Issue 41

F. Berto, Frattura ed Integrità Strutturale, 41 (2017) 464-474; DOI: 10.3221/IGF-ESIS.41.58 469             0 0 11 1 1 0 1 1 0 1 12 0 11 0 11 0 1 1 12 2 2 0 2 2 0 2 22 0 22 0 21 0 2 2 1( , *, ) 2( , *, ) 4 cos 1 cos 1 (1 ( ) ( ) ( )) 2 (1 ) 4 sin 1 sin 1 (1 ( ) ( ) ( )) 2 ( 1) f f di di K K                                                                                       (14) Solving Eq. (14), with K 1  = K 1 and K 2  = K 2 , it is possible to determine the fictitious radius  f and therefrom the support factor s by using the following expression derived from Eq. (1): * f ρ s     (15) The factor s under mixed mode loading results as a function of the mode 1 and mode 2 stress intensity factors and of the crack propagation angle  0 . Eq. (14) can be solved in general terms for any ratio K 2  / K 1  of the notch stress intensity factors, the crack propagation angle  0 being dependent on this ratio. The mode ratio   is defined as follows based on the ratio  of the notch stress intensity factors K 2  and K 1  : 2 arctan M    (16) 2 2 1 1 f K K         (17) Pure mode 1 loading results in M = 0 and pure mode 2 loading in M = 1. The dimension of the notch stress intensity factors being dependent on the relevant eigenvalues  2 or  1 , which are not identical in general, a length parameter ℓ f is introduced in Eq. (17) with the condition ℓ f =1. In the case of pointed notches,  = 0, Eq. (14) can be applied by replacing K    by K 1 and K   by K 2 while  0 remains the angle of crack propagation. Pointed notches are relevant in worst case considerations of strength assessments. The following expression is valid for   , updating Eq. (17): 2 2 1 1 f K K       (18) The simplified hypothesis, K 1 = K 1   and       which has been inherently assumed for the analytical calculations, avoids an iterative procedure for the determination of s and then of  f . This is the hypothesis implicitly used by Neuber without the explicit introduction of the stress intensity factors [1], but is approximately true only when the fictitiously enlarged notch root radius is sufficiently small. In the reality, for large  f , K 1  and K 2  do not match K 1 and K 2 [20,34]. E VALUATION OF THE SUPPORT FACTOR UNDER MIXED MODE LOADING CONDITIONS y considering Eq. (14) and solving it for different values of the mode ratio M , it is possible to obtain the support factor s as a function of the mode ratio M . To evaluate the values of s , the condition   ≈ 0 approximating pointed notches has been used. This choice was made for two specific reasons. The first reason is that pointed notches are the most critical ones in strength assessments. In many practical cases, the real notch radius is not equal to zero, but it is very small and it can be approximated in the safe direction by the worst case condition  = 0. The second reason is that by considering pointed notches, it is possible to define the mode ratio M uniquely. The mode ratio M is evaluated here by using K   = K 1 and K   = K 2. Eq. (8) has been employed here to determine the crack propagation angle  0 by applying the MTS criterion to pointed V-notches . Due to the fact that Eq. (14) has been expressed in terms of the real notch radius   this radius has to remain finite. B