJ

i-)^,

F(o-) -h C = éi<r) + ==é'i<T) + rPi<^). (3.6)

u;'i(T)

It is now assumed that the integration constant C = 0, and that the boundary function F(cr) can be represented by a Fourier series

oo oo

Fi<T) = Fie)= J2 Atexi>Xike)= ^ Ak(7', (3.7)

ib= —oo ifc= —oo

where the coefficients Ak can be determined from the Fourier inversion theorem, 1 /-^^

Ak = — Fie)exp{-ike)de. (3.8)

that they may be expanded into Taylor series,

oo <^(C) = ^ a i f c C ' , (3.9) ib=i oo

^(C) = X^6fcC'. (3.10)

Here it has been assumed that 0(0) = 0, which can be done without loss of generality, because it does not affect the stresses, and it has already been as-sumed that the displacements are determined apart from some arbitrary rigid body displacement.

The boundary condition can be written as

f ; A , ^ ^ = f ; a . ^ * - f ^ f ; A : a , ^ - ^ + ^ + f ; 6 , ^ - ^ (3.11) where it has been used that a = expi—iO) = a~^. The coefficients a^ and

èjfc have to be determined from this equation. The difficulties associated with this problem can best be investigated in successive steps, by considering various examples.

3.3 D i s p l a c e m e n t b o u n d a r y c o n d i t i o n s

As mentioned before the problem with given boundary values for the displace-ments is very similar to the problem with given surface tractions. Actually, if the displacements are given the basic equation is eq. (2.95). If the given quantity 2n{ux +2Uj,) along the boundary is denoted as G(o-), and this is again represented by a Fourier series,

oo

2/i(u^-t-iuy) = G(o-)= ^ Bkc\ (3.12)

ifc=-oo

the system of equations will be, in analogy with (3.11),

CO CO ^ X o o CO

^ B^a' = KY^a^a' - ^^Y.^ak<r-'+' -Y^b^a-K (3.13) The coefficients ak and 6jb must be determined from this equation.

4. Problems for a circular region

In this chapter some problems for a circular disk are discussed. This is the simplest possible type of problem.

For a circular region, of radius R, the mapping function is

z=u;{0 = RC, (4.1)

so that

a^'(C) = R- (4.2) In this case we have

^ = ^. (4.3)

uj'ia)

4.1 Surface traction boundary conditions

If the surface tractions are given along the boundary, the boundary condition is

f ; A^a' = Y.aka' + f2kaka-'^' + f2~^t<T-', (4.4) or J t = - o o ib=l k=l k=0 oo oo oo oo ^ A,(7* = ^ a f c 0 - * + ^ ( A : - h 2 ) a , + 2 ^ - * + ^ 6 f c ^ - ' , (4.5) A : = - o o i b = l fc=-l Jb=0 or c» oo oo

Y^ AiO-* = ^aifcO'*-haia-f-aicr + ^[6ifc-h(^-h2)afc+2]o-^ (4.6)

it=-oo k=2 ib=0

In the right hand member the various terms have now been grouped together such that each term apphes only to a single power of a. By requiring that the coefficients of ail powers of <r must be equal in the left and right hand members the coefficients can be solved successively, starting with large positive powers of

a, and then going down to large negative powers of c. The result is

ak=Ak, -fc = 2 , 3 , 4 , . . . , (4.7)

ai = ^Ai, (4.8)

6jk = A^k -ik + 2)ak+2, k = 0,1,2,.... (4.9) To derive eq. (4.8) it has been assumed that Ai is real. This can be shown to

be equivalent to the condition that the resulting moment on the body is zero. Furthermore the imaginary part of ai has been set equal to zero, for definiteness.

4 . 2 D i s p l a c e m e n t b o u n d a r y c o n d i t i o n s

For the second boundary value problem, with given displacements, the system of equations can be established in a similar way, starting from eq. (3.13). The result is

oo oo oo

Y ^jkO-* = /c^ai<7*-H/caio--aiö'-^[èifc-{-(^-|-2)ajfc+2]o- *". (4.10)

ib = - o o _{k=2 k=0}

If the coefficients Bk are known, the coefficients a^ and bk can be determined from this equation. In this case the solution is

Bk Gjb = — , k = 2 , 3 , 4 , . . . , K KBI + Bi bk = -B.k -ik + 2)afc+2, k = 0,\,2, (4.11) (4.12) (4.13) 4.3 E x a m p l e s 4.3.1 E x a m p l e 1: Uniform t e n s i o n

As a first example consider the simple case of a circular region under uniform tension, see figure 4.1. This is a stsmdard problem from the theory of elasticity. In this case the surface tractions are tx =i cos6 and ty = t s i n ^ , so that

* - X

Figure 4.1. Circle under uniform tension.

or

F = tR(7. (4.15)

An eventual constant integration factor has been omitted, on the understanding that this will only affect the value of ^(0), and can be incorporated into the rigid body displacement. The Fourier series representation of the function F((T) is very simple in this case,

Ai=tR, (4.16)

with all other coefficients Ak being zero. We now obtain from eqs. (4.7)-(4.9)

ak = 0, Ar = 2 , 3 , 4 , . . . , (4.17)

ai = \tR, (4.18) èfc = 0, ^ = 0 , 1 , 2 , . . . . (4.19)

Hence the functions éiO s-iid ^(C) a-re

éiQ = \tRQ, (4.20)

V'(C) = 0. (4.21) Because z = R^ it follows t h a t

Hz) = ^tz, (4.22) viz) = 0. (4.23)

The stresses are, with (2.93) and (2.94),

axx+<^yy=2t, (4.24)

o-yy - CTxx + 2i<rxy = 0. (4.25) Hence

axx = t, Cyy =t, (Txy = 0. (4.26)

This is the correct solution of the problem, with a constant isotropic stress in the entire disk.

