Finite element simulation of ultrasonic wave propagation in anisotropic polycrystalline aggregate

Transcription

1 2 nd Workshop on Laser Ultrasonics for Metallurgy (April th 2016, Vancouver, Canada) Finite element simulation of ultrasonic wave propagation in anisotropic polycrystalline aggregate Thomas Garcin The University of British Columbia Acknowledgments: Quentin Puydt, Bai Xue, Bing Tie, Alyssa Shinbine, Chad Sinclair, Warren Poole, Matthias Militzer, Jean Hubert Schmidt. 1

2 Introduction What is the best tool to simulate the propagation of ultrasonic wave in metals? Multi-scale problem (6 orders of magnitude): Sample (~ 10 mm) Wavelength (~ 0.5mm) Microstructure ( ~ mm) Solving linear elasticity in anisotropic medium The equations are complex for real microstructure. Could we validate LUMet measurement using Finite Element Simulation? 2

3 Explicit Dynamic Analysis in Abaqus Dynamic equilibrium equations written with inertial forces isolated from other forces For dynamic force, equation reduce to static form of equilibrium Central difference algorithm : Unknown values are obtained from information already known 3

4 Stability criteria Mesh size 4 µm: i) at least 20 elements in one wavelength ii) Multiple elements in one grain λ min = v min = 4000m/s 10 f max 20 50MHz = 4 μm Time step 2 ns : at least 4 time steps to move a node by a distance of one mesh size Δt = λ min 4 v max = 4 μm m/s = 0.2 ns 4

5 2 mm or 8 mm Model definition and geometry 2D simulation with plain strain conditions Element type (4 nodes), about 8x10 6 elements Square grid in the area of interest Rectangular geometry 10 mm or 40 mm 5

6 Initial step Imposed displacement at top surface (2 mm) Gaussian Spatial distribution Double Ricker signal in time 6

7 Propagation and detection Average displacement of nodes (or velocity) over a 2 mm segment at a free surface Either at top or at bottom surface 7

8 How to generate a grain structure? Centroidal Voronoi Tesselation. (GNU GPL license) Iterations with initial assignment of generators. Generate a network of points uniformly distributed. 8

9 Iterative homogenization code Tend toward an optimal partition corresponding to an optimal distribution of generators. All cell have 6 faces but the final structure is not ordered 9

10 Frequency count Shape of the grain size distribution Grain size distribution is symmetric with respect to a mean value. Width is function of the number of iteration Red: 5 iterations Black : 1 iteration Grain size, EQAD (µm) 10

11 Comparison of grain size distributions Width is 10% of the mean value after 100 iterations Cumulative function validate the self similarity of each distribution 11

12 Material properties Single crystal stiffness tensor FCC iron at 1423 K (Zarestky et al., 1987, Phys.Rev. B 35(9), pp.4500) Single crystal elastic constant: c 11 = 154 GPa c 12 = 122 GPa c 44 = 77 Gpa Zener Anisotropy factor c 44 /c = 4.8 Crystallographic orientation 12

13 Selection of crystal orientation Generation of weak texture, all orientations are almost equally likely to be found. Random generation of Rodrigue vectors (Matlab code written by C. Sinclair & G. Lefebvre) 13

14 Rotation of elastic tensor Each grain must be allocated with one elastic tensor rotated according to its crystal orientation Pressure wave velocity (vertical propagation direction) 14

15 Meshing of the grain structure All the simulation are conducted with the same mesh size of 4 µm Effect of 4 µm square mesh on the morphology of the grain boundaries D = 30 µm D = 100 µm D = 200 µm 15

16 Are we sampling enough grain? For large grain size, the number of grain through thickness decreases drastically Different set of orientation at each experiments. D = 30 µm D = 300 µm 2 mm 2 mm 16

17 Quantify the polycrystalline anisotropy In material science, we like to use poles figures D = 30 µm D = 300 µm 17

18 Degree of anisotropy In the ultrasonic community, one can refer to the degree of anisotropy (I = J = 1 for a compressional wave) Stanke & Kino provide solution for weak anisotropy 18

19 Repeatability experiments Each simulation is repeated 6 times with the same grain structure and different initial texture. The degree of anisotropy is used for comparison 19

20 Displacement field Example of a displacement field for sample with a mean grain size of 100 µm in sample of 2 mm thickness. 20

21 Reference (isotropic) material Random ODF (Volume Fraction of orientation V) Weighted average on elastic tensor (T) 21

22 Selection of appropriate averaging Velocity in the small grain size sample should be close to satisfy the isotropic condition D = 30 um => V = ± D = 100 um => V = ±

23 Attenuation measurements Evaluation of ultrasound attenuation using the isotropic sample as reference. (example: 100 um) 23

24 Results: Attenuation spectrum 30 µm 100 µm 200 µm 300 µm 24

25 8 mm Simulation on wider sample Evaluate the attenuation in the sample with large grain with better statistic. 300 µm 25

26 Results: Attenuation spectrum 30 µm 100 µm 200 µm 26

27 Calibration for the austenite grain size Grain size calibration developed by the NRC at the Timken company Calibration from optical metallography and Laser ultrasound measurements Valid for the austenite grain size in low alloyed steel α(f) = b*f^3 27

28 Application to FEM spectrum By selecting appropriate frequency range, the calibration provide satisfying agreement with FEM generated attenuation spectrum 28

29 Scattering regime Mathematical solution exists for 3D materials with equiaxed grain and untextured. Stanke & Kino proposed the unified formalism covering the whole range of scattering regime 29

30 Normalized attenuation Normalized frequency x ol = 2π d f v Normalized attenuation per unit length α[1/m] = α[db/m]

31 Conclusions FEM can be implemented to simulate the wave propagation in anisotropic aggregate On thin sample with large grain size, dispersion of the measured attenuation spectrum Simulation validated by applying the calibration for austenite grain size (Gives quantitative values) Application to two phases material, there is more to say about averaging methods (A. Shinbine, C. Sinclair) 31

32 Alyssa Shinbine

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