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)