The displacements are, with (2.95),

2fiiux •¥iuy) = {l- 2i/)tz -I- constant. (4.27)

The constant can be assumed to be zero, if the origin is assumed to be fixed. Thus

l - 2 i /

Ux = - ^ i ^ ^ (4-28)

l - 2 r /

uy = ^ ^ * y - (4.29) These are also well known formulas for the displacements in a disk under

con-stant stress. It may be noted that the coefficient 2/i/(l —2z/) may also be written as 2(A-|-/i).

4 . 3 . 2 E x a m p l e 2: Uniform s t r e t c h i n g

As an alternative we may consider the case that the boundary of the circular region undergoes a uniform radial displacement. In this case the boundary condition is

2 = Rexp{i0) : G = 2niux + iuy) = 2/iUoexp(fö), (4.30) or

G(o-) = 2nuoa. (4.31) Equation (4.10) now gives

oo oo

2iiuoo- = « ^ ak(T^ + Kai<7 - 01 (7 - ^ p i -h (^ -f- 2)ak+2]{T~''• (4.32)

/b=2 ib=0

Assuming that ai is real we now find that all coefficients are zero, except 2/z oi = — ^ u o = (A + ii)uQ. (4.33) Hence <?^(0 = (>^ +/^)«oC, (4.34) or <ji(2) = (A + / i ) ^ 2 . (4.35) The other function is zero,

^ ( 2 ) = 0. (4.36) The stresses are now found to be

O-xx = O-yy = 2(A + / i ) ^ , <7xy = 0. (4.37) This solution is in agreement with the previous one, and with the solution known

5. Problems for a half plane

In this chapter elasticity problems for the half plane Im(2) < 0 will be

con-sidered. The region R in the complex 2-pIane is mapped conformally onto the

y

- ^ X

Figure 5.1. Mapping of half plane on unit circle.

interior of the unit circle j in the complex <^-pléine, see figure 5.1. In this case

the conformal transformation is

z = uj{Q = -iY—.

Differentiation with respect to C gives

, 2i

"(«

=

-(wF-On the boundary C = <^ and C = o""^- This gives

^ ( < ^ ) _ 1 1 - 2 = 7 7 = ^ - 2 - 2^^ •

u!'ia)

5.1 Surface traction boundsury conditions

In this case the boundary condition (3.11) is

f ; Aka' = f2^'^^' + iJ2k^>:<r-'^'

(5.1)

(5.2)

(5.3)

- i ^ i b a . c r - ^ - i - h ^ ï . ^ - ^

(5.4)

This can also be written as

oo oo

Y ^k(r^ = Yl^^^^ "^ (^0 + è^i) + (^1 + 02)0-"^

i k = - o o fc=l

0 0

+ I ^ [ ^ * + i(ir + l)a;k+i - \{k - l)afc_l]a-^ (5-5)

fc=2

Because now in both the left hand and the right hand members all terms have

been arranged in powers of a the coefficients a^ and 6^ céin be determined,

successively. The solution of the system of equations is

ak = Ak, i = 1,2,3,..., (5.6)

6o = A o - i a i , (5.7)

ti = A _ i - a 2 , (5.8)

hk = A.k - \ik + l)afc+i + \ik - l)afc_i, ^ = 2 , 3 , 4 , . . . . (5.9)

Actually, the expression (5.8) can also be covered by equation (5.9) if this is

considered valid also for A: = 1.

5.1.1 Example: Flamant^s problem

As an example consider the problem of a concentrated point load on a half plcuie

(Flamant's problem). In this case the surface y = 0 is free from stress, except at

the origin, where a point load of magnitude P is applied, in negative y-direction,

see figure 5.1. In this case

F^ij(U*iiM^={\ llll (5.10)

or, in terms of the coordinate 6 along the unit circle in the ^-plane,

F = l °' ^ ^ < ' ' ' r5in

This function can be expanded into a Fourier series,

0 0

F ( ö ) = Y AkexpiikO), (5.12)

where now

Ak = ^ l exp(-kie)d9. (5.13)

The result is

Ao = | F , (5.14)

Ai = ^ , k = ±l,±Z,±b,..., (5.15)

Ak = 0, k = ±2,±A,±6,.... (5.16)

We now find, from (5.6) - (5.9),

0* = ^ , A: = 1,3,5,..., (5.17)

ajk = 0, ^ = 2 , 4 , 6 , . . . , (5.18)

èo = è ^ - ^ , (5.19)

6;b = ^ , A:= 1,2,3 (5.20)

bk = 0, k = 2,4,6,.... (5.21)

If we disregard the constant 6o, which can always be corrected by adding a rigid

body displacement, and which does not affect the stresses, we have

^(C) = V E T- (5-22)

^(c) = ^ E f - (^-23)

Jfc=I,3

A well known series is

l n j ^ = 2{C + Y + y + ---}> (5.24)

so that

Because {1 -\- Q/il — Q = iz it now follows that, apart from a constant factor,

iPiz)=—\nz, (5.28)

The derivatives are

*'(^) = ^ = 5 ê = ' ' P ( - ' * ' ' (5-29)

^"(z) = - £ j = - £ j e x p ( - 2 i « ) . (5.30)

^ ' ( ^ > = ^ = 5^^^P(-'*)- (5-21)

The Kolosov-Muskhelishvili expressions for the stresses now give

2F .

