PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1226–1236 Author name / Structural Integrity Procedia 00 (2019) 000–000

1231

6

et in T T T   .

(21)

The complementary strain energy stored in the un-cracked portion of the rod is written as

R

2

  0 l a u )

   (

2

U

RdR



,

(22)

3

uc

0

where 

uc u 0 is the complementary strain energy density. By using (11), 

uc u 0 is expressed as

2

n



1

n uc

0  u B uc



D



  uc

.

(23)

2 1

n



The distribution of the shear strain, 

uc  , is written as



ft

R

uc  

,

(24)

R

2

where

2 0 R R   .

(25)

ft  , at the periphery of the un-cracked portion of the rod is found by using the equation for

The shear strain,

equilibrium of the cross-section

R

2

 0

2 2  

R dR T 

,

(26)

uc

where uc  is the shear stress. After substituting of (7), (8) and (24) in (26), the equation for equilibrium is solved with respect to ft  by using the MatLab computer program. Finally, by substituting of (4), (5), (11), (12), (16), (18), (22) and (24) in (3), one derives the following solution to the strain energy release rate:

R

B R 1

2

 2 2 1  in

  

  

2

  

  

  

B

n

n



2



 0

 RdR D n 1 

 RdR D n in 1 

G









2 1

R

n

n





1

R

1

B R 2

 2 2 1  uc

  

  

  

n

 

 RdR D n uc 1 

.

(27)

n



0

The integration in (28) is performed by using the MatLab computer program. In order to verify (27), the strain energy release rate is obtained also by considering the balance of the energy. For this purpose, a small increase, a  , of the crack length is assumed. The balance of the energy is written as