PSI - Issue 41

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

129

5

2 3 L L  and

3 4 L L  

function of





 l l a  

L L

L L

a







,

(19)

2 3

3 4

1

R

R

2

2

where 2 3 L L  is the shear strain at the surface of the beam in portion, 2 3 L L . Formula (17) is found by applying the integrals of Maxwell-Mohr for calculating of  .

Fig. 3. Evolution of the strain energy release rate with time.

The equation of equilibrium of the elementary forces in the cross-section of the external crack arm (the latter represents a beam of length, a , with a ring-shaped cross-section of internal and external radiuses, 1 R and 2 R , respectively) is written as R dR T L L R R 2 2 2 3 2 1     , (20) where 2 3 L L  is the shear stress in the external crack arm. After substituting of stresses in (15), (16) and (20), equations (12), (15), (16), (19) and (20) are solved with respect to 1 3 L L  , 2 3 L L  , 3 4 L L  , 1 L T and T by using the MatLab computer program. After resolving the static indeterminacy, one can calculate the strain energy release rate, G , for the lengthwise crack in the beam under consideration (Fig. 2). The strain energy release rate is found as

dA G dU * 

,

(21)

where * U is the complementary strain energy. The elementary increase, dA , of the crack area is written as dA R da 1 2   . (22) By substituting of (22) in (21), one obtains