ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 9 pot

Section A.2 Evaluation of the Coefficients in the Potentials 85

(due to the fact that Cauchy’s Integral Theorem, which guarantees path inde￾pendence in a simply-connected region, is not necessarily satisfied for contours

around the hole). It follows that



L

∗

(z) dz = Cc, (A.19)

where Cc is some complex constant (the same reasoning applies here as was

used to show that the value Bc in (A.13) is constant).

Proceeding as before (see the reasoning used to determine equation (A.17)),

the integral of ∗

(z) can be written as

z

z0

∗

(z) dz = Cc

2πi

log(z − zc) + ∗∗ϕ (z), (A.20)

where ∗∗ϕ (z) is a single-valued analytic function. Substituting (A.20) in (A.18)

and expanding gives

ϕ(z) = Acz log(z − zc) − Aczc log(z − zc) − Ac(z − zc)

+

Cc

2πi

log(z − zc) + ∗∗ϕ (z) + C. (A.21)

Combining logarithmic terms and incorporating all single-valued analytic terms

in a new analytic function, ∗

ϕ(z), results in

ϕ(z) = Acz log(z − zc) + γc log(z − zc) + ∗

ϕ(z), (A.22)

where the constant γc and the new single-valued analytic function ∗

ϕ(z) have

been introduced for convenience.

The multi-valued nature of the potential ψ(z) can be determined by noting

that (z) = ψ

(z) is single-valued (this is because the stresses on the left-hand

side of (2.5) and the function ϕ(z) on the right-hand side of the same equation

are all single valued). With the same reasoning used to determine (A.17) and

(A.20), it can be shown that the integral of (z) can be written as

ψ(z) =

z

z0

(z) dz = γ

c log(z − zc) + ∗

ψ(z). (A.23)

where γ

c is a complex constant and ∗

ψ(z) is a single-valued analytic function.

§A.2 Evaluation of the Coefficients in the Potentials

All that remains for a full determination of the multi-valued nature of the poten￾tials in a region with a hole is to determine the unknown coefficients in (A.22)

and (A.23). This is accomplished as follows.

In order to ensure that the displacements associated with the multi-valued

potentials given by (A.22) and (A.23) are single-valued, equation (2.1) must not