Issue 37

C. Riess et alii, Frattura ed Integrità Strutturale, 37 (2016) 52-59; DOI: 10.3221/IGF-ESIS37.08 53 N ON -P ROPORTIONALITY F ACTOR FOR S TRESS T IME H ISTORIES irst ideas to describe the non-proportionality of local stress paths based on moments of inertia (MOI) stem back to Chu [7]. Bishop [3] seized this suggestion and introduced the first inertia based out-of-phase measure. The inertia based methods differ in the way the MOI is evaluated. Bishop calculates MOI with respect to the perimeter centroid (PC) of the path. For discrete data the evaluation is done using a weighted sum with the length of each segment as weighting factor. Another inertia based method is proposed by Gaier [4]. In contrast to Bishop, Gaier calculates MOI with respect to the origin. Another difference is that Gaier doesn’t use the length of a segment as weighting factor. Instead every stress state has the same mass. This formulation is only suitable for stress paths which are equally spaced in time. A small rainflow projection (RP) filter [8] may have large effects on the NPF. Bolchoun [5] introduced a method without the use of MOI. The formulation is based on the correlation coefficient Cor f g ( , ) of two functions f and g . The correlation coefficient of the time history of normal stress     x x t ' ' ( , ) and the time history of shear stress     xy xy t ' ' ( , ) is evaluated in all cutting planes  . In order to make the NPF ( B np f ) invariant with respect to the coordinate system (CS), an average over all cutting planes is computed. A disadvantage of the NPFs according to Bishop, Gaier und Bolchoun is, that they wrongly predict a high NPF for the stress path     x y t sin( ) ,   xy t cos( ) . Though, for this special case of equi-biaxial tension with out-of-phase torsion the directions of principal axes remain constant and therefore the planes of maximum shear stress do not change [6]. Figure 1 : Interpretation of the tresca-diagramm (left) and non-proportional path (right). That is why Meggiolaro [6] proposes to evaluate NPFs independent from hydrostatic stresses. According to Meggiolaro for plane state of stresses the evaluation of MOI should be based on a     x y xy 3 {( )| } stress-space. As a result of choosing this stress-space, the NPF is dependent on the choice of the CS. It is therefore suggested to use a tresca-stress- space     x y xy 2 {( )| } in order to make the NPF invariant with respect to the CS. In the tresca-diagram (see Fig. 1) every line through the origin is a line with constant principal axis (and constant angle of the maximum shear plane). Furthermore, the norm of a point in the diagram is equal to the double maximum shear stress  max 2 . Choosing the tresca-diagram, the calculation of the NPF is reduced to a geometrical 2D problem. The evaluation is performed according to the MOI method by Meggiolaro [9] on the basis of pseudo-elastic stress paths. By means of the tresca-diagram, MOIs O xx I , O yy I and O xy I are calculated with respect to the origin of the diagram as contour integrals along the stress path                         2 2 O O O xx xy yy x y xy xy x y 1 1 1 I 2 I 2 dp, d L L p I dp L , (1)   xy 2    x y ( )    x y ( )   xy 2 proportional non-proportional   45°   22 5° .   0°       xy x y 2 2 tan( )            max 2 2 max x y xy 2 2 r 2 | | ( ) ( )  r F