Issue 33

T. Itoh et alii, Frattura ed Integrità Strutturale, 33 (2015) 289-301; DOI: 10.3221/IGF-ESIS.33.33 293  1 ( t ),  3 ( t ) O O O t  1  max  min  3 A B C A’ B’ C’ 90  t  ( t )/2,  ( t ) O O O  ( t )/2  ( t ) t O O O  I ( t ) A B C A’ B’ C’ O A A’  B B’ C C’ 3  A’ A   I  I 1 O  I max  I min C’ C B’ B  I 3 S I mean loading stages of AA’, BB’ and CC’, Fig. 4 (d). Finally, the straight loading lines are obtained in the polar figure as shown in Fig. 4 (e). In the proportional loadings in Fig. 4 (a), the directions of  I ( t ) are fixed (  =0 or 180  ) and  =0  . The maximum strain range and mean strain are expressed as I I I max min 1 3 ( ) ( ') k k           (8)     Imean I I max min 1 3 1 1 ( ) ( ') 2 2 k k          (9) as shown in Fig. 4 (e), where k takes A, B or C and k ’ A’, B’ and C’. The results agree well with the principal strain range and mean strain we have considered by putting a sign with human operation. (a) (b) (e) (c) (d) Figure 4 : Variations of  i (t),  I ( t ) and  ( t )/2 in proportional straining: (a) Strain paths on    /  3 plot, (b) Variation of  i ( t ), (c) Variation of  I ( t ), (d) Variation of  ( t ) and  ( t ), (e) Strain paths on polar figure. Non-proportional Loading Fig. 5 shows non-proportional loading cases of the cruciform and box shapes. In the figure,  Imax is assumed to be given by  I (A) and  I (a) in the cruciform and box shape loadings, respectively. The principal strains and the direction of  I ( t ) in each stage in the figure are expressed as follows. Principal strains: Stage OA, OB, OD and DdAaB   I I I I I A A a a                                   2 1 2 2 2 3 1 1 3 max 1 max 1 ( ) 1 1 1 ( ) ( ) 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t Since of t t for cruciform loading for box shaped loading (10.a) Stage OC and BbCcD   I 2 1 2 2 2 3 3 1 3 ( ) 1 1 1 ( ) ( ) 2 2 ( ) ( ) ( ) ( ) t t t t t Since of t t                        (10.b)