PSI - Issue 26

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

70

8

1 N F EX = (Fig. 2). After substituting of

EX N , is obtained as

EX N in

EX  and

and (19) in (18). The axial force,

1 r x  , that is

1 + p x  . The strain,

(13), the MatLab computer program is used to solve the equation with respect to

UC  , is expressed as a function of

1 r x 

involved in (21) is found by equilibrium equation (15). For this purpose,

and R by substituting of

1 N F F b UC = −

1 r x  and (19) in (18). The axial force, UC N , is obtained as

(Fig. 2). The

UC  and UC N in (15) is solved with respect to

1 + p x  by using the MatLab

equation obtained by substituting of

computer program. The strain energy density,

IN u 0 , is found by substituting of (18) in (8) and replacing of  with

int x  . The result is

     

  

Q L

Q L

Q L

Q L

  

 +  

  

1

u

ln

ln

x 

x 

=

−

+

.

(22)

IN

0

Q

int

int

int x  is replaced with

1 0 + p EX u . For this purpose,

Formula (22) is applied also to derive the strain energy density,

1 + p x  . The strain energy density, 1 r x  in (22). The integration in (21) is performed by using the MatLab computer program. It should be noted that int x  , 1 + p x  , 1 r x  , R , IN u 0 , 1 0 + p EX u and 1 0 UC u which are involved in (21) are obtained by (12), (13), (15), (19) and (22) at x a = . In order to verify (21), the strain energy release rate is derived also by considering the complementary strain energy, * U , in the cantilever beam (Fig. 2). For this purpose, the following formula is applied (Rizov (2019)): 1 0 UC u , is found by replacing of int x  with

*

G dU cf =

,

(23)

l da

l

2  = , expression R

where da is an elementary increase of the crack length. Since the length of the crack front is

cf

3

(23) takes the form

G dU 3 * 2  =

.

(24)

R da

The complementary strain energy in the cantilever beam is written as (Fig. 2)

R

=   3 0 2 0 0 R a

a

2

 u R dxdR d *



  

*

* u R dxdR d IN A 0 A



+

U



+

EX

A

A

0

p

1

+

l

R

0

1

3

   + l a R 0 2 0



A UC A u R dxdR d * 0



,

(25)

1

where *

and * 0 1 UC u are the complementary strain energy densities in the internal and external crack

0 IN u ,

* 0 1 + p EX u

arms, and the un-cracked part of the beam, respectively.