PSI - Issue 28

1088 I.J. Sánchez-Arce et al. / Procedia Structural Integrity 28 (2020) 1084–1093 Sánchez-Arce et al. / Structural Integrity Procedia 00 (2019) 000–000 �10� Where is the normal, ̅ is the tension applied to the natural boundary ( � ), and � is the displacement applied to the essential boundary ( � ) [12]. Then, the weak-Galerkin form can be expressed in a matrixial form, as shown in (11), where is the stiffness matrix. Therefore, the boundary conditions and loading can be applied to it following the Hooke law [12]. δ � � � �� � 0 �11� 3.2. SLJ models Similarly to the experimental work described in Section 2.1, the joints were analyzed numerically employing various two-dimensional geometrical models solved using the FEM and NNRPIM methods, from which the number of elements and nodes are listed in Table 2. For comparison purposes the same number of nodes and node location was used in the models for both FEM and NNRPIM; however, NNRPIM does not require the elements and could provide accurate results with a coarser nodal distribution [11]. 5 � � � ̅ � � Although the number of nodes and nodal distributions were kept constant between the FEM and NNRPIM solutions, the latter had more two times the number of integration (Gauss) points in the adhesive layer, providing a more detailed stress distribution. The  yy (peel) and τ xy (shear) stresses were compared with those obtained with the FEM (as a benchmark). The combination of method and � gave a total of eight models. A MATLAB (The MathWorks; Natick, Massachusetts, USA) script was written for creating all the geometries parametrically, with similar nodal distributions, loads and boundary conditions, and so allowing consistency in the comparisons. The geometrical dimensions were equal to those employed in the experimental part. Then, all the models were solved using both FEM and NNRPIM techniques by using a custom-written MATLAB script for each method. In all cases, a displacement (δ=0.1 mm) was applied to the right side of the joint, simulating the pulling exerted by the UTM. The left side of the joint was fixed (U x =U y =U z =0). The material properties of the adherends and adhesive were taken from previous experimental work (see Section 2.1). In this work, the material properties were considered as isotropic and linear-elastic, and so, these results can be compared with predictions derived from theoretical models. Once all the models were solved the  yy and τ xy stresses distributions, in the mid-adhesive layer, were extracted and analyzed. Each � combination resulted in its own stress (both  yy and τ xy ) magnitude, although the stress patterns should be similar. To compare them, the data were normalized. A normalized � was considered from 0 to 1, where 0 represents the left end whilst 1 the right end; the normalization was achieved by dividing all the points by � . Similarly, the stress magnitudes (shear and peel) were normalized with respect to the average shear stress (τ ave ), as described by Crocombe [22]. Table 2. Number of nodes and elements used for the FEM models. Same nodes were used on the NNRPIM models. L O (mm) Number of nodes Number of Elements 12.5 7835 7444 25.0 11035 10560 37.5 11835 11360 50.0 11151 10680

4. Results and discussion 4.1. Experimental behavior

All the specimens presented cohesive failure in the adhesive layer. Moreover, no plastic deformation of the adherends was observed. The joint strength ( P max ) increased proportionally with � , such increment followed a ‘linear’ trend (Figure 2). Additionally, the low variation observed in the experimental data (Figure 2) indicates a good specimen preparation. The increase of P max related to L O was also noted in previous work with different adhesive types