Issue34

F. Berto, Frattura ed Integrità Strutturale, 34 (2015) 11-26; DOI: 10.3221/IGF-ESIS.34.02 20 As opposed to the direct evaluation of the NSIFs, which needs very refined meshes, the mean value of the elastic SED on the control volume can be determined with high accuracy by using coarse meshes [70, 71]. Very refined meshes are necessary to directly determined the NSIFs from the local stress distributions. Refined meshes are not necessary when the aim of the finite element analysis is to determine the mean value of the local strain energy density on a control volume surrounding the points of stress singularity. The SED in fact can be derived directly from nodal displacements, so that also coarse meshes are able to give sufficiently accurate values for it. Some recent contributions document the weak variability of the SED as determined from very refined meshes and coarse meshes, considering some typical welded joint geometries and provide a theoretical justification to the weak dependence exhibited by the mean value of the local SED when evaluated over a control volume centred at the weld root or the weld toe. On the contrary singular stress distributions are strongly mesh dependent. The NSIFs can be estimated from the local SED value of pointed V-notches in plates subjected to mode I, Mode II or a mixed mode loading. Taking advantage of some closed-form relationships linking the local stress distributions ahead of the notch to the maximum elastic stresses at the notch tip the coarse mesh SED-based procedure is used to estimate the relevant theoretical stress concentration factor K t for blunt notches considering, in particular, a circular hole and a U-shaped notch, the former in mode I loading, the latter also in mixed, I + II, mode [70-71]. Other important advantages can be achieved by using the SED approach. The most important are as follows: -It permits consideration of the scale effect which is fully included in the Notch Stress Intensity Factor Approach -It permits consideration of the contribution of different Modes. -It permits consideration of the cycle nominal load ratio. -It overcomes the complex problem tied to the different NSIF units of measure in the case of different notch opening angles (i.e crack initiation at the toe (2  =135°) or root (2  =0°) in a welded joint) -It overcomes the complex problem of multiple crack initiation and their interaction on different planes. -It directly takes into account the T-stress and this aspect becomes fundamental when thin structures are analysed [72]. -It directly includes three-dimensional effects and out-of-plane singularities not assessed by Williams’ theory [73-77]. The mean value of the strain energy density (SED) in a circular sector of radius R 0 located at the fatigue crack initiation sites has been used to summarise fatigue strength data from steel welded joints of complex geometry (Fig. 4). S YNTHESIS BASED ON SED IN A CONTROL VOLUME ocal strain energy density W  averaged in a finite size volume surrounding weld toes and roots is a scalar quantity which can be given as a function of mode I-II NSIFs in plane problems [8] and mode I-II-III NSIFs in three dimensional problems [9]. The evaluation of the local strain energy density needs precise information about the control volume size. From a theoretical point of view the material properties in the vicinity of the weld toes and the weld roots depend on a number of parameters as residual stresses and distortions, heterogeneous metallurgical micro- structures, weld thermal cycles, heat source characteristics, load histories and so on. To device a model capable of predicting R 0 and fatigue life of welded components on the basis of all these parameters is really a task too complex. Thus, the spirit of the approach is to give a simplified method able to summarise the fatigue life of components only on the basis of geometrical information, treating all the other effects only in statistical terms, with reference to a well-defined group of welded materials and, for the time being, to arc welding processes. In a plane problem all stress and strain components in the highly stressed region are correlated to mode I and mode II NSIFs. Under a plane strain hypothesis, the strain energy included in a semicircular sector shown in Figure 2 is [7, 13] 3 1 2 2 2 2 3 3 1 1 2 2 1 1 1 0 0 0 N N N e K e K e K W E R E R E R                                          (15) where R 0 is the radius of the semicircular sector and e 1 , e 2 and e 3 are functions that depend on the opening angle 2  and the Poisson ratio  . The material parameter R 0 can be estimated by using the fatigue strength   A of the butt ground welded joints (in order to quantify the influence of the welding process, in the absence of any stress concentration effect) and the NSIF-based fatigue strength of welded joints having a V-notch angle at the weld toe constant and large enough to ensure the non singularity of mode II stress distributions. L