PAMM · Proc. Appl. Math. Mech. 11, 237 – 238 (2011) / DOI 10.1002/pamm.201110110

Asymptotic analysis of the interaction of a finite number of holes in anelastic plane or half-plane

Jan Kratochvil1,∗ and Wilfried Becker1

1 Technische Universität Darmstadt, Department of Mechanical Engineering, Chair of Structural MechanicsHochschulstr. 1, 64289 Darmstadt

The problem of the interaction of a finite number of holes in an elastic plane or half-plane is considered. The analysis is basedon the complex potential method of Muskhelishvili as well as on the theory of compound asymptotic expansions by Maz’ya.An asymptotic expansion of the complex potentials in terms of relative hole radii is constructed. This expansion is uniformlyvalid in the whole domain. The method leads to a simple procedure which does not involve any coupled system of linearequations. The successive closed-form approximations can be obtained in an iterative manner to an arbitrary order withoutany need for numerical approximation.

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Problem statement

Consider an elastic plane or half-plane in the state of plain strain orplane stress. The in-plane stress components in a Cartesian coordi-nate system and the complex force acting on an arbitrary arc CD arerelated to two holomorphic functions ϕ(z) and ψ(z) of the complexvariable z = x+ iy by the Kolosov equations [1]

σx + σy = 2(ϕ′ + ϕ′

),

σy − σx + 2iτxy = 2 (z̄ϕ′′ + ψ′) ,

f = fx + ify =[ϕ+ zϕ′ + ψ

]DC.

(1)

Fig. 1 Problem configuration for H = 2 holes.

The plane is weakened by H holes with centers cp and radii ερp (see Fig. 1), where ε is a small number and ρp is of thesame order of magnitude as the respective distances of the holes.

On the boundary of the pth hole, the resultant fp(ϑ) is prescribed as a function of the angle ϑ, the total resultant is assumedto be zero. The solution of the problem of a single circular unit hole in an infinite plane under the same loading is denoted byϕ̂p(ζp) and ψ̂p(ζp). Furthermore, the state of stress at infinity σ∞xx, σ∞yy, σ∞xy and in the case of the half-plane also the tractionf0(x) on its boundary are prescribed. The solution of the problem for a plane or half-plane without holes under this load isdenoted by ϕ̂0(z) and ψ̂0(z). In the following, only the more general case of a half-plane is discussed in detail.

2 Solution

Based on the theory of compound asymptotic expansions [2], an ansatz for the complex potentials in terms of the globalcoordinate z and local coordinates ζp =

z−cpερp

is stipulated

ϕ =

∞∑n=0

εn

[ϕn0 (z) +

H∑q=1

ϕnq (ζq)

], ψ =

∞∑n=0

εn

[ψn0 (z) +

H∑q=1

ψnq (ζq)−cqερq

dϕnqdζq

(ζq)

], (2)

where ϕn0 (z) and ψn0 (z) are outer auxiliary functions, holomorphic and single-valued on Im(z) ≤ 0 and ϕnq (ζq) and ψnq (ζq),1 ≤ q ≤ H are inner auxiliary functions, holomorphic and single-valued on |ζp| ≥ 1.

Substituting the ansatz into the boundary condition on the boundary of the half-plane, re-arranging the sums as neededand comparing like-wise powers of ε, one obtains a boundary value problem for the outer auxiliary functions on the lower

∗ Corresponding author: email kratochvil@fsm.tu-darmstadt.de, phone +49 6151 166467, fax +49 6151 166117

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

238 Section 4: Structural mechanics

half-plane (without holes) whose solution is

ϕn0 (z) =ϕ̂0(z)δn0 −H∑q=1

n∑k=1

−kan−kqk (z − cq)(z − cq)−k−1 + bn−kqk (z − cq)−k, (3a)

ψn0 (z) =ψ̂0(z)δn0 −H∑q=1

n∑k=1

an−kqk (z − cq)−k − kan−kqk z(z − cq)−k−1−

− kan−kqk (k + 1)z(z − cq)(z − cq)−k−2 + bn−kqk kz(z − cq)−k−1, (3b)

where the coefficients anqk and bnqk are given by

ϕnq (ζq) =

∞∑k=1

anqkζ−kq , ψnq (ζq) =

∞∑k=1

bnqkζ−kq .

Similarly, from the boundary condition on each of the holes for the inner auxiliary functions it follows

ϕnp (ζp) =ϕ̂p(ζp)δn1 −n∑k=1

kKn

pkζ−k+2p δk>2 +

(Ln

pk +cpρp

(k + 1)Kn+1

p,k+1

)ζ−kp , (4a)

ψnp (ζp) =ψ̂p(ζp)δn1 −n∑k=1

Kn

pkζ−kp + kK

n

pk(k − 2)ζ−kp δk>2+

− kKnpkζ

k−2p δk<2 +

(Ln

pk +cpρp

(k + 1)Kn+1

p,k+1

)kζ−k−2p , (4b)

where the coefficients dnpk, enpk, Knpk and Lnpk are given by

ϕn0 (z) =

∞∑k=0

dnpk(z − cp)k, ψn0 (z) =

∞∑k=0

enpk(z − cp)k,

Knpk = ρkp

[dn−kpk +

H∑q=1q 6=p

n−k∑l=1

an−k−lql

(−lk

)ρlq(cp − cq)−k−l

],

Lnpk = ρkp

[en−kpk +

H∑q=1q 6=p

n−k∑l=1

(−lk

)ρlq(cp − cq)−k−l

(bn−k−lql + an−k−lql (k + l)

cqcp − cq

)].

Equations (3) and (4) represent an iterative algorithm for the determination of the auxiliary functions. In the case of thefull plane, equations (3) are replaced with ϕn0 = ϕ̂n0 δn0 and ψn0 = ψ̂n0 δn0. Once the auxiliary functions have been determinedup to a chosen order of approximation, they are substituted into the ansatz (2) which yields an uniformly valid asymptoticexpansion for the complex potentials. For more details on the solution, the reader is referred to [3].

3 Example

As an example, consider a half-plane Imz ≤ 0 under uniaxial tension in x-direction containing two unloaded holes of radiusε and centers c1 = −1 − i and c2 = 1 − i. Using the above described procedure, one obtains the following approximationsfor the tension along the boundary of the half-plane and for the hoop stress on the boundary of the holes

σxxσ0

= 1−4(−64 + 48x2 + 192x4 − 104x6 − 12x8 + 3x10

)ε2

(4 + x4)3 +O(ε4)

σϑϑσ0

= 1− 2 cos(2ϑ)− ε2

4+

1

8

(− 2 cos(ϑ) + 17 cos(3ϑ)− 3 sin(ϑ)− 21 sin(3ϑ)

)ε3 +O(ε4).

This example demonstrates the kind of closed-form approximations that can be obtained using the present method.

References[1] N. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity (P. Nordhoof Ltd, 1963).[2] V. Maz’ya, S. Nazarov, and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Do-

mains (Birkhäuser, 2000).[3] J. Kratochvil and W. Becker, Asymptotic analysis of stresses in an isotropic linear elastic plane or half-plane weakened by a finite

number of holes, submitted to Archive of Applied Mechanics (2011).

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