(Txx + o-yy = 2{<j^'(2) + (^'(z)} = — sin 6, (5.32)

Trr

ö^yy - 0-xx + 2f0-xy = 2{zé"{z) -j- Xp'{z)}

2P

= — [sin ^(sin^ 6 - cos^ 6) -\- 2i cos 6 sin^ 6]. (5.33)

From these it follows that

2P

(Txx = — sin Ö cos^ 9, (5.34)

2P

(Tyy = —sin^Ö, (5.35)

2F

axy — —COS Ö sin Ö. (5.36)

7rr

6. Problems for a circular ring

In this section we will consider an elastic circular ring, under the influence of

surface tractions or prescribed displacements along the the inner and the outer

boundary, see figure 6.1. The radius of the outer boundary is R, and the radius

Figure 6.1. Circular ring.

of the inner boundary is aR, where a < 1.

6.1 Surface traction boundary conditions

Let us first consider the case that along both boundaries the surface tractions

are prescribed, and that along both the loading function F can be represented

by a Fourier series. We then have

|C| = 1: F = f ; Ak<r\

Jfc=-00

oo

iCi

= a:

F=

Y

^*^*- (6-2)

* : = - o o

Here it has been assumed that the ring in the 2-plane has been mapped

confor-mally onto a ring in the (^-plane, such that the outer radius of the ring in the

C-plane is 1.

The complex stress functions éiC) ^^^ ^(C) are analytic throughout the

ring-shaped region in the C-piane. It is assumed that they are also single-valued, so

that logarithmic singularities can be ignored. This means that they can be

represented by their Laurent series expansions,

oo 00 ifc=i fc=i oo oo

^(0 = co+Yl ^^'^^+12 ^^^"'' (6-4)

(6.1)

(6.3)

The coefficients have been given a different notation for positive and negative

powers of C,, to avoid negative indices. The series expansions will converge up

to the boundaries |C| = 1 and |C| =

cn-The derivative of the function <^(C) is

0 0

é'iO = Y,kak(i'-' -Y.^bkC''-\ (6.5)

In general the boundau-y condition for a given surface traction is given by (2.98),

- M ^

F(Co) + C = <^(Co) + = = <?^'(Co) + V'(Co). (6.6)

where Co is a point on the boundary. The conditions along the two boundaries

will be elaborated separately.

6.1.1 Outer boundary

Along the outer boundary we have Co = <?" = exp(f^), so that Co = o""^. Because

the mapping function is a;(C) = RC, it follows that in this case

==== -(io-a. (6.7)

w'(Co)

With (6.5) the second term in the boundary condition is

^ ^ 7 ( ^ = f ; kuka-'^' - f ; kbk.'^'. (6.8)

'^'(Co) S k=i

This can also be written as

jb=l fc=3

The third term in the boundary condition is

0 0

M. é'iCo) = ai<T-\-2a2 + Tik -h 2 ) a , + 2 ^ - ' - ^ ( A : - 2)6,_2<r^ (6.9)

a;'(Co) ^^

rPiCo) = Co + E c i < 7 - * -h Y '^*^*'- (6-10)

i b = l J b = l

The complete boundary condition now is, with (6.6), and assuming that on this

boundéiry C = 0,

oo oo oo

y ^ AkO-^ = y ^ Qfc<^^ + y ^ fefcg""^ + aio- -h 202 + Co

i b = - o o i t = l Jfc=l

OO oo oo oo

+ 53(^ -h 2)afc+2^'' - E ( ^ - 2)6fc_2(7^ + 12^k<T-^ + 5 ^ i ( 7 ^ (6.11)

Using this equation the coefficients Ck and dk can be expressed into the known

coefficients Ak and the other set of unknown coefficients ak and bk. The result

is as follows.

co = l o - 2 a 2 , (6.12)

Ck = A.k -ik + 2)ak+2 -bk, k=l,2,Z,..., (6.13)

d,=Ai-iai+ai), (6.14)

d2 = A2 — a2, (6.15)

dk = Ak-ak + ik-2)bk-2, Ar = 3 , 4 , 5 , . . . . (6.16)

One half of the unknown coefficients have now been expressed into the other

half.

