PSI - Issue 38

Alexander Raßloff et al. / Procedia Structural Integrity 38 (2022) 4–11 ff et al. / Structural Integrity Procedia 00 (2021) 000–000 A. Raßlo

7

4

6

30

4

20

2

10

0 600 Shortest distance to next pore surface δ S in µ m 0 Rel. number of pores 100 200 300 400 500

0 Rel. number of pores

20

40 80 100 120 140 Equivalent spherical diameter d eq in µ m 60

1

2

3

Fig. 3. (Left) ESD d eq distribution and fitted gamma PDF for three exemplary CT scans; (right) distribution of the shortest distance to the next pore surface δ S and fitted lognormal PDF.

CT for three real specimen. As can been seen, the ESD distribution can be well fitted to a gamma PDF

l Γ

s Γ

f d eq ; a , l Γ , s Γ =

exp l Γ −

a − 1

d eq

d eq

Γ − 1 ( a ) , d

eq ≥ 0 , a > 0 .

(1)

s Γ −

The parameters s and l scale and shift the function, whereas a denotes the shaping parameter and Γ ( · ) denotes the gamma function 1 . A lognormal PDF f ( x ; σ, l ln , s ln ) = δ S s ln − l ln σ √ 2 π − 1 exp    − log 2 δ S s ln − l ln 2 σ 2    , x > 0 , σ > 0 , (2) is applied to fit the distribution of the shortest distance to the nearest pore surface. Here, the shape parameter is denoted by σ . The Python package SciPy is employed for identifying the parameters by maximising a log-likelihood function. The volume fraction of pores, i.e. the porosity φ , serves as another descriptor. Reconstruction. The grain structure and pores are reconstructed separately. As the focus of the present work lies on the investigation of the influence of pores, one grain structure is reconstructed in which the pores are incorpo rated. DREAM.3D as presented by Groeber and Jackson (2014) is utilised for this task. The structure parameters for DREAM.3D are chosen so that the resultant grains represent lamellae colonies, i.e. regions of similar orientation within a prior β -grain. The mean grain ESD is set to 200 µ m with a length / width-ratio of 4 to 5, based on the LM observation. The resultant SVE of size (1200 µ m) 3 is discretised in 200 3 elements and is visualised on the left in Fig ure 4. The pores are reconstructed following a procedure of sampling random variates from the distribution functions (Equation 1, Equation 2) for given parameters. For the present study, they are chosen to resemble CT scan 1 from Fig ure 3. A Python script is used to (a) compute the ESD and number of pores by sampling from the corresponding PDF, (b) distribute the spherical pores spatially by sampling from the distance PDF and (c) compare the resultant distribu tion to the input and either restart or finish. Now, the pores are incorporated into the grain domain from DREAM.3D, yielding the final SVE for the CP simulation as described in the subsequent section. Figure 4 shows an exemplary SVE. 1 The gamma function is defined as Γ ( z ) = ∞ 0 t z − 1 e − t dt for arguments z with a positive real part.