Direct numerical prediction for the propertiesof complex microstructure materials

Andrei A. Gusev

Institute of Polymers, Department of Materials, ETH-Zürich, Switzerland

Outlook

Unstructured mesh approach

– Short fiber composites

– Clay/Polymer nanocomposites for barrier applications

– Voltage breakdown in random composites

Regular grid approach

– 3D X-ray micro-tomography

– Aluminum infiltrated graphite matrix composites

– Pglass/LDPE hybrid materials

– Cellular materials

Short fiber compositesPolymers have a stiffness of 1-3 GPa

- glass fibers 70 GPa

- carbon fibers 400 GPa

Can be processed by injection molding

on the same equipment as pure polymers

Short fiber reinforced polymers

- fiber aspect ratio 10-40- volume loading 5-15%

Acceleration pedal (Ford)

Gear wheel

0.5 mm

Local fiber orientation statesArea with a high degree of orientation

Area with a low degree of orientation

Non-uniform fiber orientation states

⇒ non-uniform local material properties• stiffness

• thermal expansion• heat conductivity, etc.

Direct finite element predictionsPeriodic Monte Carlo configurations

– with non-overlapping spheres

– with non-overlapping fibers

Unstructured meshes (PALMYRA)– periodic morphology adaptive

– 107 tetrahedral elements

AAG, J. Mech. Phys. Solids, 1997, 45, 1449

AAG, Macromolecules, 2001, 34, 3081

ValidationShort glass-fiber-polypropylene granulate

– Hoechst, Grade 2U02 (8 vol. % fibers)– injection molded circular dumbbells

Image analysis– typical image frame (700x530 µm)

Measured fiber orientation distribution– transversely isotropic– statistics of 1.5·104 fibers

Measured phase properties– polypropylene matrix

E = 1.6 GPa, ν = 0.34, α = 1.1·10-4 K-1

– glass fibersE = 72 GPa, ν = 0.2, α = 4.9·10-6 K-1

– average fiber aspect ratio a = 37.3

0

100

200

300

0 30 60 90

angle θ

freq

uen

cy

ValidationMonte Carlo computer models

– 150 non-overlapping fibers

h PJ Hine, HR Lusti, AAGComp. Sci.Tecn. 2002, 62, 1927

h AAG, PJ Hine, IM WardComp. Sci.Tecn. 2000, 60, 535

Fiber orientation distribution– compared to the measured one

Effective properties

numerical measured

E11 [GPa] 5.14 ± 0.1 5.1 ± 0.25α11 [105·K-1] 3.1 ± 0.1 3.3 ± 1.5α33 [105·K-1] 11.7 ± 0.1 12.1 ± 0.2

Single fiber– unit vector p = (p1, p2, p3)

System with N fibers– 2nd order orientation tensor

– 4th order orientation tensor

Two step procedure

lkjiijkl ppppa =

1 φ

θ

p

2

3

Step 1: System with fully aligned fibers

– numerical prediction for Cijkl, αik, εik, etc.

Step 2: System with a given fiber orientation state

– orientation averaging– quick arithmetic calculation

jiij ppa =

Orientation averagingReference system with fully aligned fibers

– transversely isotropic

Effective elastic constants

A system with given aij and aijkl

Effective elastic constants

– Bi are related to the coefficients of Cref

AAG, Heggli, Lusti, Hine Adv. Eng. Mater. 2002– Direct & orientation averaging estimates– agree within 2-3%

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

66

66

44

222312

232212

121211

ref

00000

00000

00000

000

000

000

C

C

C

CCC

CCC

CCC

C

32

1

)()(

)(

542

31

jkiljlikklijijklklij

ikjliljkjkiljlikijkl

BBaaB

aaaaBaB

δδδδδδδδδδδδ

++++

++++=C

For transversely isotropic samples

As the trace of aij is 1, one needs only a11

Similarly, aijkl is solely determined by

Maximum entropy structures

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

22

22

11

00

00

00

a

a

a

aij θ211 cos=a

θ41111 cos=a

The entropy of a system

pn is probability that the system is in state n

Maximum of S under a given a11

Comparison with image analysis data

nn

n ppS log∑−=

nnn θθθθ ∆+≤≤

Complex shape partsSteel molds (dies) are expensive

– on the order of $20k and more

Before any steel mold has been cut

– mold filling flow simulations

To optimize mold geometry & processing conditions• gate positions

• flow fronts

• local curing

• mold temperatures• cycle times

• etc.

