PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 75–85 Rizov/ Structural Integrity Procedia 00 (2019) 000 – 000

81

7

h

2

2  h

(

) 2 dz   .

J

0 −  = − + 2 2 2 u a

(32)

LW LW

2

In segment, 3  , the components of the J -integral are found as

x p

UN  = − ,

(33)

0 = y p ,

(34)

3 ds dz = ,

(35)

x u

 

 = −

,

(36)

UN

1 cos = −  ,

(37)

where UN  is the stress in the un-cracked beam portion. By using (33) – (37), formula (32) is re-written as

2 h 

(

) 3 dz   .

J

0 −  = − − 3 u

(38)

UN UN

2 h

The solution of the J -integral is obtained by substituting of (26), (32) and (38) in (20). The integration is performed by the MatLab computer program. The fact that the J -integral value obtained by (20) is exact match of the strain energy release rate found by (4) is a verification of the fracture analysis developed in the present paper. 3. Numerical results In the present section of the paper, numerical results are reported in order to evaluate the effects of material inhomogeneity and crack location along the beam height on the longitudinal fracture behaviour. For this purpose, the solution to the strain energy release rate (4) is applied. The strain energy release rate is presented in non-dimensional form by using the formula ) /( 0 G G B b N = . The material inhomogeneity and the location of the crack along the height of the beam are characterized by 0 / B B K and 1 2 / h h ratios, respectively. It is assumed that 0.020 = b m, 0.010 = h m, 40 = N N, 15 = M Nm, / 0.5 0 = D B , / 0.3 0 = H B , 0.6 = m and 0.3 = n . The effects of material inhomogeneity and crack location on the longitudinal fracture behaviour of the beam are illustrated in Fig. 2 where the strain energy release rate in non-dimensional form is presented as a function of h h / 1 ratio at three 0 / B B K ratios. The curves shown in Fig. 2 indicate that the strain energy release rate has maximum at / 0.58 1 = h h for the considered loading conditions and material behaviour.