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 basis (p) and degree of polynomial (n) can be varied to arrive at best linkage

without over-fitting the data.

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

𝐶11 = 𝑓𝑛(𝑃𝐶1, 𝑃𝐶2, … . , 𝑃𝐶𝑝)

𝐶11GPafromsimulation

𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝐶11 (GPa)

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