Issue34

F. Berto, Frattura ed Integrità Strutturale, 34 (2015) 169-179; DOI: 10.3221/IGF-ESIS.34.18 170   f =  + s  * (1) By taking advantage of the analytical frame provided by Filippi et al . [4] and Neuber [1], the FNR approach has been applied to V-notches under mode 1 and mode 3 loading, respectively [5, 6]. The support factor s was found to be highly dependent on the notch opening angle 2  . A satisfactory agreement was found between the theoretical stress concentration factor K t (  f ) evaluated at the fictitiously rounded notch and the averaging stress concentration factor K t obtained by integrating the relevant stress over the distance  * in the bisector line of the pointed V-notch. It is worth mentioning that the notch stress averaging method was originally proposed for brittle fracture problems by Wiegardt [7] and later extended by Weiss [8]. The FNR concept was mathematically formalised by Neuber [1, 3, 9], who provided a general theoretical frame of the method. The application of the concept to strength assessments was demonstrated in Ref. [2]. There is also a correspondence of the concept with the ‘critical distance approach’ proposed and successfully applied by Peterson [10]. It was also applied many years later to notched thin plates subjected to fatigue loading [11, 12] using the characteristic length a 0 derived by El-Haddad, Topper and Smith [13] from the conventional endurance limit and the threshold stress intensity factor range. Referring to Neuber’s concept, Radaj [14-17] proposed to apply the FNR method to assess the high-cycle fatigue strength of welded joints (toe or root failures). A worst case assessment for welded low-strength steels, notch radius  = 0 mm, microstructural support length  * = 0.4 mm and support factor s = 2.5, resulted in  f = 1.0 mm. This reference radius was found to be generally applicable to fatigue strength assessments of welded joints in structural steels and aluminium alloys and has become a standardised procedure within the design recommendations of the International Institute of Welding (IIW) [18]. A recent work has been devoted to the application of the FNR approach to notches with root hole subjected to pure mode 1 loading [19, 20]. Some analytical expressions have been derived for the fictitious notch radius  f and the support factor s taking advantage of some closed form expressions specifically derived for V-notches with root hole [21]. An extension to pure mode 3 loading has been carried out in [22] on the basis of the same set of equations [21]. The case of in-plane shear loading (mode 2) is more complex to analyse than mode 1 and mode 3 loading. The reason for the higher complexity is the fact that out-of-bisector crack propagation is observed, because the maximum notch stress occurs outside the notch bisector line. The mode 2 problem was considered from a theoretical point of view resulting in closed-form solutions for elliptical notches and in numerical solutions for keyhole notches [23]. In a recent paper, pointed V-notches subjected to pure mode 2 loading were investigated [24]. Due to the complexity of the analytical developments, the support factor s was determined numerically using the FE method. The key problem was the choice of the crack path direction for stress averaging over the microstructural support length. Two criteria available from the literature were used to determine the angle of most probable crack propagation, the maximum tangential stress (MTS) criterion according to Erdogan-Sih [25] and the minimum strain energy density (MSED) criterion according to Sih [26]. Taking advantage of the closed form equations provided in Ref. [21] for V-notches with root hole, the FNR concept has been mathematically formalised for this notch shape in the case of in-plane shear loading [27]. Two analytical methods and one numerical method have been proposed in the quoted contribution for determining the fictitious notch radius  f and therefrom the support factor s dependent on the notch opening angle 2  . In all three methods, the crack propagation angle has been determined based on the MTS criterion. A state-of-the art review of the FNR approach comprising also the recent developments just mentioned has been carried out in Refs [28, 29]. While the application of the FNR approach to the pure loading modes is well developed, in-plane mixed mode loading remains an open problem which is difficult to solve for various reasons. The main difficulty is that the most probable crack propagation angle depends on the mode ratio ranging from pure mode 1 to pure mode 2 loading. The aim of the present paper is to provide a theoretically founded basis for the application of the FNR approach to in- plane mixed mode loading. Using the newly developed analytical frame [21], the values of the support factor s as a function of the mode ratio M and the notch opening angle 2  are obtained. As implied by Eq. (1), the support factor s is considered to be independent of the microstructural support length  *, which is only approximately the case. But all the s values reported in the present contribution are on the safe side when used for strength assessments. The analytical procedure given by Neuber [1, 3] for determining the factor s is applied in edge-normal directions outside the bisector line. This is an extension of what has been done by the authors [5, 6] in the cases of pure mode 1, mode 2 and mode 3 loading. By this method, V-notches with root hole, both in the real and in the fictitious configuration, are considered. Pointed notches are the limit case obtained when the real notch root radius tends to zero. The proposed method requires a numerical solution of the rather complex governing equation to determine the values of s , but easy-to- survey tables and diagrams present these values as a function of the mode ratio M for various notch opening angles 2  in