Issue 41

T. Morishita et alii, Frattura ed Integrità Strutturale, 41 (2017) 45-53; DOI: 10.3221/IGF-ESIS.41.07 48 axes in arbitrary directions. The rotation angle of  (t)/2 is the angle between the SI max and SI(t) directions and the rotation angle of  (t) is the angle of SI(t) from the Y-axis in X-plane. The two angles of  (t)/2 and  (t) are written as,                       t t t t t I 0 I I 0 I 1 cos 2 S S S S             2 2 0 t (7)                  Y I Z I 1 tan e S e S t t t       2 0 t (8) where dots in Eqs 7 and 8 denote the inner product, t 0 is the time to take SI max and e Y and e Z are the unit vectors in Y and Z directions, respectively. S I (t) is a vector of principal value and the subscript i takes 1 or 3, e.g., i takes 3 when S Imax =| S 3 ( t 0 )|. Figure 1 : Principal stress and strain in xyz coordinates. Figure 2 : Definition of principal stress and strain directions in XYZ coordinates. Definitions of principal stress and strain in polar Coordinates Fig. 3 shows the trajectory of S I ( t ) in 3D polar coordinates for a cycle where the radius is taken as the magnitude of S I ( t ), and the angles of  ( t ) and  ( t ) are the angles shown in Fig. 2. A new coordinate system is used in Fig. 3 with the three axes of S I 1 , S I 2 and S I 3 , where S I 1 -axis directs to the direction of S I ( t 0 ). The rotation angle of  ( t ) has double magnitude compared with that in the specimen shown in Fig. 2 considering the consistency of the angle between the polar coordinates and the physical plane. The figure on the right side in Fig. 3 shows the waveform of loading which is the projection value to path length of S I ( t ) projected to S I 1 -axis. The loading waveform represents the magnitude of the normal strain or stress acting on the plane ( S Imax -plane) perpendicular to the direction of S I ( t 0 ). The maximum range (  S I ) and mean value ( S Imean ) can be obtained from Fig. 3.  S I and S mean are calculated as follows,       Imin Imax I Imax I cos Max S S t tS S S       (9) x y z S 1 ( t ) S 1 ( t ) S 2 ( t ) S 2 ( t ) S 3 ( t ) S 3 ( t ) X Y Z  ( t ) / 2  ( t ) S I ( t ) S I ( t 0 ) e X e Y e Z S Imax -plane S I ( t ) S I ( t 0 )  (t) / 2 S Imax = S I ( t 0 )