Issue 41

F. Berto et alii, Frattura ed Integrità Strutturale, 41 (2017) 260-268; DOI: 10.3221/IGF-ESIS.41.35 262 A NALYTICAL BACKGROUND ollowing Williams’ approach a generic term of the stress function expressed as a series expansion can be written in the following form [1]:                                                        2 / 2 1 χ( r, θ) r sin 1 θ sin 1 θ cos 1 θ cos 1 θ 2 2 2 2 2 i j n n n n n n a a n (1) By explicitly writing the terms of the series (1), the stress function becomes:                                                                               2 2 3/2 2 1 2 3 3 5/2 4 5 2 3 6 7 7/2 8 9 4 θ θ θ ( r, ) r sin sin 3 cos 2 r sin θ 3 2 2 2 4 θ θ 5 r 3 2cosθ sin cos cos θ 5 2 2 2 4 r sin θ sin θ 3 cosθ 3 3 3 7 3 7 r sin θ sin θ cos θ cos θ 2 7 2 2 2 a a a a a a a a a                                   2 4 10 11 9/2 12 13 2r sin θ sin 2θ 1 2cos 2θ 5 5 9 5 9 r sin θ sin θ cos θ cos θ 2 9 2 2 2 a a a a (2) Here a 1 , a 2 , a 3 , a 4 , etc. are the undetermined parameters whereas  is the angle shown in Fig. 1 (with  . As well known, there is a precise link between the three first coefficients of Eq. (2) which control the stress intensity factors (K I K II ) and the T-stress acconding to the expressions:      1 2 3 / 2 / 2 / 4 I II a K a K a T (3) In the contribution by Williams (1957), although fundamental and pioneering, some typos were present where higher order terms were presented by splitting into even and odd parts the initial stress function (ibid. Eqs 8 and 9). In order to avoid misunderstandings, all stress components are determined here by directly using the stress function approach, according to which:                                2 2 2 2 1 1 r, ψ r, ψ 1 1 r, ψ rr r r r r r r r r (4) The generic n-th terms of the stress distributions are then as follows:                                                                       /2 1 2 2 r, ψ,n ( 2)sin ( 2)cos 4 2 2 2 2 6 sin 6 cos 2 2 n rr i j i j n n n r a n a n n n a n a n (5a) F