Issue 71

Ch. F. Markides et alii, Fracture and Structural Integrity, 71 (2025) 302-316; DOI: 10.3221/IGF-ESIS.71.22

(a) (b) Figure 5: (a) “Problem 3” of the strip with the upper notch; (b) “Problem 4” of the strip with the lower notch.

The complex potentials Φ 4 , Ψ 4 , which solve “Problem 4”, in turn, are easily obtained from Φ 3 , Ψ 3 (by means of the non interacting notches assumption), by modifying Φ 3 , Ψ 3 , by changing properly the coordinate axes (Fig. 6). Namely, starting from the solution Φ 3 , Ψ 3 , of “Problem 3” in Fig. 5a (or equivalently in Fig. 6a), then by a 180 o clockwise rotation, and an upwards translation of the coordinate axes by an amount 2(h–c), one obtains the solution Φ 4 , Ψ 4 of “Problem 4” (Fig. 6c) in terms of Φ 3 and Ψ 3 as:



    Φ (z) Φ ( z i2(h c)),

         Ψ (z) Ψ ( z i2(h c)) i2(h c) Φ ( z i2(h c))

(6)

4

3

4

3

3

where prime in Φ 3 denotes the first order derivative with respect to z. Clearly, the right-hand sides of Eqns.(6) are directly obtained from Eqns.(4) and (5) substituting in the latter z by –z–i2(h–c).

(a)

(b)

(c)

Figure 6: (a) “Problem 3” (with solution Φ 3 , Ψ 3 ); (b) An 180 of the coordinate axes, reaching the solution Φ 4 , Ψ 4 , of “Problem 4”.

o clockwise rotation of the coordinate axes; (c) A 2(h–c) upwards translation

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