PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 428–442 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

439

12

1  z dA t A D  ( )

0



,

(37)







dA

dA

dA

t 









,

(38)

Ut

Uc

( ) D A

( ) Ut A

( ) Uc A

h h z dA      1 1 2 2   







z dA 2

z dA 2





t 





,

(39)

Ut

Uc

A ( ) Ut

A ( ) Uc

A ( ) D

where t  is the stress in the lower crack arm, Ut  and Uc  are, respectively, the stresses in the tension and compression zones of the un-cracked beam portion. The areas of the tension and compression zones are denoted by Ut A and Uc A , respectively. Formula (13) is applied to obtain t  . For this purpose, E and  are replaced with t E and t  , respectively. The stresses, Ut  , is calculated by replacing of E and  with t E and t  in (17). Formula (17) is used also to calculate Uc  by replacing of E and  with c E and c  , respectively. After substituting of t  , Ut  and Uc  in (37), (38) and (39), the three equations are solved together with equation (21) with respect to the curvatures and the coordinates of the neutral axes by using the MatLab computer program. The strain energy in the beam is written as

l a u dA Ut A Ut 0 ( ) ) (  

  

  U a u dA Dt A D 0 ( )

l a u dA Uc A Uc 0 ( ) ) (



.

(40)

The axial force in the lower crack arm is found by applying formula (22). For this purpose,  is replaced with t  . By substituting of (21), (22) and (40) in (23), one derives the following time-dependent solution to the strain energy release rate for the case of material with different viscoelastic behaviour in tension and compression:

  

  

b 1

h h z

  

  

   

   

 dA t 

G

z

 









1

D n 1

2

U

n

2 2

1

n

A ( ) D

b ( ) 1     A D

   .





u dA Dt 0

u dA Ut 0

u dA oUc







(41)

Ut ( ) A

Uc ( ) A

The integration in (41) is performed by the MatLab computer program. The compliance method is applied in order to verify (41). For this purpose, (21) and (22) are substituted in (26). The result is

  

  

1

h h z

  

  

   

   



G

z

dA





t 

 



1

,

(42)

D n 1

2

U

n

2

2 2

b

1

n

A ( ) D

where the curvatures and the coordinates of the neutral axes are obtained from equations (21) (37), (38) and (39). The strain energy release rate found by (43) matches exactly that obtained by (41) which is a verification of the analysis for the case of material with different behaviour in tension and compression.