Composite Properties and Microstructure I:

Effective Properties

Robert LiptonLouisiana State University

Composites: Where Mathematics Meets Industry

February 7, 2005IMA-Institute for Mathematics and its

Applications

Supported in part by: AFOSR and NSF

Composite structures

Fiber reinforced epoxy Horizontal StabelizerBoeing 777Racing Sail

Bone

Multi-scale problem statementIt is supposed that the length scale of the composite microstructure is

significantly smaller than the length scale of the load.

The goal is to characterize the behavior of the composite structure when the characteristic length scale of the loading is much larger than the composite microstructure.

0

x

Effective properties such as:

1) Effective stiffness2) Effective thermal conductivity3) Effective dielectric constant

Are introduced within the context of homogenization theory.

Homogenization theory provides a way to calculateaverage strain measured by a strain gauge in microstructured

media.

ε= Length scale of microstructurerelative to characteristic dimension of loading

Strain gauge

Set covered bystrain gauge is S

Strain gauge measuresaverage strain over S

Boundary value problem in periodically microstructured elastic media

)(xuε

)(xεσ)(xεε

10 <<< εLength scale of microstructure

Elastic displacement field2/)( ,,

εεεε ijji uu +=Stress tensor

)()()( xxCx εεε εσ =

iCxC =)(ε

Strain tensor

Ci is the elasticity of ith material

Traction BC on ΓNand displacement BC on ΓD

fdiv =− )( εσ

gxn =• )(εσUu =ε ΓD

ΓN

RVE

1Local elastic tensor

Constitutive relation

Equation of elastic equilibrium

Boundary conditions:

Homogenization theory;convergence of averages

dxxuS

dxxuS S

M

S∫∫ → )(

||1)(

||1 ε

dxxS

dxxS S

M

S

)(||

1)(||

1∫∫ → εε ε

dxxS

dxxS S

M

S∫∫ → )(

||1)(

||1 σσ ε

For any subset S:

= Average ofhomogenized

strain

Average strain over S:

)()( xCx MEM εσ =Homogenized constitutive relation

2/)( ,,M

ijMji

M uu +=ε

fdiv M =− σHomogenized boundaryvalue problem

UuM =gxn M =• )(σ

Traction BC on ΓNand displacement BC on ΓD

Formula for CE

Load unit RVE with homogeneous strainLocal elastic tensor in microstructure = C(y)Micro-problem in RVE is to solve for periodic strain fluctuationLocal periodic strain fluctuation e(y) solution of:

CE given by:

__ε

,0)))()((( =+ εyeyCdiv

dyyeyCCQ

E ))()((∫ += εε

1RVE:y

Here: Q denotes the RVE

∫ =Q

dyye 0)(

Physical interpretation of the effective elastic tensor CE

Local stress inside RVE

)))()((()( εσ += yeyCy

εε =+∫ dyyeQ

))((

εσ E

Q

Cdyy =∫ )(

Average strain over RVE

Effective elasticity relatesaverage stress to average strain

Approximate formulas for effective elastic tensors

Given only a partial statistical descriptionof the microstructure the goal is to construct an approximation to the effective tensor

Two well known examples are considered

1) Low volume fraction expansions for dilute suspensions.J.C Maxwell (1873), A. Einstein (1906)2) Low contrast expansions.W.F. Brown (1955)

Effective elastic tensor for a dilute suspension of spheroids

Data: C1=elastic modulus of spheroidsC2=elastic modulus of matrix, θ= volume fraction of spheroids

Expand CE in a Taylor series with respect to volume fraction

j

jj

E BCC θ∑∞

=

+=1

2

The coefficients Bj encode geometry of the distribution of the particles in the suspension, eg., they reflectclumping together of the particles, etc.

B1 is associated with the single particle boundary value problemin an infinite matrix subject to applied field averagedover all orientations

B2 associated with a two particle boundary value problemin infinite matrix averaged over all two particle configurations

Bj associated with j particle boundary value problem in infinite matrix.

