PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 95–104

101

loading. The reduction gives

            0 2 h 2 h

  



cos

sin(

)  

cos(

)   

N

q

q

q

dz dx





 

,

(30)

1

5 D x

5 D z

1

1

L



    

2 h



      0

(

)sin

(

)cos(

)   

M

q

x

q

x













1

1

1

5 D x

1

y

L



2 h

,

(31)

  



(

)sin(

)   

q

x

dz dx







5 D z

1

1

1

where

q q q

ma ma ma



,

(32) (33)

1

1

L

L





5  5 

,

D x 5

D x

.

(34)

D z 5

D z

In the above equations the following notations are used:  - distance of the cross-section from point, 1 D ;  1 L q , D x q 5 and D z q 5 - intensities of the inertia load induced by accelerations,  1 L a , D x a 5 and D z a 5 , in point, 1 L , of bar, 1 2 DD ;  - angle between 5 1 D L and bar, 1 5 DD (angle  is shown in Fig. 1). The MatLab is used for determining of 1  and n z 1 from Eqs. (28) and (29). The approach for obtaining the complementary strain energies in the upper and lower arms of the lengthwise crack in portion, 2 3 D D , of the bar, 2 4 D D , the intact portion, 3 4 DD , of bar, 2 4 D D , and bar, 4 5 D D , is analogous to one used for * 1 U (the integration is performed by the MatLab). The thus obtained complementary strain energies are substituted in Eq. (23) to derive * U . Then SERR is found by substituting * U in Eq. (22). The solution of SERR is checked by applying the J -integral (Broek (1986)). The integration contour, T , has three portions, namely 1 T , 2 T and 3 T (Fig. 1). Thus, J is written as

3 T T J J J J    . 2 1 T

(35)

The integration over 1 T , 2 T and 3 T is performed by the MatLab to obtain 1 T J ,

3 T J . The J -integral

2 T J and

agrees with the SERR, proving that the latter's solution is correct. 3. Parametric investigation

The parametric investigation performed in this section of the paper aims primarily to examine how SERR is affected by the geometric shape of the moving frame-like component. The effects of the change of parameters related to the material inhomogeneity across the thickness of the bars on SERR are examined too. Below are three figures showing the behavior of SERR when changing the geometric shape and inhomogeneity parameters. The data used in this investigation are summarized as follows: 0.015  b m, 0.025  h m, 0.800 1  l m, 1.500 2  l m,