PSI - Issue 33

A.M. Mirzaei et al. / Procedia Structural Integrity 33 (2021) 982–988 Author name / Structural Integrity Procedia 00 (2019) 000–000

985

4

  

  

 

r 

 

 

1

tanh

,



 

  

lim



r    

F

(10)

F

c  

c



F







c

1   

,

  

  

lim

lim



2.3. Cohesive crack mode: Rigid-Linear Softening Model (CCM) According to CCM, the interface is characterized by a linear softening from  c to  r as the relative displacement s increases from 0 to s f :

s

   

(     

)

s s 

c

c

r

f

s

(11)

[ ] s





f

s s 

r 

f

where:

2

 G

s



IIC

(12)

f

 

c

r

Using no-slip and traction-free conditions at x =0, s [0]=0 and s '[0]=0, and employing Eq. (11) for s < s f , the shear stress along the interface is:   c ch r [ ] cos 1 x x l            (13) The bond length required for a fully developed softening zone, l eff , can be calculated using the condition of  [ l eff ] =  r . Thus:   r eff eff c r h Arccos 1 l l       (14) In Fig. 2, shear stresses for different stages of debonding based on the CCM model are plotted.

F

(a)

F

F

(h)

F

 c

 c

(b)

(i)

 r  c

 c

(c)

 r

(j)

l eff

 r  c

 c

(d)

(k)

 r

l eff

a

 r  c

 c

(e)

(l)

 r

a

l eff

 r  c

(m)

 r

(f)



a

(g)



 r

Fig. 2. Distribution of shear stress for different stages of debonding based on the CCM. (a) to (g) for long, and (h) to (m) for short bond lengths.