PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 63–74 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

67

5

The strain energy in the external crack arm is found as

l

R

R

a

2

2





i p =

i

   

  

.

(9)

U

u R dxdR d EX A 0 A

u R dxdR d

=



+



EX

EX

A

A

0

i

p

1

+

i

1

=

l

R 1 3

l

R

0

0

i

p

3

−

where i EX u 0 is the strain energy density in the i -th portion of the external crack arm, 1 0

+ p EX u is the strain energy

density in portion, CD , of the external crack arm. In the un-cracked part of the beam, the strain energy is expressed as

s

s

R

R

2

2





= i q

i

1

   

,

(10)

  

U

A UC A u R dxdR d 0

A UC A u R dxdR d 0

=



+



UC

i

1

i

2

=

a

s

0

− i 0 1

0

0

1 0 UC u is the strain energy density in portion, DH , of the un-cracked part of the beam,

i UC u 0 is the strain

where

energy density the i -th portion of the un-cracked part of beam (Fig. 1). By substituting of (4), (5), (6), (7), (9) and (10) in (3), one obtains the following general expression for the strain energy release rate for the longitudinal crack in the inhomogeneous beam loaded by axial forces (Fig. 1): ( ) ( ) − − + − = +  = = 1 int 1 1 3 3 1 2 2 r r p F DH x b x b x iDH x iCD i i n i N N R F N N R F G      

R

R

R

2



2

2





  

  

1

3

.

(11)

 

 

 

0 IN A A u R dR d

0 A A u R dR d EX

0 UC A A u R dR d

−

+

−







R

2



p

1

1

+

3

R

0

0

0

0

0

3

int x  , that that is involved in (11) is found from the following equation of equilibrium of the cross

The strain,

section of the internal crack arm (Fig. 1):

  = 3 0 2 0 R



,

(12)

F

A A R dR d





b

where the normal stress,  , is presented as a function of of σ in (12), the equation should be solved with respect to int x  by using the MatLab computer program. The following equation for equilibrium of the cross-section of the external crack arm is used in to determine the strain, 1 + p x  , that is involved in (11): int x  by using the stress-strain relation. After substituting

  = R R 2 0



,

(13)

N

R dR d





EX

EX A A

3

EX N , in portion, CD , of the external crack arm is obtained as (Fig. 1)

where the axial force,

 = = i p i i EX N F 1 =

.

(14)