PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 115–124 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

118 4

  0 0 

ij i  

,

(11)

The integration constant is found as 0 1  C . Therefore, (10) takes the form

(12)

s



ij

1 1

s

   

s      1 ij

s t s ij

v

ij



i 



.

(13)

j 

ij 

ij

The strain in the model in Fig. 2 is written as        j m j i iE ij ij 1     ,

(14)

where m is the number of springs and dashpots. By substituting of (8) and (13) in (14), one derives

s



ij

1 1

1

s

   

     s ij ij 1

s t s

v

  

ij E v t 

  

  

    ij r

    j m j 1 

ij



.

(15)

ij 

ij

Relationship (15) is applied for treating of the non-linear viscoelastic behaviour of the beam depicted in Fig. 1. As already mentioned, the maim aim of this paper is to obtain the strain energy release rate, G , for the delamination problem in Fig. 1. By analysing the balance of the energy, the strain energy release rate is found as

b a b M    1 

u

a U





b G F 



F

,

(16)

a





where F u and  are the axial displacement of the application point of the external force, F , and the angle of

rotation of the free end of the lower crack arm, U is the strain energy in the beam. First, the strain energy is found. By integrating of the strain energy density, one derives        01 1 1 1 1 1 1 u dz U ab i z z i n i i i        2 02 1 2 1 2 1 u dz ab i z z i n n i i i 03 3 1 3 1 3 ) ( u dz l a b i z z i n i i i       ,

(17)

where 1 n is the number of layers in the lower crack arm,

i u 01 is the strain energy density in the i -th layer, 1 z is the

vertical central axis of the lower crack arm, i z 1 and 1 1  i z are the coordinates of the upper and lower surface of the layer. The designations in the second and third term in (17) are analogical. The strain energy density, i u 01 , is obtained as        0 01 d u i . (18) The distribution of the strain along the thickness of the lower crack arm is written as   n z z 1 1 1     , (19) where 1  is the curvature, n z 1 is the coordinate of the neutral axis. The following equations of equilibrium are used to determine 1  and n z 1 :

z

1 i       1 1 i 1 1 i z i n i

F b 

dz

,

(20)

1