PSI - Issue 51

Victor Rizov et al. / Procedia Structural Integrity 51 (2023) 206–212 V. Rizov / Structural Integrity Procedia 00 (2022) 000–000

210

5

2 h

2 h

2 h

y

y

y

1 i         1 1 i 1 1 y i n i

1 i         2 1 i 2 2 y i n i

1 i         3 1 i 3 y i n i

1 1 1 z dy dz i

2 2 2 z dy dz ri

3 3 3 z dy dz ui





,

(9)

2 h

2 h

2 h

i y 2 , 2 1  i y ,

i y 3 and 3 1  i y are the

where 2 n and n are the number of layers, ri  and ui  are the normal stresses,

coordinates, 2 y , 2 z , 3 y and 3 z are the centroidal axes. The stress, i  , is obtained by (2). The strain that is ri  by replacing of  with r  . The strain, r  , is found by replacing 1  and 1 z with 2  and 2 z in (7). The stress ui  is obtained by replacing  with u  in (2) where u  is found by replacing 1  and 1 z with 3  and 3 z in (7). Further two equations are written by expressing the rotations of the two delamination arms by integrals of Maxwell Mohr   a l a    3 1 1    ,   a l a    3 2 2    . (10) The equation for equilibrium of the bending moments and equations (10) are used to determine 1  , 2  and 3  . Formula (5) is applied also to obtain TDSE cumulated in the left-hand delamination arm by performing the necessary replacements. Here, the time-dependent SE density, i u 02 , is found by replacing of i  and  with ri  and r  in (6). The TDSE is obtained by replacing 1 n , i y 1 , 1 1  i y , i u 01 and 1 z with n , i y 3 , 3 1  i y , i u 03 and 3 z in (5). The time-dependent SE density, i u 03 , is derived by using (6). By substituting the time-dependent SE, 1 U , 2 U and 3 U , in (4) and then in (3), one obtains involved in (2) is expressed by (7). The relationship (2) is used also to obtain

2 h

h

h

        2 1 2 2 2 2 y y i n i i i

y

       2 1 3 1 3 y y i n i i i

1 i        1 1 i 1 1 y i n i

1 h

  

  

2 2 u dy dz i 02

3 3 u dy dz i 03

G

1 1 u dy dz i 01





.

(11)

2 h

2 h

2 h

The MatLab is used to solve (11). The TDSERR for the delamination in Fig. 1 is found also by considering the compliance. For this purpose, beam compliances, 1 C and 2 C , are written as

1 

2 

C

C





,

,

(12)

1

2

M

M

1

2

1 M and

1 M are obtained as

where

2 h

2 h

y

y

1 i         1 1 i 1 1 i y i n i

1 i         2 1 i 2 2 y i n i

M

z dy dz

M

z dy dz





.

(13)

1

1 1 1

2

2 2 2

ri

2 h

2 h