statistical continuum mechanics as a tool for materials … · statistical continuum mechanics as a...
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Statistical Continuum Mechanics as a Tool for Materials Design
Hamid Garmestani, G. Saheli, D. L. Li School of Materials Science and engineering
Georgia Institute of Technology
ARO/GATECH Workshop2
Ti-6Al-4V
• (L. Semiatin, AFRL, C. Hartley, AFOR)• Scott Schoenfeld (Army Research Lab)Alpha (HCP), Beta (BCC)• Effect of texture for each phase and in combination at different Temperatures
0
20
40
60
80
100
700 800 900 1000
Temperature (oC)
Vo
lum
e F
ract
ion
of
Bet
a (P
ct.)
Present Work (Microprobe)Present Work (Quant. Metall.)Castro & Seraphin
50 µmTi-6Al-4V - Predictions of stress behavior at diferent temperatures (range 830-950 Celsius)
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Volume fraction of Beta phase (%)
upper bound
statistical
lower bound
910 C
950 C830 C
600 C
RT
ARO/GATECH Workshop3
Challenge
• What are the optimal properties necessary for design of a component:
• For Ti-6A-4V: What are? • Textures
• Grain boundary morphology
• Distribution of the two phases
• …
• Inverse Methodologies
ARO/GATECH Workshop4
Methodology
Desired MicrostructureProcessing
Desired Properties
A Methodology to Link the Property and
Microstructure
Characterize the Microstructure by Probabilities Functions
Applying Statistical Continuum Mechanics Theory
Vary Texture and phase distribution
50 µm
ARO/GATECH Workshop5
Microstructure RepresentationVolume fractions as first order
Higher order probability functions
Homogenization relations(Statistical continuum Mechanics)
Microstructural Sensitive Design
Property
Microstructure
Statistical Continuum Mechanics
DistributionFunctions
ARO/GATECH Workshop6
General Strategy for MSD
1-Microstructure representation: The microstructure and its detail is represented as a set of orthogonal Basis Functions (Microstructure Hull).
2- Properties and Constraints: are represented in the same orthogonal space
3-Coupling: The properties and constraints will now represent hyper planes in the material Hull 4- Designer Materials: Intersection of these planes defines the set of materials appropriate for design similar to Ashby’s Diagrams.
F(χn,Cn ) = Cnχn
n
∑
P(χn, pn ) = pnχn
n
∑
ARO/GATECH Workshop7
Inverse Methodology at the Heart of MSD
Property, Pn
Microstructure
Statistical Continuum Mechanics
DistributionFunctions
F(χn,Cn ) = Cnχn
n
∑
P(χn, pn ) = pnχn
n
∑
χn
Cn
Both Microstructure and properties are
represented using the same set of
orthonormal basis functions
ARO/GATECH Workshop8
Transformation to Fourier Space(Adams, Kalidindi,..)
Ý Ý k l
* µ (φ ,β )
-Material Hull: υ i = 1i∑
v iÝ Ý k l
*µ (φ i ,βi )i∑
One point statistics-Series representation:
-Material Set:
(all φ, β)
f(φ,β ) = Flµ
µ =1
M(l)
∑l =0
∞
∑ Ý Ý k l
µ(φ,β )
)|,(2 rhhf ′2-point statistics: ),,( gh φλ= ),,( gh ′′′=′ φλ
ARO/GATECH Workshop9
One Point Probability
• The number of points in one phase compared to the rest defines a total probability function P(Φ1) and P(Φ2)
Schematic micrograph of a composite Enlarged area for the measurement of onepoint probability
v1 = P(φ1), v2 = P(φ2)
P(φ1) + P(φ2) = 1.0
ARO/GATECH Workshop10
• Attach a vector r=r0 to these random points. • Now find out the probability of a specific phase at the head of
the vector given the phase at the tale of the vector P(r){1,2|1,2}
Enlarged area for the measurement oftwo-point probability
Two Point Probability as a Conditional Probability
ARO/GATECH Workshop11
Definition of Orientation
e1
e2
e3
a2
a1
a3
OIM Scan data
Orientation g Orientation g’
a3
a1
a2
ARO/GATECH Workshop12
50 µm
Two Point distribution distribution function for a two phase structure
i=1, 2
r
Φi Φj
f2(Φ,r) = Flµn
n
∑µ∑
l
∑ e− inr / rc Ý Ý k l* µ (φ ,β)
ARO/GATECH Workshop13
Empirical Forms of Two point Probability Functions
• Anisotropic form
Pij (r) = α ij + βij exp(−cijrnij )
For a highly anisotropic materials, the limit for P12 and P21 at a particular angle, θ=θ0 is zero. Let’s take θ0 =0, Then,c12(0,k)=ac012, and n12= n0
12.