The expansion when only the volume fraction and orientation distribution is known

)( 212 θθ OBCC E ++=

1211 )( TCCB −=

dyyeE E

))((||

11 εε += ∫ ∞T

0)))()((( =+∞ εyeyCdiv

0|)(||)(|

1lim 2

)(

=∫ ∞

∞→dyye

RB RBR

Here brackets denote orientationaverage and T1 is theWu strain tensor and E is the spheroid domain.

The fluctuating part of thestrain solves a boundary valueproblem on the whole space:

B(R) is a Ball of radius R and

Low volume expansion forspheres

C1 isotropic and specified by µ1 κ1

C2 isotropic and specified by µ2 κ2

Material 1 incompressible: µ1/κ1= 0Material 2 incompressible: µ2/ κ2= 0

)(32

)(5 2

21

212 θθ

µµµµµµ OE +

+−

+=

Established rigorously using variational methods byJ.B. Keller, L. Rubenfeld, and J. Molyneux (1967)

Low volume expansion for the effectiveshear modulus

For rigid spheres µ1/ µ2=∞ have Einstein’s formula (1906)

)(25 2

2 θθµµ OE ++=

Formulas for higher order terms in expansion of elastic tensor see, Chen and Acrivos (1978)

For complete references and detailed derivationswe refer to the recent manuscripts of of G. Milton (2002)and S. Torquato (2002)

Low contrast expansions

Contrast given by δC=C2-C1

W.F. Brown (1955)

Expand effective elastic tensor in a TaylorSeries with respect to contrast

j

jj

E CACC )(1

2 δ∑∞

=

+=

The geometry is encoded in the coefficients Ajthese are given in terms of N-point correlationfunctions.

One and two point correlationfunctions

One point correlation Two point correlation

r

Probability that both endsof a segment lies in yellowphase when thrown into

composite randomly

Probability of a pointlying inside the yellow phase

(Volume fraction of the yellow phase)

N-point correlationfunctions

N-point correlation function = probability that N pointschosen at random lie inside yellow phase

An example of three point correlation.

Probability that all verticesof a triangle lies in yellowphase when thrown into

composite randomly

One, two, and three-point correlation Functions

These statistical functions can be computed using techniques from

image analysis. J. Berryman 1988.

Coefficients Aj of contrast expansion

,)( 21 CCyC −=δSet for y in material one

,0)( =yCδε=)(0 ye

for y in material two

Set

And iterate according to

,...2,1,0)),()(())(( 12 =−=+ jyeyCdivyeCdiv jj δ

dyyeyCAQ

jj )()(∫= δε

Aj is given in terms of the j-point correlation function

And for j=1,2,… set

W.F. Brown (1955), M. J. Beran (1965)

An Example: Bulk modulus for isotropic effective elastic tensor

12 µµδµ −=12 κκδκ −=

23

213

32

212 )()()( AAAAE δκδµδκδκδκκκ ++++=

)43)(1()43()1(3

22112 µκθµκθ

θθ+−++

−−=A

θ=1A

22211

1

)]43)(1()43([)21)(1(9

3 µκθµκθθθθ

+−++−−

=A

22211

12

)]43)(1()43([))(1(12

3 µκθµκθθζθθ+−++

−−=A

G. Milton (1982)

Here is a geometric parameter given in terms of the three point correlation function

1ζ

10 1 ≤≤ζ

The parameter is computed for cubic arrays of spheres1ζin McPhedran and Milton (1981)

Extensive references to the literature and detailed analysis can be found in the recent books byG. Milton (2002) and S. Torquato (2002)

Bounds on effective properties:An example from heat conduction inside

composite mediaConsider the ensemble of composite samplesmade from two isotropic heat conductors withsame volume fractions – having an isotropic effectiveconductivity ΣE = σEI.

Subject each sample to the same imposedtemperature gradient.

Find lower and upper bounds on the effectivethermal conductivity σE.

Identify realizations that attain the bounds.

Setting

RVE Cube filled with compositemade from two heat conductors.The volume fractions of eachis prescribed.

0=∆T

TnTn ∇•=∇• 21 σσ

Conductivities:Volume fractions:

σ1 > σ2θ1, θ2

Inside each phase

On interface

xET •−and is periodic on the cubeET =∇

Effective Conductivity

σ(y) is the local conductivity taking the value σ1 inside material one and σ2inside material two.

