SecondOrder Microstructure Sensitive Design

1
Second-Order Microstructure Sensitive Design
Brent L. Adams Department of Mechanical
Engineering Brigham Young University Work
sponsored by the Army Research Office. February
2006
2
Collaborations
David Fullwood, Carl Gao, Jordan Cox Brigham
Young University Surya Kalidindi, Gwenaelle
Proust, Max Binci Drexel University
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MSD Spaces
Designed Processing Routes
Microstructure Hull
Property Closure
g
FE Model
g
Design
4
First-Order Linkage Between MSD Spaces
Texture Hull
Property Closure
Homogenization Techniques
e.g. First-Order Bounds for Elastic Stiffness
1-point statistics f(g) and local properties
Cijkl
Global properties Cijkl
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First-order linkage

• Volume average, f(g), of each state (e.g.
  orientation, g) known
• Fourier representation of f(g) used to preserve
  the symmetry of the material
• Bounding theories give max and min for global
  properties
• Wide range of possible microstructure designs for
  each chosen f(g), may give wide range of
  properties

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Second-OrderLinkage Between MSD Spaces
Microstructure Hull
Property Closure
Homogenization Techniques
2-point statistics f(g,gr) and local
properties Cijkl
Global properties Cijkl
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Second-order linkage

• Two-point correlation function, f2(g,gr) gives
  probability of finding orientations at particular
  separation r
• Texture function, M(x,g), gives texture at each
  point and used as basis for mapping
• Fourier representations used to discretize real
  and orientation spaces and the linking map
• An algebraic instantiation relation is found
  linking the Fourier coefficients of M and f2.
• Perturbation theory used to give single property
  for each microstructure
• Optimization theory finds boundary of closure

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Common Framework For Field Theories
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Second-Order Theories
Non-local constitutive relations (e.g. elasticity)
Low contrast expansion (polycrystals)
2nd order theory for texture design introduces
the 2-point spatial correlations of local lattice
orientation combining crystallographic and
morphologic texture.
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Explicit form of the second-order correction

• Second-order correction Greens function
  convolution

• Correlation tensors / 2-point orientation
  correlation functions

• Full second-order correction for statistically
  homogeneous polycrystals

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Recovery of 2-point spatial (pair) correlations
of orientation
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Experimental Recovery of the Pair Correlation
Functions PCFs
Scan 1
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Experimental Recovery of the Pair Correlation
Functions PCFs
Scan 1
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Typical (renormalized) Pair Correlation Functions

• The PCF can be normalized such that

g.s. dist.
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Coherence Limit and the Coherence Plateau
The Coherence Plateau increases as grain
size/sample volume increases, and as the
tesselation of the fundamental zone coarsens.
CP
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Central Problem Can we define the set (hull) of
all physically-realizable 2-point orientation
correlation functions?
The problem of r-interdependence in
statistically-homogeneous microstructures
Upshot no method is known for proceeding
directly to the hull of 2-point orientation
correlation functions.
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Introducing the Texture Function (TF)
The texture function M(x,g) is the volume density
of material in an infinitesimal neighborhood of
material point x that associates with lattice
orientation in an infinitesimal neighborhood of
measure dg of orientation g.
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Introduce piecewise continuous rectangular models
The influence of the spatial correlations of
components of microstructure is obtained by
evaluating the Greens function convolutions over
pairs of cubical boxes. The overall effect is
then obtained by summing over all pairs.
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Primitive representations (piecewise-continuous)
of the Texture Function
Primitive basis indicator functions
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Fundamental relationship between the TF and the
2-point correlation functions
In continuous form
In finite Fourier form
Instantiation relation -gt
Note the algebraic (quadratic) dependence between
the coefficients of the TFs and those of the PCFs!
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COMPLETENESS Construction of the
Hull of Texture Functions
Define eigen-texture functions
Each sub-cell has orientation from only one
sub-region
All possible microstructures are convex
combinations of the eigen-texture functions
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Geometrical Interpretation of the Hull of Texture
Functions
Defines a hyper-cube in the SN-dimensional
Fourier space
Set of S constraining hyper-planes intersecting
the SN-dimensional hyper-cube to define an SN-S
dimensional polytope, also known as a pyramidal
polytope
The corners (extreme points) of the polytope are
the NS eigen-texture functions
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Sequence of Polytopes for S4, N2
V 1/4
E
D
V 1/8
V 0
B
C
A
J
V 1/2
K
V 3/8
G
I
V 5/8
H
P
L
O
V 1
M
V 7/8
N
V 3/4
Letters locate vertices (eigen-microstructures).
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Eigen-textures for S4, N2
A
C
D
B
F
E
G
L
K
N
O
M
P
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Explicit form of the second-order correction in
the primitive basis
The Aleph coefficients are constants defined by
the basic elastic constants of the phase, and by
the Greens functions appropriate to the boundary
conditions of the problem. Note the quadratic
dependence on the coefficients of the texture
functions
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Link between Hull and Closure

• Properties are quadratic functions of discretized
  texture function
• Standard optimization techniques may be used to
  determine closure

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Exploring the closure

• Points near the property closure boundary are
  hard to find
• This figure shows properties for 10,000 random
  microstructures plotted on the full closure

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Exploring the closure

• Generalized Pareto front techniques are used to
  find the sides of the closure
• These are new methods, and use quadratic
  programming (QP) and sequential QP.

Generalized weighted sum
Adaptive normal boundary interception
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Second order closure

• First-order closure (o) is larger than second
  order (?)
• The second order closure does not give rigorous
  bounds
• But some parts of the first order closure may not
  be realizable

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Genetic Algorithms

• Genetic algorithms may be used to find the
  closure boundary
• The maximin algorithm has been used to spread out
  the points
• The corner points must be found first and
  continually re-inserted into the search since
  random microstructures are rarely near the
  boundary
• This method is generally good for large problems

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Linearizing the quadratic solution to find
eigen-microstructures

• Eigen-micro-structures are found on the boundary
• The linearized solution (o) is close to the full
  solution (?)

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Eigen-microstructures around C11 / C66 closure
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Set of eigen-texture functions for a 4-component
model of size 43

• Many eigen-microstruct-ures on the boundary have
  strong symmetry

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Hole in plate example

• Minimize stress concentration factor (assuming
  anisotropic material)

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Design Optimization

• The closure is searched using convex combinations
  of boundary points
• Once the boundary points are available, this is a
  rapid process

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Optimal Design
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Summary and Conclusions

• Second-order homogenization relations give rise
  to coupling of crystallographic and morphologic
  texture.
• Microstructure hulls of texture functions
  eliminate the complex r-interdependence of the
  PCFs.
• An algebraic instantiation relation is obtained,
  defining the set of RVEs associated with fixed
  PCFs.
• When expressed in terms of the primitive
  representations, the microstructure hull is a
  convex polytope, whose vertices are eigen-texture
  functions.
• The boundaries of the properties closure are
  readily explored by Pareto front optimization
  methods facilitated by QP and SQP methods.