PSI - Issue 33

R.V.F. Faria et al. / Procedia Structural Integrity 33 (2021) 673–684 Faria et al. / Structural Integrity Procedia 00 (2019) 000 – 000

677

5

where t n and t s correspond to the normal and shear stresses, and t n 0 and t s 0 to the respective material strengths. The same nomenclature applies to the strains in the MAXE equation. Finally, the QUADS and quadratic nominal strain (QUADE) criteria are established by, respectively

2

2

2

2

t t

n         s t     +

s              +

n  

or

f

f

=

=

.

(3)

0

0 s     t

0

0 s     

n    

n    

For all initiation criteria except MAXPS and MAXPE, it is possible for the user to define the direction of crack propagation from one of two perpendicular axes (main model axes). In this work, crack propagation was selected along the x axis (shown in Fig. 1 ). Thus, the crack grows along the adhesive layer’s length. On the other hand, crack growth initiates when f in the aforementioned criteria attains 1. Damage propagation in Abaqus ® is based on the phantom nodes approach. This technique’s main principle states that phantom nodes can be created in the FEM models such that elements can be partitioned in two elements (to simulate a crack and corresponding separation of the crack faces). When the damage initiation criterion triggers, the phantom nodes are created, initially with the same coordinates of the real nodes and begin linked with a damage law. Fig. 2 shows an element having nodes n 1 to n 4 . When the element is divided by a crack ( Г C ), two sub-domains are created ( Ω A and Ω B ). This division is accomplished due to the creation of phantom nodes ( ñ 1 to ñ 4 ) on top of the mesh nodes ( n 1 to n 4 ). After element cracking, the new sub-domains will contain mesh nodes (corresponding to the cracked part) and phantom nodes (which no longer belong to the respective part of the original element). The newly formed elements are completely independent regarding to the nodal displacements, which replace the original element, and they are constituted by nodes ñ 1 , ñ 2 , n 3 and n 4 ( Ω A ) and n 1 , n 2 , ñ 3 and ñ 4 (Ω B ).

Fig. 2. Scheme of XFEM crack growth: (a) before and (b) after creation of the sub-domains.

After this process, the nodal pairs formed by a real and an initially overlapping phantom node become linked by the selected damage law until reaching complete separation. When this happens, both nodes separate and become unlinked, thus emulating crack propagation. The initial concept uses a linear damage law, and mixed-mode loading evaluated by a power law criterion as presented in the expression

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

1.

(4)

In this expression, G I and G II are the tensile and shear fracture energies, respectively, at a given instant during loading, while α corresponds to the damage exponent ( α =1 if linear softening is chosen). 3.2. Numerical modelling The Abaqus ® software was used to predict the strength, by XFEM, of T-peel joints (welded, bonded and hybrid). The formulation described in the former section was used to induce crack growth in the models. Geometrical non linearities were activated because of the significant rotations induced to the joints due to the peel load. All analyses were three-dimensional, which is mandatory due to the circular geometry of the weld-nugget. However, symmetry conditions were used wherever possible. Before the strength prediction, σ y stresses were evaluated in models