But, P12=0 for any r or,
c ij θ,k( )= c ij0 a + 1− a( )sinθ( ) nij θ,k( )= nij
0 1− 1− a( )sinθ( )
P12(r) = α12 + β12exp(−ac120 rn12
0
)
ARO/GATECH Workshop14
Two Point distribution distribution function for the orientation space
OIM Scan data
a
3
a
1 a
2
e1
e2
e3
a2
a1
a3
f2(g, ′ g | r ) = Flλµnσρ(r ) Ý Ý T l
µη(g) Ý Ý T λσρ ( ′ g )
ρ∑
σ∑
λ∑
η∑
µ∑
l∑
r
A 3+3+3=9 parameter equation
ARO/GATECH Workshop15
Fourier representation
Construction of the above function from a set of Dirac-like distributions ( ):
f2(g, ′ g |r
r ) = Fl ′ l µ ′ µ n ′ n (
r
r )′ l ′ µ ′ n
∑lµn
∑ Ý Ý T lµn (g) Ý Ý T ′ l
′ µ ′ n ( ′ g )
f2(g, ′ g |r
r ) = ν j (r
r )δ(g − g j )δ( ′ g − ′ g j )j
∑ ( ν j (r
r ) =1)j
∑
δ(g − g j )δ( ′ g − ′ g j )
ARO/GATECH Workshop16
Statistical MechanicsComposite Formulation
• Materials’ response under deformation is represented by a unified stress strain relationship
εσ C=Stiffness matrix represents the mechanical properties of the composite
Stiffness matrix represents the mechanical properties of the composite
Phase 1
Phase 2
ARO/GATECH Workshop17
Statistical Mechanics Analysis
• Hill’s Criteria:
• is defined to present the heterogeneity such as:
• Equilibrium Eq:
• The local moduli and compliance :
)()()(
0,
xxcx klijklij
jij
εσσ
=
=
)(~)(
)(~)(
xssxs
xccxc
ijklijklijkl
ijklijklijkl
+=
+=
εε cC =
ijkla
εεεε a=−=~
ARO/GATECH Workshop18
Ergotic Hypothesis:
• Statistically Continuous Objects
• Random, disoriented objects
• Ensemble Average = Volume average
σij(r) =
1
∆V (r)σ
ij(r )
∆V (r)∫ dV
σij(r) =
1
Nlim σ
ij(r )
N∑
σij
( r ) = σij( r )
Volume average
Ensemble average
ARO/GATECH Workshop19
Method of the Simulation
• Measurement of two points probabilitiesin the microstructure
• Use of Perturbation Techniques to incorporate one, two and higher Statistics
• Satisfying Equilibrium Eqs by applying Hill’s Criteria
• Finally: Calculating the fluctuations of the Elastic properties relative to the average value.
ARO/GATECH Workshop20
Estimation of Effective Elastic Properties
• The effective elastic properties of the composite can be calculated by :
• is the average ensemble of elastic property
• is the average fluctuation
)()(~ xaxccC mnklijmnijklijkl +=
)()(~ xaxc mnklijmn
ijklc
ARO/GATECH Workshop21
RESULTSElastic Properties of an Isotropic Al-Lead
• Material 1: Lead ( Pb )
Gpa
Gpa
• Material 2: Aluminum
(Al)
Gpa
Gpa25
286.64
==
µλ
926.4
88.25
==
µλ
Elastic Modulus
01020
30405060
7080
0 0.2 0.4 0.6 0.8 1
volume of the second phase(Al)
E (G
pas)
E upper E lower E stat
ARO/GATECH Workshop22
Effect of porosity (isotropic distribution on Elastic Properties)
Elastic Modulus of (Pb + Porosity)
02468
10121416
0 0.1 0.2 0.3 0.4 0.5 0.6Porosity volume fraction
E(Statistical) E (Voigt) E (SC)
ARO/GATECH Workshop23
RESULTSElastic Properties of an Anisotropic Al-Lead
Property Hull of Al-Pb
20
25
30
35
40
45
50
55
30 40 50 60 70 80 90
C1111 (GPa)
C11
33 (
Gpa
)
C1133 Upper Bound C1133 Lower BoundAl:20% Al:30%Al:40%
1
2
3
The microstructure is repeated in the 1-2 plane.
a=0.061-0.093
ARO/GATECH Workshop24
Elastic Property of Ti-6Al-4VUsing 2-point function
Ti-Al6-V4, Elastic Property in different Temperature
88
90
92
94
96
98
100
102
104
106
0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00%
Volume Fraction (beta phase)
vol=20% , T=830 cent.vol=40% , T=910 cent.vol=60% , T=950 cent.
Starin rate of 0.0003
E upperE lowerE stat
Plastic Property vs. Elastic PropertyTi-6Al-4V
0
50
100
150
200
250
300
350
104.752 98.4748 91.7333
E (Gpa)
Upper boundLower boundStress (Stat.)
850 C
910 C
950 C
ARO/GATECH Workshop25
Statistical Mechanics Modeling of a Two Phase Medium
• Two Green's Function Solution (Molinari, et. al., Acta Met, 1987)• Infinite Body
• Solution
NijklR Gkmlj(r − r' ) − Hm,i (r − r' ) +δmδ(r − r' ) = 0
Gim ,i(r − r' ) = 0
vi (r) = v i + Gij(r − r' ) f jr'∈V∞∫ (r' )d3r'
p(r) = p + Hi(r − r' ) f jr'∈V∞∫ (r' )d3r'
Localization Relations
With the same procedure, the Modulus N can also be expanded in Taylor series around L
˜ N (L) = ˜ N (L ) +∂ ˜ N (L )
∂L[ L − L ] + ...
˜ N = ˜ N + ˜ N ' [L − L ] + ...
˜ N ijkl(L) = ˜ N ijkl (L ) +∂ ˜ N ijkl(L )
∂Lmn
[ Lmn − L mn ] + ...
Compact Form
More compact
ARO/GATECH Workshop27
Polycrystalline MaterialsDevelopment of Texture during Rolling
0.0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1
{112}<111>{123}<634>{110}<112>
Taylor
{112}<111>{123}<634>{110}<112>
Statistical
Vol
ume
frac
tion
Eeq
CopperS
Brass
Equivalent strain
RD
ND
TD
(110)
[112]
Brass component
(100) (111) (110)