PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 125–133 Author name / Structural Integrity Procedia 00 (2019) 000 – 00

130 6

G dU 1 * 2  

.

(23)

R da

The complementary strain energy in the beam is determined as  l a u RdR a u U R R   2 ) 2 ( * * 2 1   

R

2



0 



0 

* 0

*

2

l l a u RdR   

RdR



,

(24)

1

0

1

0

L L

L L

2 3

3 4

R

1

* 0 u is the complementary strain energy density in the beam portion, 1 2 L L , and in the internal crack arm.

where

Fig. 4. Evolution of the strain energy release rate with increase of 1 f (curve 1 – for statically undetermined beam and curve 2 – for statically determinate beam). The complementary strain energy density is found as     d u    0 * 0 . (25) The complementary strain energy densities, * 0 2 3 L L u and * 0 3 4 L L u , are derived by replacing of  with 2 3 L L  and 3 4 L L  in (25). By combining of (23) and (24), one obtains

R

R

R

1

  

   .

1

2

2





 0

u RdR *

u RdR L L * 0 2 3

u RdR L L * 0 3 4

G







(26)

0

R

1 0

R

1

The integration in (26) is performed by the MatLab computer program. The strain energy release rate is determined also by analysing the balance of the energy in order to verify (26). The analysis yields the following expression:

1

a U







G T 



,

(27)

2

R a R  2  



1

1

where is the strain energy, U , in the beam is derived as