PSI - Issue 26

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

79

5

* 1 U and

* 2 U are the complementary strain energies in the upper and lower crack arms, respectively.

where

The complementary strain energy in the upper crack arm is written as

h

1

2

* 1 U ab u dz a h  = 0 * 1 1

.

(15)

1

−

2

In formula (15), *

0 1 a u is found by (7) where  is calculated by (8). By applying (2), the distribution of material

property, B , in the cross-section of the upper crack arm is expressed as       + − = + 1 0 1 0 2 h z h B B B B K ,

(16)

where

1 1 h z h −  

1

.

(17)

2

2

The complementary strain energy in the lower crack arm is written as

h

2

2

* 2 U ab u dz a h  = 0 * 2 2

.

(18)

2

−

2

1  , n z 1 , 1 N and

* U in (10) – (13), the four equations are solved with respect to

After substituting of  and

1 M by using the MatLab computer program. The complementary strain energy density in the un-cracked beam portion ahead of the left-hand crack tip is obtained by replacing of  with UN  in (7). The distribution of the strains, UN  , in the un-cracked beam portion is expressed by replacing of 1  , n z 1 and 1 z with 3  , n z 3 and 3 z in (8). Here, 3  and n z 3 are the curvature and the coordinate of the neutral axis in the un-cracked beam portion. Equations (10) and (11) are used to determine 3  and n z 3 . For this purpose, 1 N , 1 M , 1 h , 1  and n z 1 are replaced with N , M , h , 3  and n z 3 , respectively. Then, equations (10) and (11) are solved with respect to 3  and n z 3 by the MatLab computer program. The strain energy release rate is obtained by substituting of * 0 1 a u , * 0 2 a u and * 0 u in (4). The integration is performed by the MatLab computer program. Formula (4) can be used also to calculate the strain energy release rate when increase of the crack length at the right-hand crack tip is assumed because the beam cross-section, the axial force, the bending moment and the material properties are constants along the length of the beam. In order to verify the solution to the strain energy release rate, the longitudinal fracture is analyzed also by applying the J -integral approach. The J -integral is solved along the integration contour,  , shown by the dashed line in Fig. 1. The J -integral solution is written as

  = + + J J J J , 

(19)

1

2

3