Issue 33

T. Itoh et alii, Frattura ed Integrità Strutturale, 33 (2015) 289-301; DOI: 10.3221/IGF-ESIS.33.33

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 

1 

( ) ( ') k k

(8)

max

min

3

1 2

1 ( )



 



Imean (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. I  I  max min   1  3  ( ') 2 k k   



( t ),  3

( t )

 1

3

A

C

 max

A’ B’ C’

B

 I

1

 1

 I max

B C



O t

O

A

B

A’

O A

O

C

 3

B’

 min

C’

S I mean

  I

(a) (b)

 I

3

O

( t )

 I

 ( t )/2,  ( t )

C’

A

B’

A’

A’ B’ C’

 ( t )/2

 I min

90 

B C

 ( t )

t

t

(e)

O O

O O

O

O

(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               2 1 2 2 ( ) 1 1 1 t

  

I   I  2

 3

( ) ( ) t t

( ) t

2 2

(10.a)

 1

 1 t for cruciform loading for box shaped loading  3   ( ) t ( )



( ) t

Since of

A A a a   1 ( ) ( )

I      ( ) ( ) I 

max

I 

max

1

Stage OC and BbCcD

1 

( ) ( ) t t

1      

 

 1 1  



2

2

2









2 

3 

( ) ( ) t t

2 2

(10.b)

I 

3 

1 

3 





( ) t

Since of

( ) t

( ) t

293