Issue 41

F.V. Antunes et alii, Frattura ed Integrità Strutturale, 41 (2017) 149-156; DOI: 10.3221/IGF-ESIS.41.21 151 where  xx ,  yy ,  zz ,  xy ,  xz and  yz are the components of the effective stress tensor (   Σ σ X ) defined in the orthotropic frame; and F=0.5998, G=0.5862, H=0.4138, L=1.2654, M=1.2654, and N=1.2654 are the coefficients that characterize the material orthotropic behaviour. The material constants, determined for 6016-T4 aluminium alloy, are:  ys =124 MPa, R sat =291 MPa, n  = 9.5, C x = 146.5 and X sat = 34.90 MPa. For the 6082-T6 aluminium alloy, the material constants obtained were ܻ 0 = 238.15 ܯ ܲܽ , ܴ ݏ ܽ ݐ = 249.37 ܯ ܲܽ , n= 0.01, ܥ ݔ = 244.44, and ܺ ݏ ܽ ݐ = 83.18 ܯ ܲܽ . The finite element model of the M(T) specimen had a total number of 6639 linear isoparametric elements and 13586 nodes. The finite element mesh was refined near the crack tip, having 8  8  m 2 elements there. Only one layer of elements was considered along the thickness. Crack propagation was simulated by successive debonding of nodes at minimum load. Each crack increment corresponded to one finite element and two load cycles were applied between increments. In each cycle, the crack propagated uniformly over the thickness by releasing both current crack front nodes. A total number of 320 load cycles were applied, corresponding to a total crack propagation of  a=(160-1)  8  m=1272  m. Note that the first two load cycles were applied without crack increment, i.e., at a=5 mm. A wide range of constant amplitude tests was considered. The remote stresses can be obtained by dividing the loads by the area of cross section, i.e.,  =F/A, being A=30  0.1 mm 2 . The numerical simulations were performed with the Three-Dimensional Elasto-plastic Finite Element program (DD3IMP). This software was originally developed to model deep drawing, and was adapted to study PICC due to its great competence in the modeling of plastic deformation. The CTOD was measured at the first node behind crack tip, i.e., 8  m from crack tip. Further details of the numerical procedure may be found in previous publications of the authors  1,6  . N UMERICAL RESULTS Effect of measurement point ig. 1a presents typical results of CTOD versus load. The CTOD was measured after 160 crack propagations (  a=1.272 mm), at nodes 1 and 5 behind crack tip. The load is quantified by the remote stress. The first node behind crack tip (node 1) is closed at minimum load (A) and only opens when the load reaches point B, which is the crack opening load. After opening, CTOD increases linearly with load, but after point C, there is some deviation from linearity, which indicates the occurrence of plastic deformation. The maximum CTOD occurs at point D, which corresponds to the maximum applied load. The fifth node behind crack tip (node 5) has higher levels of CTOD as could be expected. The crack opens at point E, at a load lower than observed for node 1. In fact, the crack opens progressively, therefore node 1 is the last node to open. In the region E-F, the crack opens progressively from node 5 to node 1, and only after point E it is totally open. After the opening of node 1 there is a second linear region, but with a slope higher than in region EF, which indicates a lower rigidity. The plastic deformation starts at point G, increasing progressively up to the maximum load (H). After the maximum load, there is also a linear variation of CTOD with load decrease. With subsequent load decrease, reversed plastic deformation starts and the crack closes again. Matos and Nowell  7  studied nodes 1 and 2 behind crack tip. The displacements obtained at node 2 were higher than those obtained with node 1, as could be expected. The same global aspect was observed, however the plastic deformation obtained by them is significantly higher, since the elastic regimes are relatively short. They studied the Ti-6AL-4V titanium alloy, assuming an elastic perfect plastic behaviour, and used 5 or 10  m elements near the crack tip. However, only the plastic variation of CTOD is relevant for the study of fatigue crack propagation, since this phenomenon is linked to irreversible mechanisms. Fig. 1b plots the variation of CTOD p for different nodes behind crack tip. For each node, there is a progressive increase of plastic CTOD up to the maximum load. The maximum CTOD p decreases with the departure of measurement point from crack tip. The load corresponding to the onset of plastic deformation can be used to calculate the fatigue threshold. As can be seen in Fig. 1b, the same fatigue threshold is obtained at different measurement points behind crack tip. Fig. 2 presents the variation of plastic CTOD range with distance to crack tip, d. There is a sharp decrease of  CTOD p immediately behind crack tip. This variation is particularly relevant up to a distance of 100  m behind crack tip. The increase of d reduces the rate of variation, however, a slight decrease is always observed, at least for the range of d values studied. Note also that there is a great difference between the first value, obtained at a distance of 8  m behind crack tip (  CTOD p =0.35  m), and the value measured at a distance of 640  m behind crack tip (  CTOD p =0.086  m). In other words, and predictably, the numerical predictions of  CTOD p are quite sensitive to the position of measurement point relatively to the crack tip. F