PSI - Issue 33

R.F.P. Resende et al. / Procedia Structural Integrity 33 (2021) 126–137 Resende et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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a) Nodal distribution, yellow is the adhesive layer

b) Geometry as a surface

d) Close-up of the right corner of the adhesive layer and substrates

c) Close-up of the overlap area

Fig. 2. Geometry as a nodal distribution and its details as surface. The 12.5 mm case shown as an example.

Table 3. Number of divisions and biases selected for the nodal distribution (dimensions in mm).

Number of divisions along the dimension

Bias in % along the dimension

L O

t P

t A

L S 80 74 60 48

L O

L S 80 90 90 80

t P

L O 20 40 40 40

12.5 25.0 37.5 50.0

10 10 10 10

4 4 4 4

102 188 220 254

80 80 80 80

The essential boundary conditions numerically replicated the experimental work described in Section 2.2. In summary, the left boundary was constrained in both vertical and horizontal directions ( U x = U y = U z =0) whilst a horizontal displacement was imposed to the right boundary ( U x =  ; U y = U z =0), in which δ corresponds to an experimental observation, as shown in Fig. 3.

U x =  U y = U z = 0

U x = U y = U z = 0

Fig. 3. Definition of boundary conditions on the geometry.

3.2. Elasto-plasticity Ductile adhesives present non-linear behavior under tensile loading. Consequently, elastic-plastic models are used to describe this behavior , which represents the material’s stress -strain relationship. In these models, the behavior is linear-elastic until the yield strength (  y ) is reached. Then, the behavior can be linear, multilinear, or polynomial, to name a few. These behaviors are intended to describe the material’s stress -strain relationship as close as possible to the experimental. As described in the previous Section, here bilinear elastic-plastic material models were used. Thus, beyond  y , the slope of the stress-strain relationship is lower than in the first linear part; also, the slope is related to the material’s hardening parameter k (  ). Therefore, the yield surface ( F ) is a function of k (  ) as well as from the stress tensor,  ; hence F (  ,  ). The von Mises yield criterion is often used to describe the yielding onset in adhesives and adhesive joints. This criterion, when represented in a three-dimensional space, describes a cylinder, where its circular cross-section corresponds to the deviatoric plane. The yield function, F (  ,  ), can be related to the second invariant of the deviatoric stress tensor, J 2 , as follows