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.