PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 86–96 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

92

7

0 1 = N ,

(31)

1 M M y =

,

(32)

1

0 1 = z M .

(33)

After substituting of  in (28), (29) and (30), the equations of equilibrium are solved with respect to 1 C  , 1 y  and 1 z  by using the MatLab computer program. By substituting of and (2) in (1), one arrives at

     

    

    

    

2 h

2 b

2 h

2 b

2 b

    

    

    

    

h

1 b

 

 

  − − h

* u dy dz

* u dy dz

4 4 u dy dz R * 0

G

=

+

−

.

(34)

L

U

0

1

1

0

2

2

2 h

2 b

2 h

2 b

2 b

− −

− −

The integration in (34) is performed by using the MatLab computer program. It should be noted that the solution (34) is time-dependent since the complementary strain energy densities which are involved in (34) are functions of time. The solution (34) can be used to calculate the strain energy release rate at various values of time. The strain energy release rate is derived also by analyzing the balance of the energy in order to verify (34). For this purpose, a small increase, a  , of the crack length is assumed and the balance of the energy is written as



a M M U    = + 2 2 1 1

a Gb a   +

,

(35)

where 1  and 2  are the angles of rotation of the free ends of the lower and upper crack arms, U is the strain energy cumulated in the beam. From (35), one drives the following expression to the strain energy release rate:

b a b M −   1 2 2 

a U







G M =

+

1 1

.

(36)

b a 



The angles of rotation are obtained by using the integrals of Maxwell-Mohr

l

 a 0

 +

x dx 1 3 3

x dx 2 3 3

1 

=





,

(37)

a

l

 a 0

 +

x dx 3 3 3

x dx 2 3 3

2 

=





,

(38)

a

where 2  and 3  are the curvatures of the un-cracked beam portion and the upper crack arm, respectively. The strain energy in the beam is obtained by replacing of the complementary strain energy densities with the strain energy densities in (2). The strain energy release rates calculated by (36) are exact matches of these found by using (34). This fact is a verification of the analysis of the strain energy release rate developed in the present paper with taking into account the aging of the material.