Issue 48

F. V. Antunes et alii, Frattura ed Integrità Strutturale, 48 (2019) 666-675; DOI: 10.3221/IGF-ESIS.48.63 667 fatigue crack growth (FCG). Crack closure was proposed  3  and has been used as a complementary parameter to explain the effect of plastic deformation on FCG relations. It has been used to explain the effects of stress ratio, overloads, short cracks and specimen thickness. The T-stress is another concept which has been used to explain the effect of specimen geometry  4  . The concept of partial crack closure, proposed by Donald and Paris  5  and Kujawski  6  , assumes that the contact of crack flanks does not occur immediately behind crack tip and there is, therefore, a contribution of the load range below crack opening to fatigue damage. Christopher et al.  7  proposed a new model based on four parameters to describe the stresses around the crack tip: K F (opening mode stress intensity factor), K S (shear stress intensity factor), K R (retardation stress intensity factor) and the T-stress. However, these new concepts only mitigate the problem, without attacking the real source of the problem. In authors’ opinion, the linear elastic  K parameter must be replaced by non- linear crack tip parameters, able to quantify effectively the crack tip plastic deformation which is supposed to control fatigue crack growth rate (FCGR). The effect of physical parameters like thickness, stress state, specimen geometry and/or overloads is naturally accommodated by the non-linear parameters. Antunes et al.  8  used non-linear parameters to validate the crack closure concept and to identify the best crack closure parameter. The non-linear parameters identified in the literature review made by Antunes et al.  8  were the range of cyclic plastic strain, the size of reversed plastic zone, the total plastic dissipation per cycle and the crack opening displacement. Note that these non-linear crack tip parameters, and also the J integral, usually replace K when the LEFM is no longer valid. The crack opening displacement (COD) is a classical parameter in elastic-plastic fracture mechanics, still widely used nowadays  9  . It has a direct physical meaning and can be measured experimentally. Crack tip blunting and re-sharpening has been used to explain fatigue crack propagation under cyclic loading  10  . This cyclic blunting and re-sharpening was modeled by different authors in order to predict the FCGR  11  . Additionally, it was demonstrated that there is a relationship between COD and striation spacing, and between this and the crack propagation rate  12  . The experimental measurement of COD has been done using different strategies. In Compact-Tension (CT) specimens, an extensometer with blades is used to measure the opening of the specimen at the edge. Therefore, this parameter is usually called crack mouth opening displacement (CMOD). In Middle-Tension (M(T)) specimens, a pin extensometer is placed at the center of the specimen, fixed in two small holes to avoid sliding. The resulting force versus displacement curves are typically used to calculate the crack closure level. Digital Image Correlation (DIC) has been used to define a virtual extensometer, therefore COD can be obtained at different positions relatively to the crack tip. However, analytical or numerical approaches are required to measure the crack tip opening displacement, CTOD. In the finite element method (FEM) studies the CTOD is usually measured at the first node behind crack tip. The use of CTOD in the study of FCG in notched samples has not been seen in literature. One of the well-known approaches to deal with the notch effect in fatigue problems was proposed by Neuber [13] who stated that the geometric mean value of the stress concentration factor and strain concentration factor is constant and equal to the elastic stress concentration factor. Glinka [14  developed an energy-based approach, known as equivalent strain energy density, based on the assumption that the elastic-plastic strain energy density of the material in the yielded zone of the material is equal to the strain energy density assuming an elastic behaviour. Other current and important energy-based approaches are, for example, the control-volume technique introduced by Lazzarin [15] and the total strain energy density concept formulated by Ellyin [16  . It should be also mentioned the Theory of Critical Distances which combines a set of alternative approaches which have in common the fact that the effective stresses at the process zone is estimated on the basis of a characteristic material length, also called critical distance [17-19]. However, these methodologies have some limitations. The main objective here is to study numerically the effect of notches on fatigue crack growth rate using the plastic CTOD concept recently introduced by Antunes et al. [20]. The material studied was the 7050-T6 aluminium alloy, in the form of Single-Edge Notch Tension specimens (SENT). The radius of the notch was varied, keeping constant the total depth of the notch. An automatic elastic-plastic finite-element procedure was developed and the simulations were computed assuming both plane stress and plane strain conditions. N UMERICAL MODEL ig. 1a shows the geometry of the notched samples. Four different notches were considered, with radius of 8, 4, 2 and 1 mm, as illustrated by Figs. 1c, 1d, 1e and 1f, respectively. The total notch depth was always 8 mm, as indicated. The initial crack length was a 0 =8.096 mm including the notch, i.e., the initial crack extended 96  m ahead of the notch. The specimens had a thickness of 0.2 mm in order to simulate pure plane stress state. Additionally, plane strain state was modeled imposing boundary conditions which avoid out-of-plane plane deformation. Only ¼ of the F