Issue 33

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

Direction of  I ( t ): Stage OA, OB, OD for cruciform loading ( ) 0 , ( ) 0 ( ) 90 , ( ) 90 ( ) 90 , ( ) 90 t t along OA t t along OB t t along OD                   

(11.a

Stage OC for cruciform loading

( ) 180 , ( ) 0 t t along OC      

(11.b)

Stage aBbCc for box-shaped loading 0 ( ) 180 , ( ) 90 t t        Stage cDdAa for box-shaped loading 0 ( ) 180 , ( ) 90 t t        

(11.c)

(11.d)



( t ),  3

( t )

 1

( t ),  3

( t )

 1

Cruciform

Box

B 3

a

b

 1

 1

OC DO

t O

A

C

t

 I

1

 I

1



B a b

a

A cC D d

- 

O O

B A



 Imax

O

a

 Imax

A

 3

 3

A

d

c

D

B

d

D

(a) (b)

B

 I

3

 I

3

  I

  I

O

b

D

C

( t )

Cruciform

 I

 ( t )/2,  ( t )

 ( t )/2,  ( t )

Box

Box

 ( t )/2

c

a b

C D d

c

C

a

 Imin

Box

 Imin

90 

90 

 ( t )/2

A

(e)

B

t

t

t

BA O C DO O O

O

B a b

A cC D d a

Cruciform

-90 

-90 

 ( t )

 ( t )

(c) (d)

Figure 5 : Variations of  i

(t),  I ( t ) and  ( t )/2 in non-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.

In the cruciform loading,  I ( t ) makes four triangles shown in Figs 5 (b) and (c). The angle of  ( t ) rotates in the range of 0   ( t )  180  , Fig. 5 (d), and the angle of  ( t ) changes by 90  at the origin O in a cycle. The principal strain path in the polar figure became a cruciform shape as shown in Fig. 5 (e). In the box loading,  1 ( t ) takes positive and  3 ( t ) negative values without changing sign in a cycle as shown in Fig. 5 (b). Then,  I ( t ) in the box loading has large mean value with a small fluctuation during a cycle, Fig. 5 (c). The principal strain direction rotates continually in the range of 0   ( t )  180  and  ( t )=  90  and the principal strain path forms rhombus shape, Figs 5 (d) and (e). The maximum strain ranges are

I            I  1  I  I  a ( ) max 1 



I A C max min ( ) ( ) 

for cruciform loading

(12.a)

3

I   3

min

for box-shaped loading

(12.b)

c ( )

The mean strain in the two cases is 0.

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