Asymptotic Homogenization of 3-D Microstructures Simulated Either by the Multi-phase
Field Method or by Cellular Automata
G. Laschet1, M. Spekowius2, M. Apel1 and Ch. Hopmann2
1ACCESS
2Institute
e.V, RWTH Aachen University, Germany
of Plastics Processing, RWTH Aachen University,
Germany
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Multiscale Simulation of Manufacturing Processes – Linepipe Tube
Heating
Multi-Pass Rolling
Cooling
U-O-Forming
UP-Welding
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Calibration
Multiscale Analysis Tools
asymptotic expansion
interface tools
microscale
homogenization
HOMAT
Mesh2Homat
Mesh2XXX
effect. thermoelastic properties:
● Hooke matrix, eigenstrain,
● thermal expansion coeff
effect. thermal conductivity
Mesh2Abaqus
effect. Darcy permeability
virtual uniaxial tests
■ 3-D microstructures simulated by:
●
ABAQUS
multi-phase field method for metals
● cellular automata for semi-crystalline plastics
effect. elastoplastic flow curves
■ real 3-D microstructures digitalized via CT
■ synthetic 3-D microstructures
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Features of HOMAT version 5.1
● Several homogenizations in one run: f. ex. at different specified T
● Multilayer or monolayer homogenization for selected layers
● different homogenization analyses can be achieved in one run: thermal, mechanical, …
● homogenization models: 3-D (default), 2-D (plane strain) or 2-D (plane stress)
● Output: - microscopic displacements and implicit fluxes / forces on the initial or deformed
RVE (VTK files viewed in Paraview )
- effective properties in Abaqus format (.mat file)
● HOMAT detects automatically the double, period. nodes on the wedges and fixes
directly the rigid body motions
● Localization analysis in critical macroscale regions
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Application 1: Austenite to Ferrite Transformation
Austenite
Ferrite
High temperature phase fcc-Fe (austenite) transforms into bcc-Fe (ferrite) during cooling
C diffusion and/or interface kinetics controls the process rate
The molar volume of ferrite is larger than austenite (low C steels)
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Austenite to Ferrite Transformation - Problem Definition (I)
low carbon steel: Fe-0.023C-0.17Mn-0.009Si [wt %]
■
Cooling curve extracted from the thermal analysis of
sand casting of a graded steel disk
TC 1
■
Statistical homogenization procedure1 determines
the RVE size of a random grain microstructure with
no specific texture
■
RVE: microstructure generated by Voronoi tesselation:
1543cells,
non-uniform cooling
213 grains
with - cell spacing: ∆x = 0.5 µm
- mean grain size: <rgrain = 8 µm>
quasi-isotropic, random orientation distribution via
a novel procedure
1G.
Laschet, P. Hul and M. Apel, WCCM-14, Barcelona, July 2014
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Austenite to Ferrite Transformation: Problem Definition (II)
■ Phase properties:
● temperature dependent surfacial energies and mobilities:
cm4/(Js)
=
austenite: µ0 = 35 cm4/(Js) ; ferrite: µ0 =150 cm4/(Js)
● temp. and conc. dependent elastic constants
(cubic symmetry)1, density2 and mole volume
Zener anisotropy factor: - austenite: 3.723
- ferrite:
1Ghosh
2.389
and Olson, Act. Mat., vol. 50 (2002) 2655-75.
2Cho et al, Met. And Mat. Trans. A, vol. 42 (2011) 2094-2106
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Multi-Phase Field Simulation Results on the RVE
T= 1000°C, fα=0%
t=0 s
■ austenite
■ ferrite
T= 862°C, fα=9.2%
T= 844°C, fα=49.5%
T= 699°C, fa=99.6%
t = 38 s
t = 42 s
t = 120 s
■ transformation kinetics is simulated as C diffusion process
■ due to low carbon content, the austenite phase transforms into ferrite without
cementite formation but with residual austenite (high C concentration)
final state: 981 ferrite + 98 residual austenite grains
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<r>=4.47 µm
Variation of Effective Young and Shear Modules During Phase Transformation
Ehom
Ghom
■ Ehom (T) and Ghom(T) decrease strongly in the range 826°C -856°C
■ Eii(T) / Gij(T) present max. deviation from Em / Gm for pure austenite state:
∆Em,2= -0.95% ; ∆Gm,13= -1.80%
quasi-isotropic global behaviour predicted
■ Em (T) and Gm (T) ) transferred to the macrosimulation
result of practical interest
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Effective Volumetric Eigenstrain and Thermal Expansion Coefficients
αhom
κhom
■ T >860°C: eigenstrain κii decreases due to thermal contraction of austenite (slope = αii)
■ 860°C < T< 825°C: phase transformation κii increases as Vmol,fer > Vmol,aust
■ T < 825°C: κii diminishes again due to αfer
■ residual orthotropy of αhom: Dα11 max. for pure austenite (-16.88%); reduces to -5.43% for ferrite
1st Int. Workshop on Software Solutions for ICME
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Variation of Effective Poisson Coefficients During Phase Transformation
■ mean Poisson coeff. varies strongly from austenite (νm ≈0.341) to ferrite (νm ≈ 0.280)
1st Int. Workshop on Software Solutions for ICME
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macroscale
Example 2: Integrative Multiscale Simulation of the Injection Moulding Process
structural analysis with
effective local material
properties with ABAQUS
Multi-phase flow and heat transfer
σ, ε
simulation with COMSOL
microscale
T, v, p
homogenisation
3-D prediction of PP
with HOMAT
microstructure with
Eh, νh, αh, λh
SphaeroSim
ΦS, κ
R. Spina et al., Jnl. of Mater. Forming, 2014
Spekowius et al, WCCM-14, Barcelona, July 2014
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Zoom in the Structural Composition of a Semi-Crystalline Thermoplastic
crystalline/amorphous
lamella
folded molecular chain
molecular chain:
a,b,c: dimensions of the
crystalline mono-bloc
spherulite
sample
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Mechanical Behaviour of the Amorphous and Crystalline Phases
■
Difficulties: - amorph and crystalline phases are inseparable
- it is impossible to produce a pure amorphous / crystalline phase
quasi-impossibility to characterize each phase separetely
■
Spherulite of a 3 mm PP plate
amorphous phase: assumed to be in a rubbery state at 20°C
G = 0.3 Mpa ; E = 0.9 Mpa et ν = 0.49993
■ crystalline phase: ● Tashiro & al. (2002) achieve molecular dynamic calculations
● Sakurada & Kaji (70, 75): X-ray diffraction measurements
1.26
1.27
0
0.8
0 
 2.8
1.26
− 0.36
3.1
1.15
0
0 