6.1.2 Inner boundary

Along the inner boundary we have Co = tt<^ = aexp(fö), so that Co = aa~^. In

this case

- '^(Co)

= Co = oc(r. (6.17)

'^'(Co)

With (6.5) the second term in the boundary condition is

/ ^ \ oo 00

'MnCÖ) = f: kaka'^c-'^' - T kbka-'a'^\ (6.18)

'^'(Co) S t^i

This can also be written as

= i é'iCo) = aiacr + 2a2a'' + ^ ( i b -F 2)0^+2^*+^-*

-'(Co)

oo

- X ^ ( ^ - 2 ) 6 f c _ 2 a - * + V . (6.19)

/fc=3

The third term in the boundary condition is

oo oo

V'(Co) = Co -h ^ ClfcQ^O--^ -f ^ 5 f c C t - * = < T ^ (6.20)

J b = l i k = l

The complete boundary condition now is, with (6.6),

oo oo oo

Y ^h(T^ + ^ = 5 Z ^kCt^'a^ -H Y bkCt'^a''' -f aiacr -I- 202»^

oo oo

-f Co + Yi^ + 2)ajt+2a'=+V-* - ^(ifc - 2)ïi_2a-*+V^

Jb=3 oo oo

- h ^ C f c a V - * ^ - f - ^ 5 A a - * ( r \ (6.21)

It is perhaps most convenient to solve these equations again for the coefficients

Ck and dk- The result is

co = Bo + C-2a2a^, (6.22) Ck = B.kQ-^ -ik-\- 2)ak+2a^ - ha-^^, k=l,2,Z,..., (6.23)

di = Bia-{ai + ai)a'^, (6.24)

d2='B20c'^ -a2a^, (6,25) dk = Bkct^ - ata^* -H (fc - 2)6i_2Q^ it = 3 , 4 , 5 , . . . . (6.26)

The coefficients can now be determined successively.

First consider (6.14) and (6.24). It follows from these equations that

. 4 i - ( a i - h a i ) = S i a - ( a i - H a i ) a ^ (6.27) Hence, if it assumed that Im(ai) = 0,

Ai - aBi

From (6.12) and (6.22) it follows that

l o - 2a2 = Bo + C - 202^2. (6.29) Hence

"'^ 2(1-U • («-^O)

From (6.15) and (6.25) it follows that

A2- a2 = B2op- - a2a'*. (6.31)

Hence

A2 - o?B2

^ 2 = i _ ^ 4 • (6-32) It follows from (6.30) and (6.32) that the value of the constant C must be

C = Ao-Bo- ' ' • (6.33)

1 -1- Q-^

The value of the integration constant C appears to follow from the analysis. From (6.13) and (6.23) it follows that

Hence

(1 - a-2*)6i +ik + 2)(1 - a^)ak+2 = A.k - a-*B_fc, ^ = 1 , 2 , 3 , . . . (6.35) Furthermore, it follows from (6.16) and (6.26) that

Ak-ak-\-ik- 2)bk-2 = Bka^ - ata^^ +ik- 2)bk-2a^• (6.36)

Hence

-kil - a-^)bk + (1 - a2*+^)ai+2 = Ak+2 - a ^ + ^ B , ^ ^ , A: = 1 , 2 , 3 , . . . (6.37)

The coefficient bk can be eüminated from (6.35) and (6.37). This gives

kil - a^)iA.k - a-'B.k) + (1 - a-^')iAk+2 - a'^^Bk+2)

(1 - a2*+4)(l _ Q-2t) + kik -h 2)(1 - a2)2

ib = 1,2,3,... (6.38)

ait+2 =

All coefficients a^ have now been determined. The coefficients bk can then be determined from (6.35) or (6.37). The coefficients Ck can then be determined from (6.13) or (6.23), and the coefficients dk can be determined from (6.16) or (6.26). The problem has now been solved in a general form.

6.1.3 Example; Ring u n d e r constant pressures

As an example we will consider the case of a ring loaded by a uniform pressure

P2 along its outer boundairy and a uniform pressure pi along its inner boundary.

Along the outer boundary we then have

tx + ity = -P2expii9), (6.39)

Because along this boundary the length element is ds = Rd9 it follows that

F = i jitx + ity)ds = -p2Rexpii9) = -p2Ra. (6.40) Comparison with (6.1) shows that all coefficients Ak are zero, except

Ai = -P2R. (6.41)

Along the inner boundary we have

tx + ity=piexpii9), (6.42)

Because along this boundary the length element is ds = —aRd9 it follows that

F = i jitx + ity)ds = -piaRexpii9) = -apiRa. (6.43) Comparison with (6.2) shows that all coefficients Bk are zero, except

Bi = -apiR. (6.44)

It now follows that all coefficients ak and bk are zero, except

_ iP2-o^'^Pi)R .- ._.

" ^ - - 2 ( l - a 2 ) • (^-^^^

The constant C appesirs to be zero, from (6.33). The coefficients Ck are all zero

ïiiso, and of the coefficients dk the only non-zero one is

_ ( P 2 - P i ) a ^ f l .„ . . .

^1 - ( 1 - ^ 2 ) • (^-^6)

The complex stress functions now are

^iO='^f^{ (6.48)

Because the conformal mapping function in this case is 2 = i2C it follows that

These expressions are in agreement with the results given by Sokolnikoff (1956),

p. 300.

7. Elastic half plane with circular cavity

In this chapter and the next we will study the problem of an elastic half plane with a circular cavity, see figure 7.1. The upper boundary of the half plane is assumed to be free of stress, and loading takes place along the boundary of the

Mr

IlL

/^x:x:;:\s ,

\^x:;::;:

Figure 7.1. Conformal transformation.

circular cavity, in the form of a given stress distribution or a given displacement distribution.

