PSI - Issue 25

R.M.D. Machado et al. / Procedia Structural Integrity 25 (2020) 71–78 Machado et al. / Structural Integrity Procedia 00 (2019) 000 – 000

75

5

t t

   

    

    

n     s 0 0 ,

    

t

n

(2)

s

f

f

max , 

or

max

=

=

0

0

t

n

s

n

s

t n and t s are the current normal and shear traction components to the cracked surface. t n 0 and t s 0 represent the respective limiting values. The strain parameters have identical significance. The QUADS and QUADE criteria are based on the introduction of the following functions, respectively

2

2

2

2

t t

n         s t     +

s              +

n  

f

f

or

=

=

(3)

0

0 s     t

0

0 s     

n    

n    

For the MAXS, MAXE, QUADS and QUADE criteria the user can select between horizontal or vertical crack growth. In this work, horizontal growth was selected for the horizontal adhesive portions, and vertical growth was considered for the vertically oriented adhesive portions at the step-ends. All the six aforementioned criteria are fulfilled, and damage initiates, when f reaches unity. For damage growth, the fundamental expression of the displacement vector u , including the displacements enrichment, is written as (Abaqus® 2013)

N

=  u

( )

( )

 + u  i

a

N x

H x

  i .

(4)

i

i

1

=

N i ( x ) and u i relate to the conventional Finite Element formulation, corresponding to the nodal shape functions and nodal displacement vector linked to the continuous part of the formulation, respectively. The second term between brackets, H ( x ) a i , is only active in the nodes for which any relating shape function is cut by the crack and can be expressed by the product of the nodal enriched degree of freedom vector including the mentioned nodes, a i , with the associated discontinuous shape function, H ( x ), across the crack surfaces. The parameters to feed the models in ABAQUS ® were taken from Table 1. A linear softening XFEM law was initially considered with an energetic failure power law criterion of the following type, where α is the power law parameter

I G G G G       + =         II IC IIC

(5)

1.

4. Results

4.1. Failure modes

In the experimental tests, a cohesive failure of the adhesive layer was found on all joints, although with adherend plasticization in some of the cases. No conclusion could be experimentally found regarding the failure path, since failure took place abruptly in the adhesive bond. Thus, it can be concluded that the bonding procedure was correctly accomplished and that the adhesive layer is always the weakest link in the joints. This is a relevant information for the XFEM analysis to the joints, since the XFEM aims to promote crack growth in the adhesive to simulate this failure mode.

4.2. Stress analysis in the elastic domain

Both  y and  xy stress distributions along the lengthwise adhesive portions and at the adhesive mid-thickness are presented and discussed. This study considers all range of L O values, hence attaining a better insight regarding the behavior of the different joint geometries. Moreover, this study will support the joint strength evaluation that will follow.  y and  xy stresses are divided by  avg , the average  xy for each L O , and were captured while the adherends and adhesive are in the elastic domain. All stresses are depicted as a function of x / L O (0≤ x ≤ L O ). Fig. 3 shows the  y (a)