Issue 72

M. Perrella et alii, Fracture and Structural Integrity, 72 (2025) 236-246; DOI: 10.3221/IGF-ESIS.72.17

   s



J

  s



 

(2)



 s

Successively, the traction-displacement law can be expressed by means of a single function or a system of equations, which best interpolates the data obtained by Eqn. (2). Fig. 5.b highlights how adhesive layer behavior can be described by means of three analytical equations, which globally and properly fit the experimental results. These formulas can be easily implemented in a FEM code for predictive purposes. More specifically, the mode II cohesive traction-separation obtained via DIR1 method is expressed by the system:



    ,

K

      

s

s

s max _

0

 s



  s

 



 A e



 

 

(3)

,

d

s max _

s

s tr _







  K

  

     s tr s _

,

tr

s

s tr _

s f _

f

where the term    max s max K 0 _ / is the cohesive elastic stiffness and  s max _ is the displacement at the maximum shear stress  max , whilst the slip displacement  s f _ corresponds to the full damage condition when cohesive stress have null value. The softening behaviour is expressed by coupling an exponential stage, described by the parameters A and d , and a linear branch that better represents the brittle failure of bonded joint. The transition between the two branches of the softening stage occurs at the displacement  s tr _ corresponding to the stress  tr . The coefficient K f is the final softening rate:

    tr s f _



K f

(4)





s tr _

The values of CZM parameters resulting from identification are reported in Tab. 2. In addition, the critical energy release rate J c represented by the area under the curve formula (3), calculated using integration between   s 0 and    s s f _ , is listed into the same table.

 s max _ 0.024

 max

A

d

K o

K f

J c

Method

DIR1

45.5165 0.0492 1165.20

27.96

2441.77

1.39

Table 2: Traction-separation law parameters by DIR1 method.

Direct method DIR2: evaluation of exact fracture toughness by cohesive forces The DIR2 method is instead based on the exact solution for CZM identification of adhesive layers subject to shear load and on the concept of cohesive forces, formulated by Cricrì [21]. Experimental data provided by the load cell and the linear variable displacement transducer (LVDT) of testing machine in addition to results, in terms of displacements and strains,

from DIC analysis are necessary. The flowchart of DIR2 approach is depicted in Fig. 6. From the equilibrium of the cohesive energy density acting on an ENF specimen it follows:

2













2

     P a x M 16

 P a



   sa sb

P

9

3 8





   sa Q Q 

   sb

b

o

 

 



(5)

3

3

h

16

EBh

EBh

16

where    sa Q and    sb Q are respectively the cohesive energy density at the crack tip xa and in a point x b and  x is the horizontal distance between the points at x a and x b . The location of the point x b should be selected sufficiently far from the point x a , in order to best enhance the accuracy of the approach proposed by Cricrì [21], but far enough away from the mid span, i.e. the loading point, in which there is a significant local deformation occurrence.

240