ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 3 pot

Section 3.2 Complex Potentials for a Half-Plane with Holes 13

where the notation []xb

xa denotes the increase undergone by the expression inside

the brackets along the integration path from xa to xb. The integral in (3.18) is

path independent in portions of R beyond the expansion circle. Substituting

(3.14) and (3.15) in (3.18) gives, for values of xa and xb outside a sufficiently

large circle centered at the origin and containing all of the holes,

i

xb

xa

(tx + ity ) ds = (ϒ + ϒ

)ln |xa|

|xb|

+ i(ϒ − ϒ)π

+ ϕ0(xb) + xbϕ

0(xb) + ψ0(xb)

− ϕ0(xa) − xaϕ

0(xa) − ψ0(xa).

(3.19)

Since (3.19) must remain finite when xa and xb approach infinite values inde￾pendently, the coefficient of the real-valued logarithm must vanish. When xa

and xb each approach infinite values simultaneously the integral must approach

the sum of all external forces on the half-plane. It follows that we must have

ϒ + ϒ = 0 and (ϒ − ϒ)π = T

Fx + i

T

Fy , (3.20)

where total resultant force

T

Fx + i

T

Fy = s

Fx + i

s

Fy +

h

Fx + i

h

Fy (3.21)

is given by the sum of the resultant forces acting on the surface ( s

Fx + i

s

Fy ) and

on the holes (

h

Fx + i

h

Fy ). The corresponding values of ϒ and ϒ agree with the

coefficients of the logarithms obtained in [10] for the far-field behavior of the

potentials in a semi-infinite plane with holes and vanishing stresses at infinity.

Final Form of the Complex Potentials

Using (3.20) and (3.16) to calculate γ and γ and substituting the results in (3.8)

and (3.9) allows us to obtain the general form of the potentials for a half-plane

with holes, valid for all values of z in R. Equations (3.1) and (3.2) become

ϕ(z) = − s

Fx + i

s

Fy

2π

+ κ(

h

Fx + i

h

Fy )

2π(1 + κ) 

log(z − zc)

− m

k=1

k

Fx + i

k

Fy

2π(1 + κ) log(z − zk) +

∞

σxx

4 z + ϕ0(z), (3.22)