Date post: | 16-Jul-2015 |
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Novel protocol for developing Structure-Property linkages for Polycrystalline materials
ME8883/CSE8883 : Material Informatics
Group Members:Dipen Patel
Akash Gupta
Polycrystalline Material• Polycrystalline microstructure can include all features of internal
structure of heterogeneous materials at different length scales
▫ e.g.: phase, grain size, crystal orientation, dislocation, voids, interatomic spacing, etc.
The crystal orientation, g, can be defined by a set of three ordered rotations (φ1, Φ, φ2) that relates the crystal frame to the sample frame.
Spatial distribution of the crystal lattice orientations at the micro scale plays an important role in controlling their effective properties.
Objective/Motivation
• Advance materials are inherently anisotropic
▫ Spatial distribution of the crystal lattice orientations at the micro scale plays an important role in controlling their effective properties.
• Develop protocols for structure-property linkages to tailor materials that meets the functionality and design requirements.
▫ Homogenization: communicating the local properties to higher length scales.
▫ Linkages will ultimately be helpful in process-design
Framework
Generating Synthetic Dataset
• Fundamental Zone (FZ) for cubic crystal lattice
• 3D 21 x 21 x 21 microstructures to simulate elastic deformation
▫ 222 distinct orientation were selected on the surface of FZ
▫ Selected orientation were assigned to each class of microstructure
1200 microstructures of each class were included in the calibration dataset
Finite Element Simulations – Property Calculations• Periodic boundary conditions* are
applied to elastic deformation model
*Landi, Giacomo, A novel spectral approach to multiscale modelling, PhD Thesis
Effective elastic property, 𝜎𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙𝜀𝑘𝑙
Step 1: Generation of Calibration Dataset
Generate synthetic representative
microstructures
Obtain mechanical response for each
microstructure using an established
numerical model
Step 3: Establishment of Structure-Property Linkages
Generate linkages using regression
methods on structure and property data
Validate linkages using Leave-One-Out-
Cross-Validation (LOOCV)
Step 2: Reduced Order Quantification of Microstructure
Low-dimensional representation of
microstructure based on Principal
Components Analysis
Quantify microstructure using a desired
subset of n-point correlations
Conventional approach
123
m 𝑔 = 𝑚(𝜑1, Φ, 𝜑2 ) = ℎ 𝑖𝑓 𝑔 = (𝜑1, Φ, 𝜑2 ) ∈ ℎ
• For each bin, indicator basis function is defined as:
𝐻 − 𝑙𝑜𝑐𝑎𝑙 𝑠𝑡𝑎𝑡𝑒𝑠
where the local state space is divided into H bins, ℎ = 1,2, … , 𝐻.
Binning of orientation space (FZ)
Building Microstructure Function:
𝜑1
𝜑2
Φ
Conventional approach
where the total spatial bin is divided into S bins, s = 1,2, … , 𝑆.
2-Point Statistics using indicator basis:
𝑓𝑡ℎℎ′ =1
𝑆
𝑠=0
𝑆
𝑚𝑠ℎ𝑚𝑠+𝑡ℎ′
1
2
1630
28
S
New ApproachBuilding Microstructure Function using continuous basis function:
𝑚(𝑔) =
𝐿
𝑎𝐿 𝑇𝐿(𝑔)
𝑓𝑡(𝑔, 𝑔′) =1
𝑆
𝑠=0
𝑆−1
)𝑚𝑠(𝑔)𝑚𝑠+𝑡(𝑔′
g
g
t
where 𝑇𝐿 𝑔 is generalized spherical harmonics basis functions weighted with appropriate coefficients.
2-Point Statistics using continuous basis:
𝑓𝑡(𝑔, 𝑔′) =
𝐿
𝐿′
𝑘′
𝑎𝑘′𝐿 𝑇𝐿(𝑔
∗
𝑎𝑘′𝐿′𝑇𝐿
′(𝑔′)𝑒
2𝜋𝑖𝑘′𝑡𝑆
DFTs
• 2-Points Statistics using indicator basis function
▫ Primitive binning of the local state space is computationally highly inefficient
Binning of FZ leads to large number of discrete local state space for orientation representation
Example:
H (50 bin) = 72 X 9 X 9 = 5832
▫ Not compact for representing orientation
Increase the total number of statistics for higher discretization of the local state
• 2-Points Statistics using GSH basis function
▫ GSH basis allows continuous representation over orientation space
▫ Compact representation of the local state space.
Only 10 local states are required to represent the entire orientation space
Advantages of New Approach
𝑓𝑡ℎℎ′ = 𝑆𝐻2
Quantification of Delta Microstructure• Plots of Product of Fourier coefficients and their conjugates in real space
𝑓𝑡(𝑔, 𝑔′) =
𝐿
𝐿′
𝑘′
𝑎𝑘′𝐿 𝑇𝐿(𝑔
∗
𝑎𝑘′𝐿′𝑇𝐿
′(𝑔′)𝑒
2𝜋𝑖𝑘′𝑡𝑆
𝑎𝑘′𝐿 ∗𝑎𝑘′
𝐿′ = 𝐹𝑘𝐿,𝐿′ 𝑖𝑓𝑓𝑡 𝐹𝑡
𝐿,𝐿′
5 10 15 20
5
10
15
20
Quantification of Fiber Microstructure
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
0
0.05
0.1
0.15
0.2
0.25
Auto Fiber
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Cross Fiber
Cross Random
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Quantification of Random Microstructure
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10 0
1
2
3
4
5
6
Auto Random
PCA for all 3600 microstructuresDimensionality of each microstructure reduced from 231525 (SH2) to 25 (significant PCs)
0 5 10 15 20 25 300
10
20
30
40
50
PCs
Expla
ined V
ariance
Preliminary Results: Regression/LOOCV analysis for linkages• Number of PC and power of polynomial (n) can be varied to arrive at best linkage without overfit
• Preliminary results does not show good linkages
All microstructures. Number of PC = 5 , Power of polynomial (n) = 2
165 170 175 180 185
165
170
175
180
185
yhat
y
Goodness of Fit Scatter Plot
Conclusions
• Novel protocol is presented for efficiently capturing structure-property linkages for polycrystalline material.
• GSH provides a continuous basis function for compact representation of crystal orientation
• PCA results look promising as they were able to separate out different class of microstructures
• Structure-property linkages for elastic response of polycrystalline material have been developed but linkages needs further improvement.
• Further extension of structure-property linkages to capture plastic response.
Collaboration/Acknowledgement
▫ Yuksel Yabansu, GT (code for generation of microstructure dataset)
▫ David Brough, GT (for discussions on GSH )
▫ Ahmet Cecen, GT (for Low Rank Approx. to compute PCA)
▫ Course instructors: Dr. Kalidindi and Dr. Fast