Issue 37

C. Riess et alii, Frattura ed Integrità Strutturale, 37 (2016) 52-59; DOI: 10.3221/IGF-ESIS37.08 54 where L is the total length of the stress path and dp denotes the length of an infinitesimal segment of the path. A detailed description for the evaluation of contour integrals of discrete 2D paths can be found in [8]. Principal moments of inertia  O 1 and  O 2 (    O O 1 2 ) are calculated by transformation into the principal coordinate system:           2 O O O O 2 xx yy xx yy O O 1,2 xy I 4 I I 2 I I (2) Finally, the NPF is defined as root of the ratio of both principal MOI:    O O np 2 1 f / (3) Similar to the herein proposed method, the NPF may also be calculated using Bishop’s method and the tresca-diagram. Then, the only difference is the evaluation of the MOIs with respect to the perimeter centroid and not with respect to the origin. This NPF is named PC np f and is only capable to describe the the out-of-phase extent of a stress path. Examples in engineering applications The non-proportionality at the crack initiation sites of several tests with 3 load channels and variable amplitude loadings is examined. Quite similar results are obtained using PC np f and B np f , because both NPFs only describe the out-of-phase extend of the stress path. In some cases there is an obvious difference to the results of np f . Figure 2 : Tresca-diagram (left), changes of maximum shear planes (middle) and corresponding crack initiation site (right). Four significant examples are extracted from the internal database and discussed in detail. The local stress paths at the crack initiation sites are displayed on the left of Fig. 2. The normalized maximum shear     max max t t ˆ ( )/ max( ( )) over the corresponding angle  is plotted in the middle of this figure. Examples a and b indicate an almost uniaxial stress state with small changes in principal directions, which corresponds to small rotation of the planes of maximum shear. The NPFs at the crack initiation sites are 0.016 and 0.046, respectively. The crack initiation sites are located at a milled-out portion (a) and a die-cast rib (b). At examples c and d the crack initiation sites are located at a notch between two flanges. Both of them have almost the same out-of-phase extent (  PC np f 0 3 . ). However, there are large differences in the NPF