PSI - Issue 33

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Riccardo Caivano et al. / Procedia Structural Integrity 33 (2021) 1095–1102 Riccardo Caivano et al./ Structural Integrity Procedia 00 (2019) 000–000

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Similarly, through Eq.(4) it is possible to plot the maximum first principal stress limit with respect to R as reported in Fig. 2b. This plot can be used to set the Murakami fatigue limit in the TO HyperWorks environment. It is wort noting that the fatigue limit estimated with Eq. (3) must be limited to the fatigue limit without defects, according to [30]. A constant stress constraint over the quasi-static structural limit must be also set [32]. On the contrary, it is worth remembering that, if the considered stress ratio is smaller than 0, two different load cases must be imposed in the TO setup, one for the different senses of the applied force. Finally, Fig. 2c and 2d show an example of the LEVD cumulative distribution function and of the Gumbel plot for the defect size within the final optimised part, respectively. As for the finite element analysis, square second-order elements have been employed with a side length of 2 mm, i.e. 8850 elements. The filtering radius for the optimisation is set to 4 mm, i.e. 2 times the element size. It has been verified that a smaller value of the filtering radius may lead to very tiny side structures in the final topology. On the other hand, a filtering radius greater than 3 times the element size may increase remarkably the amount of element with intermediate densities at the end of the optimisation, affecting the final topology reliability. Overall, for this type of optimisation, the filtering radius should belong to the range between 1.5 and 3 times the element size. As for the convergence parameters, with a maximum number of cycles equal to 500 and a convergence checker of 10 -4 , the final topology results to be almost clear and well defined. Four different stress ratio conditions are selected to carry out the TO in HyperWorks. First, the stress ratio R is set equal to -1, i.e. fully reversed tension-compression. This condition is characterised by the lowest value of the limit on the maximum allowable stress of 116 MPa according to Eq.(4) and as visible in Fig. 2a. As already mentioned, two different load cases are set, the first one with ��� downward and the second with ��� equal to - ��� (upward). Concurrently a limit on the total volume fraction of 30% and the maximum admissible von Mises stress equal to 260 MPa are imposed. Fig. 3a shows the result of the optimisation where all the constraints have been satisfied. Therefore, the structure can be considered safe even if the predicted greatest defect, according to the LEVD distribution, would accidentally lay in the highest tensile-stressed portion of the material in any of the different load cases.

b) R=-0.5

c) R=0

d) R=0.1

a) R=-1

����� �� � � ������� ����� �� � � ������� ����� � ������� � ������� �� �� � ������� ����� � ������� � ������� Fig. 3 – Final topologies obtained under different stress ratio conditions: a) R = -1; b) R=-0.5; c) R=0; d) R=0.1. Th term stands for the mean compliance of the final topology. ����� �� � � ��� ��� ����� �� � � ������� ����� � ������� � ��� ��� �� �� � ��� ��� ����� � ������� � ��� ���