PSI - Issue 38

Michaela Zeißig et al. / Procedia Structural Integrity 38 (2022) 60–69 Zeißig, Jablonski / Structural Integrity Procedia 00 (2021) 000–000

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4.2 Weibull distribution As the SLM process and thus the properties of SLM specimens are based on stochastic elements, a probability based concept is chosen. Fatigue properties are often calculated based on a fundamental approach by Weibull (1939). It is assumed that specimens initially include randomly distributed defects, hence the number of defects is related to the specimen size. Crack initiation occurs if the local stress exceeds the local material endurance. The Weibull distribution is a probability density distribution and has been applied to various problems, including the field of fatigue, see Weibull (1951). In the context of the fatigue assessment of SLM specimens, the approach allows for the inclusion of porosity as well as mean stresses and residual stresses while the likely failure mode may be included via the applied equivalent stress hypothesis. Furthermore, the distinction between failure from an internal defect and failure from a surface defect or surface roughness can easily be incorporated. Here, we apply the Weibull distribution to 316L and use a linear elastic isotropic material model which allows scaling of external loads until a survival probability of 50% is reached, see Koehler et al. (2012). The fatigue data for the following calculations is taken from Liang (2020). The calculation of the survival probability ��������,∆� of an individual volume element ∆ , as applied in Bomas et al. (1997), is given by    V 0 Mises WV / / survival,ΔV 2 m V V P      (4) � represents the total specimen volume, the equivalent stress amplitude is included via the von Mises stress ����� and �� corresponds to the local fatigue limit according to the load ratio (here R = -1). Via the Weibull exponent � , the effect of scale and density of defects may be adjusted. Integration of (4) leads to ��������,� of the whole specimen. Kahlin et al. (2017) showed that there is little deviation of hardness values throughout the thickness of a specimen. Hardness is therefore assumed to be constant and consequently properties like ultimate tensile strength and thus the local fatigue limit are also assumed constant. Based on that and the data given in Liang (2020), for a 50% survival probability according to (4), the fatigue limit is determined to 91.5 MPa and the Weibull exponent to 51. The exponent seems to be overestimated in view of the pore scattering present and it is expected that a broader experimental data basis, especially in the range of the estimated fatigue limit, will lead to smaller values. The influence of smaller Weibull exponents is shown in Fig. 2, left. The slope of the curve flattens with decreasing exponent; however, the relation is not linear. For the following calculations, the local fatigue limit value �� in the absence of residual stresses and surface influences is required. Therefore, the value is again based on the ultimate tensile stress data as previously described. The basic description in (4) can be extended to take the origins of fatigue failure, thus the different possible crack initiation sites, into account. In order to do that, the specimen needs to be divided into two zones to determine the material strengths for surface and volume separately. This approach was used e.g. by Bomas et al. (1997) for inhomogeneous materials. The survival probability of the whole specimen results from the product of the individual survival probabilities as, see Bomas et al. (1997),

survival survival,A survival,V P P P  

(5)

Another possible extension is the inclusion of residual stresses. As mentioned before, these are a characteristic of SLM specimens due to the production process. In the untreated state, they are not negligible and depend on the building