Issue 39

M. Muñiz Calvente et alii, Frattura ed Integrità Strutturale, 39 (2017) 160-165; DOI: 10.3221/IGF-ESIS.39.16 163 Finally, the results are obtained by fitting Eq.(1) to a straight line using a probabilistic paper or a Matlab subroutine [5]. Step 6: Convergence of the model: When the variation of the sum of all the parameters becomes less than a certain threshold,  , the fitting process is considered to be fulfilled.                 1 1 1 i i i i i i (6) Otherwise, the iterative process continues returning to step 3. E XAMPLE OF APPLICATION any multiaxial fatigue limits criteria, such as Sines [6] or Crossland [7] criteria, are based on the calculation of a equivalence shear stress amplitude, a J 2 , which becomes quite complex to be obtained for general multiaxial loading [8]. Some examples of models to handle the multiaxial fatigue damage phenomenon are the Maximum Circumscribe Circle (MCC) and the Maximum Circumscribe Ellipse (MCE) models, which propose the calculation of the equivalent shear stress amplitude as proposed by Papadopoulos [9] and Freitas et al. [10]. Both multiaxial fatigue criteria are based on the calculation of the curve described by the shear stress in the critical plane during a cycle. With the aim of calculating the path described by the shear stress, it is necessary to define a material plane (See Fig. 1b), Δ , passing through the selected point and assuming a biaxial loading state, i.e.:                        0 0 0 0 ) sin( 0 0 0 ) sin( 0 0 0 0 0 0 0 , , yy ayy axx yy xx wt wt       In order to evaluate the stress vector T acting on the plane Δ passing through the point considered, a local coordinate system is defined by three unit vectors:       0 ) cos( ) sin( ) sin( ) sin( ) cos( ) cos( ) cos( ) cos( ) sin( ) sin( ) cos( ) sin(                 l r n where n is the vector perpendicular to the plane and r and l are vectors in the plane, which define an orthogonal basis with the previous one.  and  are the angles between these vectors and the xyz axes . Thus, the stress vector T acting on the plane can be obtained by the Cauchy's theorem:   0 ) sin( ) sin( ) cos( ) sin( ·        yy xx n T   Then, the stress vector could be decomposed in two stresses: a normal stress, nn  , that changes in magnitude but not in direction during a cycle of loading; and a shear stress,  , that changes in magnitude and direction along each loading cycle, and can be decomposed in two directions rr  and ll  :       yy xx ll yy xx rr yy xx nn Tl Tr Tn                             ) sin( ) cos( ) sin( '· )( sin )( cos ) sin( ) cos( '· )( sin )( cos )( sin '· 2 2 2 2 2 M