It is assumed that the region in the 2-plane can be mapped conformally onto a ring in the C-plane, bounded by the circles |Cl = 1 and |Ci = a, where a < 1. The properties of the mapping function will be studied in this chapter.

7.1 T h e i n n e r b o u n d a r y The conformal transformation is

2 = a ; ( C ) =

•la-1 - C ' (7.1)

where a is a certain length. The origin in the 2-piane is mapped onto C = — 1, and the point at infinity in the 2-plane is mapped onto C = ^5 see figure 7.1.

Differentiation of (7.1) with respect to C gives

'-'(C) =

ii-cy

It will be shown that concentric circles in the C-plane are mapped on circles in the 2-plane, and the relation between the depth of the circle and its radius with the parameter a, which is the radius of the circle in the C-plane, will be derived.

C = aexp(z^), (7.3)

where a is a constant, and 9 is a variable. With (7.1) this gives

2aQ; sin 9

^ = l - h a 2 _ 2 a c o s ^ ' ^''"^^

a ( l - c t ^ )

y- n . a 2 - 2 a c o s ^ - ' ^^'^^

It is now postulated that these formulas represent a circle, at depth h, having a

radius r. This means that it is assumed that there exist constants h and r such

that

x2 + (y-H/i)2 = r2. (7.6)

In order to prove this we will demonstrate that dr"^/d9 = 0. This is the case if

£ = 2 . | | . 2 ( . . ^ ) g = 0. (7.7)

This means that

dx/d9 , ^

It follows from (7.4) that

dx_ _ il + a^)2azos9-Aa^

d9 ~ °' (l + c^2_2acosö)2 ' ^"^-^^

and from (7.5) it follows that

dy _ (l-Q^)2Q;sing

'd9~^il + oc^-2oc cos^)2 • ^'^'^^'

Substitution of these two results into (7.8) gives, after some algebraic

manipu-lations,

h = aj-—^, (7.11)

which is indeed a constant, and which also proves that r is a constant. With

(7.6) the corresponding value of r is found to be

2a

r = aj--^. (7.12)

If the covering depth of the circular cavity in the 2-plane is denoted by d, see

figure 7.1, it follows that

d=a^—-. (7.13)

The ratio of depth and cover is

h_ l-t-g^

(f ~ ( l - a ) 2 "

(7.14)

If a —^ O the radius of the circular cavity is practically zero, which indicates

a very deep tunnel, or a very large covering depth. If a —* 1 the covering

depth is very small. For every value of h/d the corresponding value of a can be

determined from (7.14).

7.2 Multiplication factor

An interesting quantity, that may be needed in elaborating certain specific

prob-lems, is the multiplication factor of the transformation. This can be investigated

by noting that

dz=^dC = u;'iOdC, (7.15)

Thus it follows that

{ ^ = k ( C ) | . (7.16)

From (7.2) it can be derived that in this case

\dz\ 2a 2a

[(id l - i - a 2 - 2 c t c o s ö l-t-a2-a(o--H0'-i)

(7.17)

where cr = expii9). It may be noted that W = c"^ so that a -\- a~^ is always

real. Eq. (7.17) permits to transform cin integration path in the 2-plane to the

C-plane.

7.3 A displacement boundary condition

A simple boundary condition along the inner boundary in the 2-plane is that

the normal stress, or the radial displacement, is constant along this boundary.

In terms of the displacement this means

Ux = -uo^, (7.18)

Uy =—uo , (719)

where UQ is the radial displacement, directed inwardly. With (7.4), (7.5) and

(7.11) this gives

Ux = —Uo Uy = —Uo

il-a^)sm9

It may be noted that for a —»• 0 this reduces to Ux + iuy = iuo exp(z^).

7.4 Fourier series e x p a n s i o n

In the complex variable method as used in this report the boundary values have to be expanded into Fourier series,

fi9)= J2 Akexpiki9),

i = - o o

where

~ STryo fi9)expi-ki9)d9. (7.23)

Some well known integrals (Gröbner & Hofreiter, 1961, section 332) are •2'^ cosik9) '0

J:

J:

i:

i:

i

i:

i:

^ d9 = - -. )fc = 0, -I- a2 — 2a cos 9 s\nik9) -}- a2 — 2a cos 9 cos 9 s\nik9) -(- a2 — 2a cos 9 sin9 sinik9) -}- a2 — 2a cos 9 sin 9 s\nik9) + a^ — 2oc cos 9 l - a 2 d9 = ^, Ar = 0 , l , 2 , . . . , d9 = {), k = Q,l,2,..., do = 7^a*~^ ^ = 1 , 2 , 3 , . . . , d9 = Q, k = Q. (7.24) (7.25) (7.26) (7.27) (7.28) (7.29) (7.30) (7.31)

Using these results it can be shown that the Fourier series expamsion of the

horizontal displacement Ux, as given by (7.20), is

oo

Ux= Yl Pkexpiki9),

where

Pk = huoil-Q^)

a*-^ ib = l , 2 , 3 , . . . ,

0, k = 0,

—a

, Ar = - l , - 2 , - 3 , . . . .

