Issue 33

E. Maggiolini et alii, Frattura ed Integrità Strutturale, 33 (2015) 183-190; DOI: 10.3221/IGF-ESIS.33.23 184 of the most useful tools for studying this kind of problem in a simple way. Ever since the 1970’s, the Brazilian disk has been commonly used for mixed-mode fracture testing of a cracked component [9, 10]; 30 years later, Torabi started to use a modified version of the original Brazilian disk, the U-notched Brazilian disc specimen, to understand the triaxial stress state around U-notches under mixed-mode loading [11, 12]. In the present article, the Authors investigate the geometries of different shaped U-notches, maintaining the same depth and only changing the angle of the notch axis (perpendicular to the applied force and at 45° to the applied force) and the height of the U-notch (from 0 to a height where the middle point of the notch tip is not affected by the proximity of the notch edge). In a subsequent step, another set of geometries were created by adding a fillet radius. U-notches characterised by these differing geometries were subjected to pure tensile load, and the notch tip was studied according to implicit gradient (IG) theory. Focusing on the position of the point of maximum first principal stress, and later IG, we investigated how its location changed with changing notch height. It is clear that the location of the point of maximum first principal stress is always in the notch tip due to the presence of a stress singularity; the situation is different for rounded notches, where it moves from the beginning of the fillet to the inside of the rounded part of the fillet. T HEORETICAL F RAMEWORK he Authors present an approach capable of estimating the fatigue life of notched structures and welded joints based on an effective stress value computed numerically by solving the Helmholtz differential equation linked to the implicit gradient method [13]. Implicit gradient (IG) was originally formulated by Neuber in the 1930’s [14]. The concept is that when it is not possible to use the local value of stress (due to the infinite value of the stress given by FEM in the crack tip, for example), the better way to calculate an effective local stress is to use the near stress field with a proper averaging function. The implicit gradient method supports the idea that the damage should be related to an average value of the stress components exerted on the body; when calculating this average, the stress values near to the critical point have a greater impact than the distant field (this suggests the possible use of a weight function). In other words, with a non-local approach we can determine the local effective stress and then compare it with the reference resistance. This theory is based on a differential equation resulting from the Helmholtz equation that includes a constant representing the characteristic length “c” for a weldable structure equal to 0.2 mm. 2 2 , , c eff IG eff IG eq       (1) 2 , n = 0 eff IG    (2) The gradient parameter “c” is a squared length; an internal length scale is therefore present in the gradient formulation (the set-up of this parameter is explained in [15]). As shown in Eq. (1), the only variable of this equation is the equivalent stress σ eq , which in our case is the first principal stress. As a result, we obtained the IG effective stress σ eff,IG . Eq. (2) is the natural boundary condition [16, 17]. It is important to consider that if the first principal stress is taken as σ eq , the entire study field is around the point of maximum tension, but that does not mean that the result (maximum value of σ eff,IG ) is necessarily in the same location. However, as will be shown below, the maximum σ eff,IG does not always match the location of maximum first principal stress, but it remains in the very near field. The FE software COMSOL MULTIPHYSICS admits the implementation of this kind of equation. G EOMETRIES AND MODEL PROPERTIES wo macrosets of notch geometry were created with variable notch height. The first set is based on a horizontal notch with a fixed depth, and the second set is based on a notch of the same depth, but with the orientation of the notch set at 45 degrees. The origin is taken in the notch upper tip. The half notch height H varies between 0 and 4. A vertical tensile unitary load is applied to the top 25mm side, while the bottom 25mm side is fixed. To make a proper model, we created a mesh with triangular elements and a mapped mesh zone over the notch tip. For the horizontal notch, the model was created with symmetry constraints in the mid-section of the notch.