Software vendors: Moldflow, Sigmasoft, etc.

– full 3D flow simulations instead of 2½ D

SigmaSoft GmbH, 2001

Computer-aided design of short fiber reinforced composite parts

MethodAAG, J. Mech. Phys. Solids, 1997, 45, 1449AAG, Macromolecules, 2001, 34, 3081

Short fibers:AAG, Hine, Ward, Comp. Sci.Tecn. 2000, 60, 535Hine, Lusti, AAG, Comp. Sci.Tecn. 2002, 62, 1445Lusti, Hine, AAG, Comp. Sci.Tecn. 2002, 62, 1927AAG, Lusti, Hine, Adv. Eng. Mater. 2002, 4, 927AAG, Heggli, Lusti, Hine, Adv. Eng. Mater. 2002, 4, 931Hine, Lusti, AAG, Comp. Sci.Tecn. 2004, 64, 1081

Spin-off company: MatSim GmbH, Zürich– Palmyra by MatSim, www.matsim.ch

Acknowledgements

– Professor U.W. Suter, ETH-Zürich

– Dr. P.J. Hine, IRC in Polymer Science & Technology, University of Leeds

– Dr. H.R. Lusti, ETH-Zürich– Professor I.M. Ward, University of Leeds

Performance of finished partAbaqus, Ansys, Nastran, etc.

Local material propertiesPalmyra + orientation averaging

Mold filling flow simulationsMoldFlow, SigmaSoft, etc.

Mold geometry,processing conditions

Voltage breakdown in random compositesDielectric parameters of pure polymers

– dielectric constants – voltage breakdown– determined by chemical composition– and processing route

Putting metal particles into polymers

– increasing dielectric constants– decreasing voltage breakdown

• local field magnifications

Periodic unstructured meshes– morphology-adaptive

Voltage breakdown mechanism– localized damage: pairs, triplets, etc– percolating path

E

Numerical procedureNumerical set up

– matrix: εM = 1& Ec = 1– inclusions: εI = 106

Apply a very small E– such that all local fields e are below Ec

Gradually increase E– when somewhere in the matrix e > Ec

– local voltage breakdown occurs– in the damaged sections, replace εM by εI

– go on

Computer model with 27 spheres– here, sphere volume fraction f = 0.3

Numerical predictions

– overall dielectric constant: εeff = 2.54– breakdown field: Eeff = 0.084– Adv. Eng. Mater. 5, 713 (2003)

E

Numerical predictionsNumerical estimates

– sphere volume fraction f = 0.1Composite voltage breakdown

– arrestingly, ensemble minimum valuesrepresentative already with only 8 spheres

Overall dielectric constants– remarkably, uniform RVE size is very small– ensemble vs. spatial averaging

Variable sphere volume loadings

Technological aspects

– relatively slow increase in εeff

– rapid decrease in Eef

0.0080.030.12Eeff

5.122.571.37εeff

0.50.30.1f

Aluminium / Graphite Composites

Squeeze casting process− graphite pre-form (porosity 14.5 vol-%)− infiltrated with aluminium melt (AlSi7Ba with 7 wt% Si & 0.25 wt% Ba )

3 phase composite: graphite (C), aluminium (Al), and pores

Al

C

Pore

3D X-ray tomography

Swiss Light Source (SLS)

Beamlines at SLS XTM (X-Ray Tomographic Microscopy)− at Materials Science beamline of SLS− beam energy of 10 keV

