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
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