The effective conductivity is given by

dyyTyEQ

e ∫ ∇=Σ )()(σ

Bounds of Hashin and Shtrikman for effective properties

Hashin Shtrikman bounds on effective conductivityfor isotropic composites (1962).

eσσσθσσσσθσ ≤−+−

+)(3

)(3

2122

21212

)(3)(3

1211

12121 σσθσ

σσσθσσ−+−

+≤e

Coated sphere assemblage ofHashin and Shtrikman

Step I: Fill the RVE witha sphere packing withsphere sizes ranging tothe infinitesimal.

Step II: Place a smaller concentric sphere inside each sphere such that the ratio of the radii of the outerand inner spheres is the same.

Optimal MicrogeometriesThe lower bound is attained by the Hashin Shtrikmancoated sphere assemblage with core of material one and coating of material two.

The upper bound is attained by the Hashin Shtrikmancoated sphere assemblage with core of material twoand coating of material one.

Bounds for effective thermalconductivity with coupled mass and thermal

transport on the two-phase interfaceGreen spheres represent pores

Matrix is ceramic

Physical Process:

A temperature gradient isplaced across porous ceramic

Impurities diffuse throughmatrix and release heat whenarrive on surface of pore.

When impurities the leave poresurface and reenter matrixthey absorb heat; this effect canenhances or decrease the effective conductivity of the porous ceramic.

Steady state coupled heat and mass transport inside the RVE

00 ,TC

sC C

Ca ⎟⎠⎞

⎜⎝⎛∂∂

=

Thermal conductivity of the matrix σm

Thermal conductivity of the pore σp : σp < σmTemperature inside the composite T=T(y)Impurity concentration inside the composite C=C(y)Impurity concentration on the pore surface Cs=Cso+αT(T-T0)+αC(C-C0)Cso , T0 , C0 are the equilibrium values and

The coefficient aT is negative since Cs decreases with temperature

The surface diffusivity as has dimensions of length x diffusivity00 ,TC

sT T

Ca ⎟⎠⎞

⎜⎝⎛∂∂

=

0=∆T 0=∆CCDCa SSSS ∇•−=∇•∇ n)(CqDTT porepmatrixm ∇•−=∇•−∇• nnn )()( σσ

q is specific heat releaseEPrescribed average temperature gradient

Boundary conditions: T- E.x is Q periodic, C is Q periodic.

inside each phase

Effective thermal conductivity

E Prescribed average temperature gradient across RVE

j Prescribed average heat flux acrossRVE

Ej EΣ=

Microgeometry known onlystatistically

It is supposed that the microgeometry is such that the effectiveconductivity is an isotropic tensor ΣE=σEI.

The microgeometry is composed of spherical pores of differentradii. It is supposed that the volume fractionand size distribution of pore phase is known.

Since the matrix is a better conductor than the pore we seekto determine conditions on the pore size distribution and material properties for which σE>σm and σE<σm

The effective conductivity with pore phase conductivitybeing perfect conductor is σ∞ . The effective conductivitywith pore phase insulator is and σ0.

Interplay between microgeometry and material properties

Average pore radii ⟨a⟩Harmonic average of pore radii ⟨a-1⟩-1

If

Then

111 1122

−∞−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

−−

≥p

CS

pm

TS

Daaaqaa

θσ

σσ

mE σσ ≤ No enhanced conduction for sufficiently small

surface to volume ratio

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

−−

≤ 1122 0

p

CS

pm

TS

Daaaqaa

θσ

σσEnhanced conduction for surface to volume ratio sufficiently large

If

mE σσ ≥Then R. Lipton (1999)

JMPS (47)1699-1736

One has upper and lower bounds on σ∞ and σ0 in terms of lower bound on particle separation, Bruno (1991) P.Roy.Soc. Lond.(433)355-381.

For monodisperse suspensions see Torquatoand Rubinstein J.App.Phys.(69)7118-712 for bounds on σ∞ and σ0 in terms of nearest neighbor distribution function.