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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1564

Concerning the notched joint, d was changed from 0.125 to 1.5 mm (with steps of 0.125 mm). In addition, two further analyses were carried out with d=2.55 mm and d=5.1 mm. In all cases, the standard mesh pattern shown in Fig. 2d was adopted. Fig. 3b shows the H * FE values calculated in the case of 2  =60°. It is seen that, for a/d ranging from 4.7 to 56, all the data fall in the ±3% scatter with respect to the mean value equal to 0.38. For each analysed s value, the mean value of H * FE was calculated and the results are summarised in Fig. 4a, by plotting H* FE vs s. While in Meneghetti and Lazzarin (2007) a constant value of K * FE was found for -s ranging from 0.5 (2  =0°) to 0.326 (2  =135°), it is seen that, by using 8-node elements and considering bi-material problems, the H * FE value depends on s.

0.0 0.2 0.4 0.6 0.8 1.0 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 H * FE s Butt joint Notched (2 elements) Notched (4 elements) 120° (4 elements) 120° (2 elements) (a)

1.0

(b)

Butt joint Notched (2 elements) Notched (4 elements)  >120 2  ≤120

H *

FE = 1.44 s + 1

0.8

H *

FE = - 1.44 s + 0.29

0.6

FE

H *

0.4

H *

FE = 2.34 s

2 + 2.71 s + 1

0.2

0.0

-0.5 -0.4 -0.3 -0.2 -0.1 0.0

s

Fig. 4. (a) H* FE values vs s for different mesh patterns and (b) best fitting curves to evaluate H* FE having s and the notch opening angle. Moreover, it is worth noting that the standard free mesh algorithm adopted by ANSYS code uses 4 elements around the notch tip for 2  ≤ 120° (see as an example Fig. 2d) and 2 elements for 2  >120°. Consequently, the value of the peak stress is influenced by the different mesh patterns, as shown in Fig. 4a, where triangular open and filled symbols refer to notch tip modelled by 4 and 2 elements, respectively. Dedicated FE analyses were carried out in the case of 2  =120° to confirm this result, by using mapped meshes to enforce 2 elements as well as free mesh with 4 elements and the different combinations of material properties listed in Table 2. The results are shown in Fig. 4a with circular filled and open symbols, respectively, and it can be seen that they fall in the relevant group of data. Therefore, with the aim to provide designers with a simple tool to evaluate H * FE , once calculated s according to Bogy (1971), different curves have been defined by fitting the FE results shown in Fig 4b, by using the least square method. Concerning butt joints and notches having 2  >120°, a quadratic curve has been adopted by imposing that H * FE =1 for s=0 (i.e. plain material). Conversely, in the case of 2  ≤120 two linear curves have been proposed depending on s, as shown in Fig 4b. It is worth noting that a mesh density ratio a/d equal or greater than 5 is required to adopt the best fitting curve shown in Fig.4b. In this paper, the numerical evaluation by means of coarse meshes of the intensity of the linear elastic stress fields close to the singularity point at the interface of bi-material corner is analysed. In particular, the concept of Generalized Stress Intensity Factor H, thought of an extension of Gross and Mendelson parameter (Gross and Mandelson 1972), was considered. In view of this, the Peak Stress Method proposed by Meneghetti and Lazzarin (2007) to estimate the Stress Intensity Factor at the tip of a geometrical singular point was extended to isotropic bi-material corners, by using 2D plain strain quadrilateral 8-node elements of ANSYS code. A design rule was proposed to estimate H as a function of the opening angle, either lower or greater than 120°, and the singularity exponent s calculated according to the open literature and dependent on the notch-opening angle and elastic material properties. 4. Conclusions