PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 416–427 Author name / Structural Int grity Procedia 00 (2019) 000–000

420

5

0

x   

l l

.

(13)

3

1

UPF E 1

UPF E 2

UP E 1

UP E 2

UPF 

UP 

In formulae (10), (11) and (12), the left-hand end of the beam,

,

and

are, respectively, the values of

,

and

at

3 x

UP E 1

UP E 2

is the longitudinal centroidal axis of the beam. The distributions of

,

UP 

F 

F 

F 

and

along the length of the beam are controlled by

,

and

, respectively.

At

, l l x     UP E 1 2( 3 1 l l UP E 2 ) 1

,

(14)

UP 

the distributions of

and

along the length are written as





  

  

  

  

l l

x

   1 l l



1

sin 2

 E E UP 1





3

,

(15)

1

UPF

F

2

1





  

  

  

  

l l

x

   1 l l



1

sin 2

 E E UP 2





3

,

(16)

2

UPF

F

2

1





  

  

  

  

l l

x

   1 l l



1

sin 2











3

.

(17)

UP

UPF

F

2

1

Formulae (10) – (17) show that the material properties are distributed symmetrically with respect to the mid span. In the present paper, the time-dependent longitudinal fracture is studied in terms of the strain energy release rate. Due to the symmetry only half of the beam is considered. First, a time-dependent solution to the strain energy release rate is obtained in the phase of loading. For this purpose, the strain-time relationship is (1) is applied. The strain energy release rate, , is written as

G

dA G dU 

,

(18)

where

bda dA 

.

(19)

From (18) and (19), one obtains

bda G dU 2  U

,

(20)

1 2 B B

where

is the strain energy cumulated in right-hand half of the lower crack arm and in beam portion,

(Fig.

da

1), is an elementary increase of the crack length. It should be mentioned that the right-hand side of (20) is doubled in view of the symmetry (Fig. 1).