Issue 35

E. Giner et alii, Frattura ed Integrità Strutturale, 35 (2016) 285-294; DOI: 10.3221/IGF-ESIS.35.33 288 all the maxima and minima are attained at the same angles (±90º for  and ±45º for  ) is indicative that the stresses evolve in a proportional way. Should the maxima and minima be shifted along the  axis, then it means that the loads are nonproportional. Fig. 4 right, shows the application of the min(  ) criterion. The same  curves of Fig. 4, left, are replotted and the maximum and minimum with time along the step 4 are marked in black. Then, the range of variation of  is computed simply as  max -  min . The minimum shear stress range criterion predicts that the prospective propagation angles are either 0º or 90º. Due to the nature of the shear stresses, there are always two prospective angles with a difference of 90º. The discrimination between both angles is done choosing the angle that also leads to the maximum normal stress  to that plane. The method sharply indicates that the predicted angle of propagation for this case is 90º, as expected in such a simple problem. Figure 3 : Left, geometry and loads of the model of a cracked bar in tension. Right, detailed view of a von Mises contour plot. -100 -80 -60 -40 -20 0 20 40 60 80 100 -15 -10 -5 0 5 10 15 20 25  (º) [MPa] 12345678  12 max  12 min  12 Figure 4 : Left, variation of the normal stress  (in blue) and shear stress  (in red) on a plane forming an angle  with respect to the horizontal surface. Stresses are evaluated at the finite element located ahead the crack tip. Right, application of the min(  ) criterion. N UMERICAL MODEL ue to symmetry conditions, a quarter 2D finite element model has been considered to represent the fretting fatigue tests, as shown in Fig. 5. The rectangle L x B corresponds to the portion of the analyzed specimen and has a length of L = 4 B = 20 mm, the half length of the indenter c is 5 mm, and the distance between the contact plane and the point of the indenter at which loads are applied is h = 10 mm. Four node, plane strain quadrilateral elements were used with a thickness t = 5 mm. The smallest element size considered is 5  m at the right end of the contact zone. The friction model assumed for the contact zone is a Coulomb model and the ABAQUS contact formulation based on Lagrange multipliers is used to model the contact between the indenter and the specimen. The friction coefficient is taken as  = 0.8 [6]. The material behaviour is assumed linear elastic, despite the high stress concentration at the contact edge. D