1.82 1.95
− 0.57
0
0 
21.4
cr
=
HPP

−
0
0
0
1.85
0
0.12


[Gpa]  0.8 − 0.36 − 0.57
0
1.735
0 


−
0
0
0
0.12
0
1.75


1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Two-Level Homogenization Scheme of Semi-Crystalline Thermoplastics
■
Model: successive homogenizations at different scales:
● nanoscale: homogenization of the amorphous-crystalline bilamina
● microscale: spherulite homogenization with a radial distribution of
equivalent bilamina around the spherulite center
■
F.E. discretization of the spherulites
● two random numbers specified per spherulite
● in each Gauss point:
r r
r
- the eff. Hooke matrix of the bilamina is adopted with PG − C = e1
r r
r
- e2 ,e3 defined with the random numbers perpendicularly to e1
3 Euler angles: θ, φ, ψ
■
HOMAT ● the orientation discontinuity at the spherulite interfaces
● the variation of the bilamina orientation in each spherulite
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Microstructure Prediction of an Injected Moulded 2mm PP Plate
■ Injection moulding process parameters:
tool = 80°C ; melt = 280°C
larger spherulites below the
thin microstructure at the skin
ΦS
5 elements
■ Mould filling and heat transfer analysis: model
almost homogeneous
microstructure
with 124264 elements
spherulite
diameter
■ 3-D Sphaerosim simulations for the selected
5 elements
also larger spherulites below the
very thin skin microstructure
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc
Homogenization of the Bilamina of isotactic PP at the Nanoscale
■
Definition of the bilamina RVE:
● length ≈ 1500 nm; thickness = 15 nm (X-ray measurements)
● width: wc = 60 nm; cristallinity degree: 46 %
wa= 62.5 nm
● 2D periodic B.C are applied - 2nd degree hexahedron elements
● challenging problem due to incompressibiliy of amorphous phase and strong
anisotropy of the crystalline phase
■
r r
r
Effective Hooke matrix [GPa] is nearly monoclinic in the local axis system e1, e 2 , e 3
eff
H bila =
[Mpa]
 10268






1910
1780
− 0.007
− 185.7
2151
1923
− 0.006
− 76.18
2469
- 0.008
342.1
62.07
0.007
S
Y
795.5
M
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
0.008 
0.009 
− 5.4 

− 0.

48.55 
0.004
2mm PP Plate: Effective Young Modules over the Thickness
■
Reference: uniaxial test on an extracted bar: EX = 1860. ; EY = 1650.
■
Em = 1755 MPa
2-level homogenization scheme predicts noticeable
orthotropic effective behaviour with
EY < EX < EZ
same in-plane orthotropy as in unxial test
● EX,hom agrees with EX,exp but EY,hom underestimates significantly EY,exp
● EY and EX are lower in the center and max.at
the borders, whereas EZ decreases over the
thickness
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2mm PP Plate: Microscopic Fields in Z Direction at the Lower Border RVE
.[N]
[µm]
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Conclusion and Outlook
■
Phase-field simulations combined with HOMAT allow the prediction of a full set of effective
thermo-elastic properties of a steel microstructure during γ-α phase transformation
■
Work in progress: ● elastoplastic flow curves
● smoothing of the grain boundaries and FE remeshing of the microstructure
■
Development of a 2-level homog. scheme for semi-crystalline thermoplastics which is able to
predict the observed anisotropy of the injected moulded plate
■
Outlook: ● extension to a 3-level homog. scheme by adding a third level:
effect. properties of folded, helicoidal molecular chains
● molecule orientation has to be taken into account
● derivation of effective viscoelastic and viscoplastic properties
Funding by the DFG as part of the Cluster of Excellence "Integrative Production Technology for HighWage Countries“ is gratefully acknowledged.
1st Int. Workshop on Software Solutions for ICME
24th-27th June 2014, Aachen / Rolduc