Compact representation of
micromechanical fields
Kaushik Bhattacharya
Gal Shmuel
Jin Yang
Chun-Jen Hsueh Md. Zubaer Hossain
Dingyi Sun
Acknowledge: Mary Comer, Charles Bouman, Richard James,
G. Ravichandran
K. Bhattacharya
Review 2015: #1
Compact representation
Microstructure
Deformation
Find compact
representation
that preserves the
links
Mechanical
properties
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Topics
Multiscale Phase-field Simulation
Digital Image Correlation
Fracture of heterogeneous materials
Coarse-grained Density Functional Theory
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Phase transformation
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Phase field method
Energy
• Resolves each interface
• Need to compute large
specimens
Brinson et al.
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Multiscale phase field method
Energy
Scaling: Length scale L, Energy scale a
Large body limit
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Measures and elastic energy
Young measure (Ball, Tartar) nx describes the one-point statistics
near the point x
• Local volume fraction
• Pole figures
H measure (Tartar) mx describes the part of the two-point statistics that
is related to the induced elastic energy. Closely related to
• Localization tensor
• Eshelby tensor
• (Structure factor)
New
translation
bounds
Postulate
Phase field model for
volume fractions
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Superelasticity
Saturation.
Incomplete transformation
Reorientation
Is there a compact
representation for these
transformation strain fields that
retains the essential physics
Initiation.
Well oriented grains
Stress concentrations
Reversal.
Last on, first off
Slight differences from (B)
Progress.
Autocatalyic strips
Some reorientation
Richards et al. (2013)
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Wavelets
Haar
Frequency
Daubechies
Space
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Compressing the results using wavelets
•
•
•
•
Simulate using FFT
At each time step, take a (Symlet 5) wavelet transform of the result
Keep only 1 % of the coefficients
Compute the macroscopic stress-strain curve1
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Wavelets also provide new insight
Inactive
Total
Active
• Relative change in # of active wavelets: ~4%
• Active wavelets are predictors of stress and strain
intensities
• Dominant wavelets are basis for model reduction
• Current active set are predictors of transformation
• Active wavelets carry information about interK. Bhattacharya
granular compatibility
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Experiments with full-field strain
Daly, Ravichandran and Bhattacharya, Acta Mater. (2007)
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Representation of Daly et al. Experiments
Compression using 1% coefficients
Total strain
1.93%
2.94%
99.8%
99.9%
1.34%
99.3%
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Digital image correlation
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Digital image correlation
f(x)
g(y)
y
x
y(x)
Typically done locally
• Very computationally efficient
But,
• Need high frequency random
pattern … large data sets
• Results can be very noisy
Can we use filtered/ compressed images for DIC?
Sutton et al., Image Correlation …, Springer
Hild and Roux, in Optical Methods for Solid Mechanics, Wiley
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DIC with compressed images
Original
20 % DCT
Since the method is local,
• we need high frequency random images
• filtered and lost during compression
Generally gives significant error
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Need to introduce global information
• Global Method and Regularization (Hild and Roux)
Reduces noise,
but expensive
Use finite element basis for y
• Local method with global constraints (Augmented Lagrangian-ADMM)
Partition of unity
Related interpolation
- Computationally efficient
(multiscale approach)
- Consistent with compression
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Results with global + compression
Original
20% compressed
Global
Local
100%
20%
Avg Eyy = 1.89 %
Avg Eyy = -1.09 %
Global
0.0071 %
0.044 %
Local
0.49 %
4.61 %
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Use of priors: Fracture
Seek the stress intensity, but the
displacements are extremely small.
Collaboration with G. Ravichandran (experimental effort supported by NSF)
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Impact/Future
•
Personnel
–
–
–
–
–
•
Gal Shmuel, Post-doc (currently Asst. Prof. Technion, Israel)
Md. Zubaer Hossain, Post-doc (partial, soon Asst. Prof. U Delaware)
Jin Yang, Graduate student
Chun-Jen Hsueh, Graduate student (partial)
Dingyi Sun, Graduate student (NDSEG)
Publications
– G. Shmuel, A.T. Thorgeirsson, and K.Bhattacharya. 2014. “Wavelet Analysis of
Microscale Strains.” Acta Materialia 76: 118–126.
– J. Yang, G. Ravichandran and K. Bhattacharya. 2015. “Data compression for digital
image correlation”. In preparation for submission to Experimental Mechanics
– 4 others
•
Provisional Patent
– C-J Hsueh, G. Ravichandran, K. Bhattacharya. 2015. “A Novel device of measuring
the fracture toughness of heterogeneous materials”
•
Future directions
– Digital Image Correlation with compression (Ravichandran)
– Scale-dependent Young measure, H-measures (James)
– Mechanical properties of Al-Si (Voorhees/Kalidinidi)
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Review 2015: #22
Multiscale understanding of materials
Actuator SpringRapid growth of digital experimental
Sun
techniques and computational
power
60-Nitinol Actuator
Cover plate
with
has given unprecedented level
ofheater
data about materials
Attach Fasteners
+
NiTinol
SMA
Composite Base
Assembly
Mabe
How can we harness this for
understanding and ultimately
Composite
designing?
substrate
Free
Fan microdoStream
we represent
Stream
How
mechanical fields?
Schryvers
Brinson et al.
James and Chu
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