This can also be written as

oo

Ux = -uoil - a^) Y2 <^''~^ sinik9).

In figure 7.2 the expression (7.20) is compared with its Fourier series expansion

(7.34), the dashed line, taking four terms only, and assuming that a = 0.5.

It appears that even for such a small number of terms the approximation is

Ux/uo

*• e

Figure 7.2. Fourier series for Ux, 4 terms.

reasonably good. By taking 10 terms or more, the two expressions become

in-distinguishable.

The Fourier series expansion of the vertical displacement Uy, as given by (7.21),

is

oo

^y = 12 Qkexpiki9),

(7.35)

where

r i ( l - a V - \ ^ = 1,2,3,...,

Qk = uo< -a, k = 0, (7.36)

[ | ( l - a > - * + i , ^ = - 1 , - 2 , - 3 , . . . .

This can also be written as

oo

Uy = - u o a -\- Uoil - Q^) Y^ ^^"^ cos(k9). (7.37)

k=i . . .

In figure 7.3 the expression (7.21) is compared with its Fourier series

expan-sion (7.37), the dashed line, taking four terms only, and assuming that a = 0.5.

Again it appears that even with four terms only, the approximation is reasonably

1 Uy/ttO

Figure 7.3. Fourier series for Uy, 4 terms.

good. By taking 10 terms or more, the two expressions become

indistinguish-able.

In the complex variable method the boundary condition is expressed in terms

of the complex variable Ux + iUy. With (7.34) and (7.37) this is found to be

oo

Ux -\- iUy = -iuoa -\- zuo(l — a^) 2_. a*~^ expiik9). (7.38)

This can also be written as

oo

Ux -h iuy = -zuoa-I-iuo(l - a ^ ) ^ a * ~ V * , (7.39)

A l t e r n a t i v e formulation

The series in (7.39) is a geometrical series, with each term being aa times the previous one. The sum of the series can easily be determined. The result is

ct — cr

Ux + iUy = —iuo- . (7.40) ^ 1 — aff

This seems a remarkably simple result.

The form (7.40) can also be established immediately from the boundary condition in its original form of eqs. (7.18) and (7.19), if this is written as

z -^ ih Ux-\-iUy = —uo , (7.41) r and 2 is written as . 1-FC . 1 + Q^fl^ .^.^. z = -la- = —la- . (7.42) 1 — C, 1 — aa

The form (7.40) may seem to be inconvenient as a boundary condition because of the factor 1 — aa in the denominator. It will later be seen, however, that it is convenient to multiply the boundary condition by precisely this same factor. Therefore it will be found that this form of the boundary condition is actually very convenient for further elaboration.

7.5 A stress b o u n d a r y condition

A simple boundary condition along the cavity boundary in which the stresses sire prescribed is the case of a uniform radial stress t. Then

i x = < ^ , (7.43)

ty = i ^ . (7.44)

According to (2.90) this must be integrated along the boundary

itx + ity)ds = it j -—— ds. (7.45)

Along the boundary of the cavity we may write z + ih = rexp(iy5), where r is a constant auid /? is a variable angle. Along that path ds = rd/?, so that

F = it I expii/3) rd/S = trexpii/3) = <(2 -}- ih). (7.46)

It may be noted that an integration constant may be added to the value of F without affecting the actual surface tractions.

Expressed into the value of C = cta along the boundary in the C-plane the expression (7.46) is found to be

„ . , 2 a a — <T , ,

F = ith . (7.47)

1 -I- a2 1 - a<7 ^ ' This is the form of the boundary stress function that will be considered in detail later.

8. First boundary value problem

In this chapter the problem of an elastic half plane with a circuléir cavity is investigated, for the case that along the boundary of the cavity the surface tractions are prescribed.

The complex stress functions <^(C) and ^(C) are analytic throughout the ring-shaped region in the C-piane. It is assumed that they are also single-valued, so that logarithmic singularities can be ignored. This means that they can be represented by their Laurent series expansions,

oo oo

<^(C) = ao+X;afcC*-H^6,C"*, (8.1)

These series expansions will converge up to the boundaries |C| = 1 and |C| = a. The coefficients ak, bk, Ck and dk must be determined from the boundary conditions.

In general the boundary condition for a given surface traction is given by (3.6),

FiCo) + C = éiCo) + ^M=éÜXÖ) + W^), (8.3)

i^ (Co)

where Co is a point on the boundary. Without loss of generality the constant C

can be assumed to be zero along one of the two boundaries. This will be done

for the outer boundary.

