PSI - Issue 54

Victor Rizov et al. / Procedia Structural Integrity 54 (2024) 475–481 Victor Rizov/ Structural Integrity Procedia 00 (2019) 000 – 000

478 4

1

1



1



    1 p t i

p





i 

i

.

(12)



i  

i

i

  

  

1

1



i 

p

i

From (6), (9) and (12), it follows that

1

1



1

    i

    p t i

1

p

     i i n i i t 1





i 

i

.

(13)

i

  

  

1

1



i 

p

i

Formula (13) is constitutive law that describes the layers behaviour. Material in each layer is continuously inhomogeneous along the beam length. The distribution of material properties in an arbitrary layer along the beam length is written as

i l E E re   i



le  

le

re

i 

i 

E E

x

x

 

 

le  

,

,

(14)

i

i

i

i

le

i

i 

l

i

i

where x l   0 . In formula (14), x is the longitudinal axis of the beam (Fig. 1), i  and i  are parameters.

/ re E E 1

0.5

/ re E E 1

1.0

1  le

1  le

Fig. 3. Normalized SERR – normalized time plots (curve 1 – at

, curve 2 – at

and curve 3 – at

/ re E E 1

2.0

1  le

).

The SERR, G , is derived as

bda dU *

G



,

(15)

where * U is the complementary strain energy (CSE), b is the beam width. Formula (16) is used to calculate the CSE.