PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 105–114

111

2 y M - bending moment; b - width of

The following notations are used in the above equations: N - axial force; i z 2 - distance of the upper surface of the layer from the centre. the beam;

The axial force and the bending moment are

z

1            up i n i z l l a

  

2 1 i





cos

sin

N

q

q

dz dx







,

(34)

2

1

Dxi

Dzi

2

i





cos

sin

sin

q z Dxi

1 q x l a Dxi  

q z Dzi













    

2

2

z

1            up i n i z l l a

  

2 1 i

M

dz dx

  

,

(35)





2

2

1

y

cos

1 q x l a Dzi  



2

i

where l is the length of the beam. The MatLab is applied to determine up  and

n z 2 from Eqs. (32) and (33).

3 L J are found in a manner similar to that applied to obtain

1 L J . The solution of J is derived by

2 L J and

inserting of 3 L J in Eq. (22) (the MatLab is used for integration). The SERR in the beam is determined for control of the J -integral solution. The SERR, G , is 1 L J , 2 L J and

bda dU *

G



.

(36)

* U - complementary strain energy; a - length of the delamination.

The notations in Eq. (36) are:

The specific complementary strain energies, *

0 i u , are integrated in the layers of intact portion, 1 2 RR , of the

beam and the two arms of the delamination by the MatLab to obtain * U . The formula for * 0 i u is    i i i i i d u     * 0 .

(37)

* U is inserted in Eq. (36) to derive the SERR. The result thus obtained coincides with the J -integral, confirming the correctness of its solution. 3. Parametric study Here we focus on the analysis of the effects of different parameters of the moving beam structure on the J integral. These effects are illustrated by various graphs presented in dimensionless form in this section of the paper. The analysis is carried-out at: 0.600  l m, 0.450  a m, 0.012  b m, 0.018  h m, 4  n , 2  up n , 0.7  i  , i z 2  60   , 1.800   and 1.045   . The first two parameters whose effect on the J -integral is investigated are the angle,  , and the ratio, 2 1 / m m . The goal is to examine how the J -integral behaves when the inclination of the beam varies. The relationship, J integral -  , is found at three 2 1 / m m ratios. These relationships are presented graphically in Fig. 3. The main conclusion which can be drawn from the graphs in Fig. 3 is that for the considered case the J -integral reaches