PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 134–144 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

140

7

values of 1 E , r E 2 , 1  , 3 E , P , 2  and D on the upper surface of the beam, respectively. The parameters, 1  , 2  , 3  , 4  , 5  , 6  and 7  control the change of 1 E , r E 2 , 1  , 3 E , P , 2  and D along the beam thickness, respectively. The strain energy release rate, G , for the crack in the beam under consideration (Fig. 1) is determined by analyzing the balance of the energy. In this way, the strain energy release rate is derived as (29) where M is the bending moment applied at the free end of the upper crack arm, U is the strain energy in the beam structure.    ,          a U a M b 1 G 

1.0

0.5

5  (curve 1 – at

6  

6  

Fig. 6. Variation of the strain energy release rate with increase of

, curve 2 – at

, and curve 3 – at

2.0

6  

).

The strain energy is determined as

h

2 h 

2

2  h





01 2 U ab u dz 

02 3 l a b u dz

 

,

(30)

2 h

2





2

where 01 u and 02 u are the strain energy densities in the upper crack arm and in the un-cracked beam portion, a x l   1 , respectively. 2 z and 3 z are the vertical centric axes of the upper crack arm and the un-cracked beam portion. The strain energy density in the upper crack arm is found as    d u   0 01 , (31) where  is obtained by using (20). The strain energy density in the un-cracked beam portion is determined by