PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 103–114 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

111

9

h

1

2

h  

M b 

z dz

,

(55)

1 1

1



2

0  N ), 1 z is the vertical centric axis of this crack

where N is the axial force in the upper crack arm (apparently,

arm.

0.2  n and curve 3 – at

0.3  n ).

0.1  n , curve 2 – at

Fig. 5. Variation of the strain energy release rate with  (curve 1 – at

The curvatures, 2  and 3  , and the coordinates of the neutral axes are obtained from equations of equilibrium of the elementary forces in the cross-sections of the lower crack arm and the un-cracked beam portion. In order to verify the strain energy release rate obtained by (51), a solution is found also by analyzing the strain energy, U , cumulated in the beam. Hence, the strain energy release rate is written as

bda G dU  .

(56)

The stain energy is determined as

V ( ) 1 

V ( ) 2 

V ( ) 3 

U

u dV 01

u dV 02

u dV 03







,

(57)

where 01 u , 02 u and 03 u are the strain energy densities in the upper and lower crack arms, and the un-cracked beam portion, 1 V , 2 V and 3 V are the volumes of upper and lower crack arms, and the un-cracked beam portion, respectively. The strain energy density, 01 u , is calculated as