The transformation function mapping the region in the 2-plane onto the interior of a circular ring in the C-plane is the same function as the mapping function for a half plane onto the unit circle,

2 = u;(C) = - z a i i | . (8.4) The origin in the 2-plane is mapped onto C = — 1, and the point at infinity in

the 2-plane is mapped onto C =

1-Differentiation of (8.4) with respect to C gives

2ia

ii~y

On a circle in the C-plane we have C = Co = po", where a = expii6). Then Co = po-~^. This gives

^(Co) _ 1 jl + p(T)i(T - p)^

Z^l^)- 2 a^l-pa) • ^^-'^

8.1 Outer boundary

On the outer boundary the radius p= 1. Then

= i(l-^-')- (8.7)

^(Co) _ w , -2>

'^'(Co)

The derivative of the function ^(C) is

00 oo

é'ic) = 12 ^°*<*"' - E ^*^c-'-\ (8.8)

so that

'é^=J2kak(T-'+' -J^f'bkcr'^', (8.9)

From (8.7) and (8.9) it follows that the second term in the boundary condition

is

# ^ 3^=i f i3.--'^' - è f: «*-'*'

oo OO

- I ^ kak<7-'-' -h i J2 kbk<r'-'. (8.10)

fc=i ifc=i

The third term in the boundary condition is

OO oo

^(Co) = CO + Y^Ck(T-'' -I- 25ifcO-^ (8.11)

ib=l i = l

The complete boundary condition now is, assuming that C = 0 along this

boundary,

ao + f^ ak<T' -h f ; bkc-' + \ f ; kaka-'^' - \ f^ ^'^>^'''^'

- l l 2 ^^k(T-^-'^ + 2 E ^bk(T''-'^ + CO + 2 CitO--* -h ^ djfcCr* = 0. (8.12)

This can also be written as

oo oo oo oo ib=l k=l k=l k=2 oo oo

- 2 1 I ( ^ ~ l)ajfc_i<T-* -h ^ ^ ( ^ -h l)6ib+iör* + ao -h | a i -f f 6i

k = l,2,3,..., k=l,2,3,...,

(8.14)

(8.15)

(8.16)

Ck = -^ifc + iik - l)ak-i - ^ik -\- l)ak+i, dk = -ak + ^ik - l)bk-i - iik -h l)bk+i,

One half of the unknown coefficients have now been expressed into the other half. It may be noted that for ^ = 1 the last two expressions each contain a non-existing term, but with a factor 0, If the coefficients Ojt and 6jfc can be found, the determination of Ck and dk is exphcit and straightforward.

8.2 Inner boundary

On the inner boundary the radius p = a, and Co = cr<T. Equation (8.6) now gives

a;(Co) -ac - (1 - 2a2) -\- a(2 - a ^ ) ^ - ! - a V ' ^

a;/(Co) 2(1 - a<7) (8.17) In contrast with the case of the boundary condition at the outer boundary, where the factor representing the conformal transformation was very simple, see (8.7), this factor appears to be a rather complicated expression at the inner boundairy, especially because of the appearance of the factor (1—acr), or (1—Co), in the denominator of (8.17). In order to eliminate this difficulty, all the terms in the boundary condition are multiplied by this factor. It may be noted that this factor is never equal to zero inside the ring in the C-plane.

The boundary condition (8.3) is now written as

F-(Co) + C ( l - Co) = ri(Co) + T2(Co) + T3(Co), (8.18) where F"(Co) = ( l - C o ) F ( C o ) , (8.19) ri(Co) = (l-Co)<?i(Co), (8.20) r2(Co) = ( l - C o ) ^ l < ? ' ( C o ) , (8.21) '^(Co) T3(Co) = ( l - C o ) ^ ( C o ) . (8.22) Each of these terms will be considered separately, before attempting to solve

the complete equation.

It is assumed that in the boundary condition (8.3) the function F(Co) can be written as

oo

F(Co) = F(aa-) = ^ 5jfc<T^ (8.23)

i r r - o o

where the coefficients Bk are given. The modified boundary function F* (Co) is written as

oo

F'iCo) = F'iaa)= J ^ Ak^', (8.24)

Jb=-oo

The coefficients Ak can easily be calculated from the coefficients Bk, using the definition (8.19). The result is

Ak = Bk-aBk-i, k = -oo,...,oo. (8.25) 8.2.1 T e r m 1

The first term in the modified boundary condition is

TiiCo) = il - aa)éiot(r) =

00 oo

= ao + J2iak - ak.i)a''<T^ - 6i -f ^ ( 6 * - èjt+i)ö-*o--*. (8.26)

i b = l k=l

If it is assumed that

6o = 0, (8.27) then eq. (8.26) can also be written as

oo oo

7^1 (Co) = ao + ]^(ajfc - afc_i)a*a* -f ^ ( 6 ^ - 6 f c + i ) a - * a - ^ (8.28)

k=l k=0

8.2.2 T e r m 2

T h e second term in the modified boundary condition is considered as a product of two terms,

T2(Co) = r2i(Co) X r22(Co), (8.29)

where

r2i(Co) = ( l - C o ) ^ l , (8.30)

'*' (Co)

and

T22(Co) = <;i'(Co). (8.31) With (8.17) the first factor of the second term can be written as

2r2i(Co) = -oö- - (1 - 2a2) + a(2 - a^)^-^ - a ^ a ' ^ . (8.32)

The derivative of the function <^(C) at C = Co is

oo

<^'(Co) = X ^ A : a f c a * - V - ^ - ^ i b 6 j k a - ^ - V - * - \ (8.33)

i = l ib=l

so that the second factor of the second term is

oo oo

T'22(Co) = 12 ma'-'a-'^' - Y, kbka-'-'a'-'K (8.34)

k=i k=i

Multiphcation of the two factors (8.32) and (8.34) leads to the following

expres-sion for the second term

2r2(Co) = - [(1 - 2a2)ai -f 2a2a2 - b^]

