PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 416–427 Author name / Structural Integrity Procedi 00 (2019) 000–000

423

8

n z 1

0 

1 

After substituting of

in (26) and (27), the two equations are solved with respect to

and

at various values

of time by using the MatLab computer program. The strain energy density, the curvature, the coordinate of the neutral axis and the strain energy in the beam portion, , are found in analogical way.

1 2 B B

/ 1  h h

0.3

/ 1  h h

0.6

F 

Fig. 5. The strain energy release rate in non-dimensional form plotted against

(curve 1 - at

, curve 2 – at

/ 1  h h

0.9

and curve 3 – at

).

It should be noted that (20) is applied to calculate the strain energy release rate at various values of time. The time-dependent strain energy release rate in the phase of loading is derived also by using the compliance of the beam. For this purpose, the strain energy release rate is written as

da dC

2

G F 2 

,

(31)

b

C

where the compliance, , of the beam is expressed as

F C w 

.

(32)

w

F

In formulae (31) and (32),

is the vertical displacement of the application point of the external force,

. By

using the integrals of Maxwell-Mohr, one obtains

2

2(

)

l l a  

l

l

l l 



1

1

1







w

M dx 

k M dx

2 3 M dx B B 



1 





3 

,

(33)

3

2

3

3



2

1 l l 

l l a  

l

l



1

1

1 2 B B

 M

1 2 B B M 

2 3 B B

2 

3 

where are the bending moments induced by the unit loading in beam portion between the supports and in beam portion, , respectively. These bending moments are found as (Fig. 1) 2 3 B B and are the curvatures of the beam portions, and , respectively. and