Issue 66

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 66 (2023) 233-260; DOI: 10.3221/IGF-ESIS.66.15

2 2 2 μ (u iv ) κφ (x) ω (x)        2 2

(17)

2 2 2 μ (u iv ) κφ (x) ω (x)        2 2

(18)

By adding Eqns.(17) and (18) it is obtained:





  

      

  

κφ (x) ω (x) 

κφ (x) ω (x) 

2

2

2

2

(19)



0

o X (x)

o X (x)

where

1





o X (x)

(20)

2   i α

2

x

are values attained by the sectionally holomorphic function:

1



o X (z)

(21)

2

2



z

α

when z approaches x on the upper (+) and lower (–) crack lips, respectively. From Eqn.(19) it is deduced that the function [ κφ 2 (z)- ω 2 (z)]/ Χ ο (z) is holomorphic (single-valued) on the crack and, in turn, everywhere on z-plane. Since X o (z) is double valued on the crack, it is deduced that κφ 2 (z)- ω 2 (z) must be zero there and, in turn, on the entire z-plane. Therefore: ω 2 (z)= κφ 2 (z) (22) Now, subtracting Eqn.(17) from Eqn.(18), and using, in addition, Eqns.(22) and (16), one obtains that:     2 2 2 2 2 2 (1 κ ) σ φ (x) φ (x) τ (1 k)sin2 β α x i δ 1 k (1 k)cos2 β α x 4 κ                 (23) For large |z|, φ 2 (z) reads as [27]: where (X, Y) is the resultant vector of external forces acting on the crack (for the problem studied here it is zero), Γ 2 is zero according to Eqn.(2) and the boundary conditions at infinity, and φ 2,o (z) is a function holomorphic on the entire plane. Thus, for large |z|, it holds that φ 2 (z)= φ 2,o (z). Taking into account that the displacement (Eqn.(6)) must be zero at infinity, it should hold that φ 2,o (z)=0 at infinity and, thus, φ 2 (z →∞ )=0. In this context, the solution of Eqn.(23) for φ 2, (z) becomes:     2 2 2 2 α 2 α (1 κ ) σ 1 dx φ (z) τ (1 k)sin2 β α x i δ 1 k (1 k)cos2 β α x 2 π i 4 κ x z                   (25) Taking advantage of familiar properties of the Cauchy-type integrals it can be written that:       2 2 2 (1 κ ) σ φ (z) δ 1 k (1 k)cos2 β i τ (1 k)sin2 β z α z 8 κ               (26) Combination of Eqns.(26) and (22) yields:       2 2 2 (1 κ ) σ ω (z) δ 1 k (1 k)cos2 β i τ (1 k)sin2 β z α z 8               (27) 2 φ (z) 2 2, o X iY 2 π ( κ 1)   ogz Γ z φ (z)    (24)

241