PSI - Issue 33

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

984

3

where  = ( E p h p t p ) / ( E b h b t b ) and s = u p  u b . Parameters E p and E b illustrate the plate and block Young modulus, h p and h b illustrate the plate and block height, while their thickness is shown by t p and t b . In Eq. (1), the bond-slip law,  [ s ], is the constitutive (cohesive) law for the interface: in the following two different models are employed. 2.1. Linear Elastic Brittle Interface Model (LEBIM) According to the LEBIM, the interface is a bed of linear springs with a stiffness equal to k . The constitutive law of the interface is mentioned in Eq. (3):

k s

s s 

  

f

(3)

[ ] s





s s 

r 

f

where s f is the final relative displacement i.e. when the shear stress drops to the residual strength. Considering Fig. 1 and Eq. (2), boundary conditions are:     0 0 0 0 s      (4)

r  F a t 

1









  l a 

 



p

s l a

r  F a t 





 



(5)

p

t h

p p p E t h

p p

where a is the debonded (crack) length with constant stress distribution equal to  r . From these boundary conditions, the maximum shear stress is:

  

coth    

  

F

l a 



(6)

max   [

] l a   

a



r 



l

t

l





ch

p

ch

where  =2 k G IIc /  c c is the interface strength. According to Griffith’s criterion, propagation occurs as G II reaches G IIc , hence: 2 2 and 

ch   l     

  

  

  

  

F

l a 

coth

a



r 

  

(7)

r

t

l

2

(

)

  

p

ch







m

ax

r

G

G

I

I

II

c

2

2

k

k

Therefore, the debonding load is:   1 tanh r r F           

 

(8)

  

 





where:

2

IIc p p E h

IIc p p E h



1 2

F

G

G

,

F t  

l

c  

c

p

ch

1

1

t









c 



c p

(9)

F

l

a

 

,

,

,

F



 

 

 

r

r



F

l

l

c

ch

ch

c

2.2. Maximum load vs. bond length for LEBIM For bond lengths higher than  lim =1/√  Arccosh[√(√  /  r +1)], by setting to zero the derivative of the debonding load with respect to the crack length and some mathematical simplification, the maximum load during debonding, F c , can be calculated as: