Issue 35

R. Citarella, Frattura ed Integrità Strutturale, 35 (2015) 523-533; DOI: 10.3221/IGF-ESIS.35.58 524 where Y is the geometric factor,  c is the critical stress and c a is the critical crack length at failure. This equation can be used to assess failure criterion for a component. Hence if a particular crack length is chosen and Y and K c are both known then Eq. 3 holds true:    c c c K Y a (3) The critical stress  c must not be exceeded by the operating stress if failure of the cracked component is to be avoided. In most cases, the critical stress will decrease as the crack length becomes longer and this must be considered in the long term assessment of working stresses. If a stress level  c is chosen then the critical crack length is given by Eq. 4:          2 1 c c c K a Y (4) The critical crack length must be much higher than the minimum detectable crack length a min so that the component can be inspected for crack growth at regular intervals. The above criterion does not take into account stable crack growth which can occur in thin sections of some materials. Under these condition the crack will only grow if the load is increasing whereas if the load is constant the crack will stop. In such cases the increase in crack driving force G is initially counterbalanced by the increase in crack growth resistance R under rising load, enabling crack growth to be stable. The instability condition is reached when G = R and  dG dR da da , i.e. when the curves of G and R versus crack length are tangent to each other. Usually R is expressed in stress intensity factor units, i.e.   , R G K ER K EG and so the instability criterion becomes  G R K K ,  G R dK dK da da . R curves have been derived for many materials; more information on R curves and their use can be found in [4-6]. P ROBLEM DESCRIPTION he lap joint proposed for numerical residual strength assessment is represented in Fig.1, with an MSD experimental scenario that is taken from [7]. In Fig. 2 intermediate experimental cracked configurations, up to reaching 393000 fatigue cycles, are reported, whereas the modelled scenario (Fig. 3) makes reference to the experimental configuration after 399620 fatigue cycles, when the fatigue cycling was stopped and the experimental residual strength test started [7]. The plates are made of Al 2024-T3 with Young’s modulus E=72000 N/mm 2 and Poisson’s ratio  =0.33. The model adopted is based on the usage of Dual Boundary Element Method (DBEM) [8-10] under the hypothesis of Linear Elastic Fracture Mechanics (LEFM). In correspondence of 399620 fatigue cycles the residual ligament between hole N. 6 and 7 and between holes N. 8-9-10 had been cut by the advancing cracks and new cracks had nucleated from left hand side of hole N. 6 and from right hand side of hole N. 10; the same residual ligament failure happened between holes N. 15 and 16 with related nucleation of a new crack from right hand side of hole N. 16. The cracked plate of the lap joint has been modeled, using the commercial code BEASY, by a 2D single plate with constant traction applied on one side and constrains in y direction applied on the rivet holes in order to model pin actions, whereas no constraints are present in x-direction in order to allow transversal plate shrinkage (Fig. 3). With such constraints, the longitudinal plate compliance in the overlapping area (between the two rows of rivets) is neglected whilst it is underestimated in transversal x-direction, introducing an element of approximation. In the critical cracked area, the pin action modeling has been improved by explicitly inserting such pins in the holes (instead of just applying displacement constraints on the hole boundary) and moving the constraints on the pin centre. In particular, traction and displacements continuity conditions are imposed on 180 degrees of the pin-hole interface area (the supposed contact area after loading application), whilst the remaining part is disconnected by internal spring of negligible stiffness (Fig. 3). T