Issue 41

T. Morishita et alii, Frattura ed Integrità Strutturale, 41 (2017) 45-53; DOI: 10.3221/IGF-ESIS.41.07

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 I 0 I I 0 I S S

  t

  t

   







  

  

S S

 1 cos







(7)

0

2

2 2

    t t

  

  



I S S I

e e



   2

  t

  t

  

(8)

1 tan

0

Z



Y

is the time to take SI max

and e Y

and e Z

are the unit vectors in Y and

where dots in Eqs 7 and 8 denote the inner product, t 0

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

( t 0

=| S 3

)|.

S Imax

( t )

S 1

( t )

S 2

( t )

S 3

x

( t )

S 2

( t )

S 3

z

( t )

S 1

y

Figure 1 : Principal stress and strain in xyz coordinates.

X

( t 0

)

S I

( t )

S I

( t 0

)

S I

 ( t ) / 2

( t )

S I

 (t) / 2

= S I

( t 0

)

S Imax

e X

-plane

S Imax

Z

e Z

e Y

 ( t )

Y

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)

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