-[a^ai -H (2 - a^)bi - 262]a"^<T

- ^ [ a 2 ( / : -h 2)ajt+2 + (1 - 2a^)ik + 1)5^+1

- ( 2 - Q^)kak + (ib - l)ai_i]a*o--^

+ J2iaHk - 2)bk-2 + (1 - 2a^)ik - l)bk-i

Jt=2

- ( 2 - oc^)kbk + (ifc -H l)6;fc+i]a-*o-^ (8.35)

It appears from this expression that there are four levels of coefficients involved

in the equation: from ak-\ to aifc+2, and from bk-2 to 6jfc+i. This is not very

encouraging, as it may lead to a rather comphcated system of equations.

i b = l

0 0

8.2.3 T e r m 3

In order to evaluate the third term it is noted that the value of the function

rb{Q at C = Co is

0 0 00

^(Co) = Co + ^ Cka^c^ + Y2 ^ifca"*<^~*. (8.36)

j f c = l J b = l

so that

00 00

Wö) = co^Y2cka^c-'' + Y2dka-^<TK (8.37)

The third term is the product of this expression and a factor (1 — aa), see

eq. (8.22). This gives

^3(Co) = [co - or^ci] - [a^co -

5i]a"^o-00 5i]a"^o-00

+ Y2[ck- a^cfc+ija'^o--* -h ^ p j t - a'^dk-i]oc-^<TK (8.38)

Using the relations (8.14), (8.15) and (8.16) this expression can be rewritten in

terms of ak and 6^. The result is

2r3(Co) = - [2ao + ai-\-bi- 2a^bi - 2a^a2]

-l-[2a2ao - 2ai -|- a^ai + a% - 262] a ~ V

0 0

+ 121-^^^ + 2a2èfc+i -ik-\- l)ak+i + ik- l)ak-i

k=l

-^Q^ik -H 2)a;k+2 - ct^kak] a^ör"*

0 0

+ 12[-'^(^k + 2Q^ak-i + ( ^ - l)6fc_i - ik-\-l)bk+i

k=2

-a^ik - 2)bk-2 + ct^kbk]Q-''a''. (8.39)

Again it appears that there are four levels of coefficients involved in the equation: from ak-i to 0^+2> and from èjfc—2 to

bk+i-8.2.4 Terms 2 a n d 3

With (8.35) and (8.39) it follows that the sum of terms 2 and 3 is

r2(Co) + r3(Co) = - a o

0 0

-h ^ [(1 - a2)Arait - (1 - a^)ik -\- l)ak+i - bk + aHk+i] a^cr"^

j t = 0

0 0

+ ^ [ i l - a2)(^ - l)bk-i - (1 - a2)fc6jt + a^afc.i - ak]a-''a'', (8.40)

J b = l

if it is again assumed that 60 = 0, see (8.27).

It now appears that in this sum of two terms only two levels of coefficients occur in the equation: from ajt-i to a t , and from 6;fc_i to bk- Two of the four levels of coefficients appear to have canceled.

8.2.5 T e r m s 1, 2 a n d 3

The sum of all three terms is, with (8.28) and (8.40), Ti(Co) + T2(Co) + Ï3(Co) =

= E [ ( i - ^')^^fc - (1 - «')(^+l)°*+i

+ J2[(^- " ' ) ( ^ - ^)bk-i - (1 - a^)kbk

It now appears that in the final expression for the sum of all three terms only two levels of coefficients occur in the equation: from ak-i to ak, and from bk-i to bk.

8.2.6 T h e outer b o u n d a r y condition

According to the modified boundary condition (8.18) the value of the quantity T l + r 2 -f r 3 - C ( l - Co) must be equal to F"(Co), which is represented by its

+00

F'i(T)= Y^ ^ * ^ ' - (8-42) k = -oo

A.kct-^, it = 1 , 2 , 3 , . . . , (8.43)

+ ( a 2 - a2*)ajk_i - (1 - a''')ak = ^ ^ a * , ^r = 2 , 3 , 4 , . . . . (8.44) From these equations the coefficients a^ and bk must be determined. The con-ditions for the coefficients of (7° and a^ must be considered separately. These conditions are

{l-a^)ai+il-a^)bi + C=-Ao, (8.45)

(1 - ct^)bi -f (1 - a2)ai - Ca^ = -A^a, (8.46) or

(1 - a^)ai -h (1 - a^)bi - Co? = -T^OL, (8.47)

It follows from (8.45) and (8.47) that

C^Co? = -Ao + ~A[a, (8.48)

which determines the integration constant C.

All the coefficients can now be determined successively, except for the con-stant ao, which remains undetermined, which represents an arbitrary rigid body displacements. Of the constants a\ and 61 only the combination a\ •\- 61 is de-termined by the conditions (8.45) and (8.47). Its complex conjugate remahis undetermined.

Fourier series expansion (8.24). Hence

ri(Co) + T2(Co) + r3(Co) - C ( l - a(7) =

It follows from (8.41) and (8.42) that

il - a^)kak - il - o?)ik ^ l)ak^i

+ ( a 2 - a - 2 * ) 6 , + i - ( l - a - 2 * ) è f c = and