Issue 35

R. Citarella, Frattura ed Integrità Strutturale, 35 (2015) 523-533; DOI: 10.3221/IGF-ESIS.35.58 527 33 integration points are used along the J-integral path (Fig. 3c) whereas the increment of accuracy with 66 points turn out to be negligible. The mesh used for the lap-joint is based on about 326 quadratic elements: a p-convergence study has been realized showing that cubic elements provide an accuracy improvement of less than 2% and that 2 quadratic elements per 90 degrees are enough on the cracked hole, except when very short cracks are present (in such case 3 elements are recommendable, possibly with a scaling ratio). After link-up of the cracks between holes and consequent development of the main crack there is no longer load transfer through the pins in the central part of such main crack even in the remote case they should not break when reached by the propagating crack. Still remaining in the theoretical framework of linear elastic fracture mechanics, SIFs evaluation can be improved by empirically taking into account the elastic-plastic effects by the Irwin correction. Such correction is useful in a residual strength analysis and suggests to prolong the considered cracks of a virtual quantity calculated as a characteristic dimension of the plastic zone at the crack tip. Alternatively a fully elastic plastic nonlinear analysis could be attempted to get more accurate results [14]. S OLUTION PROCEDURE wo approaches have been proposed for failure assessment:  Plastic collapse prediction, based on Von Mises stress exceeding 385 MPa, the average of yield ( y =330 N/mm 2 ) and rupture stress (  u =440 N/mm 2 ), in large zones of ligament;  R -curve analysis for stable and unstable crack growth assessment. With reference to the latter, it is well known that the failure criterion for plane strain structure is not valid for the case of thin metal sheet structure, because of extensive slow stable growth, under monotonic loading, prior to instability and catastrophic failure. Here rather than a single material parameter, a material curve ( R -curve or K R -curve), representing an infinity of potential failure points (the crack length at instability is not known a priori), is necessary to make an accurate failure prediction. In this case two criteria must be satisfied to get an unstable crack growth:  G R K K and  G R d d K K da da (5) In the R -curve diagram there are two important points: 1. K o is the minimum SIF to start the crack propagation; 2. K c is the critical stress intensity factor (instability point). K o (the point of initial crack propagation) is independent from the specimen thickness and has a constant value equal to nearly 30 MPa m 1/2 for the considered material [6] so that the main crack will start propagating with a remote load equal to 63 MPa (as a matter of fact the first iteration in Table 1 starts considering 63 MPa). On the contrary K c is strongly influenced from the specimen thickness: thinner specimens give higher K c values and consequently exhibit slower stable crack growth. A sufficiently thick specimen will result in full plane strain and K c will then be equal to K Ic . In order to obtain a crack driving energy (or force) curve an iterative process is needed, which is based on the following steps (Fig. 5):  the load is monotonically increased by small steps and for each of them a linear elastic analysis is performed by DBEM to calculate SIFs (when two consecutive configuration have nearly the same crack configuration it is possible to avoid the DBEM analysis, imposing a linear variation of SIFs vs. loads);  at each step cracks are prolonged by a length da i that is provided by the R -curve, as a function of the SIFs determined at the previous step; moreover, in order to provide the Irwin correction for SIFs evaluation, when the plastic effects become significant, cracks are prolonged by a virtual length r y =r p (Eq. 6);  for each crack tip, the G -curve (crack propagation driving force) is drawn and superimposed to the R -curve in order to find out the instability point, as resulting from the conditions in Eqs. 5;  during the steady crack propagation some cracks will reach a link-up condition (Fig. 6) with other cracks or holes, when the plastic zone at the crack tip together with the plastic zone of the approaching crack or hole respectively, are covering the remaining ligament (Swift criterion). T