PSI - Issue 13

Mauro Ricotta et al. / Procedia Structural Integrity 13 (2018) 1560–1565 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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where (r,  ) is a polar coordinate system centred at the notch tip (Fig.1b),    is the local stress and  1 is Williams eigen-value (Williams 1952) for symmetric loadings. It is well known that the direct evaluation of NSIFs by means of Eq. (3) through FE analyses requires very refined meshes. In order to overcome this limitation and to provide a sound approach for the rapid estimation of the mode I NSIF in isotropic materials, the following relation was proposed by Meneghetti and Lazzarin (2007): 1 1 * 1 FE peak K K d     (4) where:  K 1 is the exact mode I NSIF of the analysed V-notched geometry, as given by Eq. (3); typically, K 1 is thought of as ‘exact’ if calculated using definition (3) applied to the results of a very refined FE analysis;  σ peak is the opening  peak    0) in Fig. 1a), linear elastic peak stress, as calculated with the FE method at the node located at the V-notch tip using a fixed pattern of elements having a fixed average size, i.e. a fixed value of d;  * FE K is a coefficient equal to 1.38 for quadrilateral, four-node elements with linear shape functions, as implemented in Ansys software package. The range of applicability of Eq. (4) is soundly discussed in Meneghetti and Lazzarin (2007). It is here recalled that the minimum mesh density ratio a/d required is equal to 3, a being the notch depth. Equation (4) allows K 1 to be estimated with an accuracy of approximately 3%. The use of element types or FE patterns different from those discussed in Meneghetti and Lazzarin (2007) would lead to a * FE K coefficient different from 1.38. An extension to different FE software has been recently performed in Meneghetti et al. (2018) With the aim to extend the Peak Stress Method to bi-material corners, assuming that ij f ( 0) 1    , Eq. (4) through Eq.(1) becomes: * s FE peak H H d     (5) where s is calculated according to Bogy (1971). To evaluate the H * FE coefficient, two different geometries were considered, namely a butt joint and a notched joint, the latter characterized by a notch depth, a, equal to 7.05 mm, as shown in Fig. 2c. Concerning the butt geometry, different elastic properties of materials 1 and 2 were considered according to the data listed in Table 1, while for the notched joint, the notch opening angle 2  was increased from 0° to 165° with steps of 15°. The material elastic properties were equal to E 1 =330 MPa,  1 =0.49, E 2 =2200 MPa and  2 =0.41 (i.e.  =0.7179 and  =-0.0047). A total number of 45 different conditions were analysed. 2D-plain strain linear elastic FE analyses were performed in Ansys ® 18.1 commercial software, by using quadrilateral 8-node PLANE 183 elements with element option “ pure displacement element formulation ” and by switching the “Full Graphycs” option on, to average the nodal stresses at the interface:   i,A i,B i / 2         at the i-th node. After setting the global element size parameter d, the free mesh generation algorithm was run. 3. Finite element results

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Fig. 2. (a) butt joint and (b) relevant mesh; (c) notched joint and (d) relevant mesh.