PSI - Issue 33

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

421

6

The strain energy is found as

and 2 1 U U U   1 U 2 U

,

(21)

1 2 B B

where

are the strain energies in right-hand half of the lower crack arm and in beam portion, ,

2 3 B B

respectively. The strain energy in beam portion,

, is not involved in (21) since this strain energy does not

depend on the crack length.

Fig. 3. The strain energy release rate in non-dimensional form plotted against the non-dimensional time.

The strain energy cumulated in half of the lower crack arm is expressed as

V ( ) 1  

U

u dV 01

,

(22)

1

01 u

1 V

where

is the strain energy density,

is the volume of half the lower crack arm.

The strain energy density is expressed as

01 2 1  u

.   0

(23)

By combining of (1) and (23), one obtains the following time-dependent strain energy density:

2

 0

  

   



 



 t e 

1

1

u







.

(24)

01

E 2 1



The distribution of the strains is treated by applying the Bernoulli’s hypothesis since beams of high length to thickness ratio are considered in the present paper. Thus, the distribution of the strains along the thickness of the lower crack arm is written as

  n z z 1 1 1    

,

(25)