140 m

3D tomography microstructure

10 subvolumes:25 x 25 x 25 pixel

10 subvolumes:50 x 50 x 50 pixel

10 subvolumes:100 x 100 x 100 pixel

9 subvolumes:200 x 200 x 200 pixel

200 x 600 x 600 pixels

pixel size = 0.7 µm

Pore

Al

C

Effective conductivity

Technicalities:− serendipity family linear brick elements− iterative Krylov subspace solver− σeff from a linear response relation

Implementation− GRIDDER by MatSim GmbH

0gradφ)(σdiv =r

− for nodal potentials − with position dependent σ(r)

25 x 25 x 25 pixel model

FEM solution of Laplace’s equation

C

Al

Pore

Representative Volume Element Size

104 105 106 107 1080.01

0.1

1

10

N (number of pixels)

σ ef

f [10

6 /Ωm

] measured conductivity

upper Hashin-Shtrikman bound

lower Hashin-Shtrikman bound

Heggli, Etter, Uggowitzer, AAG, Adv. Eng. Mater. 2005 (accepted)

Pglass / Polymer hybrids

• Pglass: 0.5 SnF2 + 0.2 SnO + 0.3 P2O5− Tg ≈ 150 C− can be melt processed at ca. 200 C

• 50/50 (by volume) Pglass/LDPE hybrid− melt processed at about 200 C− J. Otaigbe of Univ. of Southern Mississippi− measured stiffness:

LDPE 0.2 GPaPglass 30 GPacomposite 1.2 GPa

Why is the composite stiffness so low ?Is the microstructure co-continuous ?

LDPEPglass

Adalja, Otaigbe, Thalacker, Polym. Eng. Sci. 2001, 41, 1055

DoM1

Slide 20

DoM1 Department of Materials, 2/1/2005

LDPE

Pglass

3D tomography microstructure

Numerical predictions for σeff assuming− either σ1 = 0 and σ2 = 1− or σ1 = 1 and σ2 = 0

Both phases percolateco-continuous microstructure

400 x 275 x 200 pixelspixel size ~ 5 µm

Property predictions (GRIDDER)

Predicted E is 7 times larger than the measured oneHypothesis: disintegration of Pglass phase

− under the influence of residual thermal stressesCritical sections:

− those with large von Mises stress & negative pressure

58

7.1

Effective

12150α [10-6/K]

300.2E [GPa]

PglassLDPE

Residual thermal stresses

( ) ( ) ( )[ ]213

232

2212

1 σσσσσστ −+−+−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

332313

232212

131211

σσσσσσσσσ

σ

( )32131 σσσ ++=p

Local stress tensor

Pressure

Von Mises stress

Pglass, ∆T = -100 K

0.01 0.1 11E-3

0.01

0.1

1

E/E

s

Volume fraction

Stiffness of closed cell foams

Real foams:− with 1 < n < 2

Kirchhoff plate theory

− n = 1 reflects wall stretching− n = 3 reflects wall bending

Sparse form solver

− implemented in GRIDDER

27 cells

1024x1024x1024 grid

1E-3 0.01 0.1 10.0

0.5

1.0

1.5

C, n

Volume fraction

n

C

nss CEE )( ρρ=

Materials with cellular microstructure

Aluminum foam3D X-ray tomography

Closed cellcurved wall foam

Open cell foam

Understanding structure – property relationships− both 3D X-ray tomography and model microstructures− mechanical, thermal, electrical, and other properties

Conclusions & Perspectives

• Unstructured mesh approach (PALMYRA)– Linear tetrahedral elements– Remarkably efficient for object based representations– Currently: spheres, spheroids, platelets, and spherocylinders– Locking problems with fluids and rubbers

• Regular grid approach (GRIDDER)– Linear brick elements– Very large grids, 109 pixels and more– Appropriate for ordinary solids, rubbers and fluids– Property estimation companion for SCFT simulations

• For more on PALMYRA & GRIDDER technologies, visit www.matsim.ch