Issue 53

V. Rizov et alii, Frattura ed Integrità Strutturale, 53 (2020) 38-50; DOI: 10.3221/IGF-ESIS.53.04

non-dimensional form is presented as a function of f in Fig. 7 at three / S T E E ratios. The curves in Fig. 7 indicate that the strain energy release rate decreases with increasing of f . One can observe also in Fig. 7 that the increase of / S T E E ratio leads to decrease of the strain energy release rate. It is interesting to investigate the effect of the law for variation of the sizes of rectangular cross-section along the beam length on the longitudinal fracture behavior. In order to elucidate this effect, a further two laws (sine and power) for continuous variation of the beam cross-section are considered. The variations of the width and height of the beam according to the sine law are written as

 l           l            3 sin 2 n t n x b b b b   3 sin 2 n t n x h h h h

, (36)

, (37)

where

3 0 x l   . (38)

Formulae (36) and (37) indicate that the width and height vary smoothly from n b and n h at the free end of the beam to t b and t h at the clamped end of the beam. The variations of thicknesses of the lower and upper crack arms when the sine law is used are expressed as

3 l        l        sin 2 2 n x 3 sin 2 2 n x

h h

t

 

1 h h

 , (39)

n

1

h h

t

 

h h

 , (40)

n

2

2

where

3 0 x a   . (41) When the power law is used, the variations of the width and height of the beam cross-section are written as

3 2

 l           3 n t n x b b b b

, (42)

3 2

 l           3 n t n x h h h h

(43)

where

3 0 x l   . (44) Correspondingly, the variations of the thicknesses of the two crack arms are obtained as

3 3 2

l  

2 

x h h

t

n

 

1 h h

    , (45)

n

1

47