1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5
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Transcript of 1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
A multi-component theory ofA multi-component theory of
solid mixtures with higher gradientssolid mixtures with higher gradients
and its application to binary alloysand its application to binary alloysby
A. Brandmair,1) T. Böhme,1),2) W. Dreyer,3) W.H. Müller1)
STAMM 2008 Symposium on Trends
in Applications of Mathematics to Mechanics
Levico Italy, September 24, 2008
1) 1) Technische Universität BerlinTechnische Universität Berlin Institut für Mechanik - LKM Institut für Mechanik - LKM Einsteinufer 5Einsteinufer 5 D-10587 BerlinD-10587 Berlin
German FederalEnvironmental Foundation
2) 2) ThyssenKrupp Steel AGThyssenKrupp Steel AG Werkstoffkompetenzzentrum Werkstoffkompetenzzentrum Physikalische Technik Physikalische Technik Kaiser-Wilhelm-Straße 100Kaiser-Wilhelm-Straße 100 D-47166 DuisburgD-47166 Duisburg
3) 3) Weierstraß-Institut fürWeierstraß-Institut für Angewandte Analysis Angewandte Analysis und Stochastik und Stochastik Mohrenstr. 39Mohrenstr. 39 D-10117 BerlinD-10117 Berlin
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
SMT SnPb solder joints: formation of interface cracks spinodal decomposition
microstructural coarsening
Ball Grid Arrays and solder ball before and after 4000 temperature cycles aging at RT after (a) 2h, (b) 17d and (c) 63d
(a) after solidification, (b) 3h and (c) 300 h at 125°CMELF miniature resistor and solder joint before and after 3000 temperature cycles
Microstructural changes in solids I
Will and, if so, how will the microstructural changeWill and, if so, how will the microstructural change influence the material properties ? influence the material properties ?
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Introduction: Microstructural changes in solids II
after solidification after 2h after 20h after 40h
Ag-Cu: aging at 1000 Kelvin
Ag-rich
Cu-rich
cracks along
the phase
boundaries
Si-chipFlip-Chipunderfill
substrate
solder-ballsSnPb solder balls
with lead-free bumps
decomposition + coarsening in the bulk
Cu
Leadfree solder, e.g., AgCu28:
Formation of Inter-MetallicCompounds IMCs (Cu6Sn5, Ag3Sn)at the interface and in the bulk
Leadfree solder, e.g., SnAg3.8, SnAg3.8Cu0.7:
thor.inemi.org/webdownload/newsroom/Articles/Lead-Free_Watch_Series/Oct05.pdf
Will and, if so, how will the microstructural changeWill and, if so, how will the microstructural change influence the material properties ? influence the material properties ?
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
• Triggered by different diffusion coefficients Cu atoms move to pile up IMCs.
• Kirkendall voids appear, e.g., between Cu substrate and thin Cu3Sn layer due to migration of Cu atoms from Cu3Sn to Cu6Sn5, which is much faster than Sn-diffusion from Cu6Sn5 towards Cu3Sn.
• Unbalanced Cu-Sn interdiffusion generates atomic vacancies at lattice sites which coalesce to voids.
• Model: vacancy diffusion
Introduction: Microstructural changes in solids III
Will and, if so, how will the microstructural changeWill and, if so, how will the microstructural change influence the material properties ? influence the material properties ?
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Experiments I: Setup & realization
material ► eutectic Ag-Cu temperature ► 970 K aging time ► 0 - 40 h etching (for Ag) ► solution of NH3-H2O2
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
► instantaneous decomposition ► coarsening (Ostwald ripening)► light: α-phase (Ag-rich) , dark: β-phase (Cu-rich)
aftersolidification
after 2haging
after 20haging
after 40haging
Experiments II: Micrographs
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Experiments III: Image analysis determination of the β-areas and the total number of β-precipitates N by
means of Digital Image Analysis (DHSTM) Attention: 2D analysis of a 3D problem (observation of one cross-section) Solution: statistical averaging
• sufficient large areas of intersection • analysis of various micrographs at each coarsening stage
iA
N
iiA
NA
1
1
pictures
AA
6
1
π2
3 A
r
235.1
A
a
(individual photo)
(individual stage)
(spherical phases)
(oblate spheroids)
(cf., Underwood, 1970)
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
faster coarsening rate for oblate spheroids with and
t 1/3- dependence well-known from LSW-theories
Experiments IV: Coarsening rates
baV 234 π
3/1ta 3/1tr
3/1024.0 tr 3/1032.0 ta
1/3
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Nomenclature I motion
displacements, velocity, deformation gradient
strains and stresses
),( tXx jii
,dd, txXxu iiiii j
iij
X
xF
0det ijFJ
gradientntdisplaceme:ijijj
iij F
X
uH
ijmjmiij cJFFC 3/2
strainslinearized:,,)( tensor)strain s(Green'21 jiijij CG
1det ijc
ij11 )()( jnmnimij FFJt
,: tensorstressCauchy
orGreen tens-Cauchyright:
shapeofn deformatiotensor,Green-Cauchyrightreduced:
tensorstressKirchhoffPiola2nd :
)(21 jiijij HH
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Nomenclature II
Primary variables:
Variables determined by partial mass balances (w/o chemical reactions), and balance of momentum and internal energy:
total mass balance
)(,)()( )def(
iiii
ii
i
i
Jx
J
xt
0,)( Def.
i
i
i
Jxt
j
iijii
i xqe
xt
e
,...,1with,)or(, euii
ijijj
i
xt
(no external forces)
constitutive equations for ijii qJ ,, sought
in a situation wheresolids separated byphase boundaries“move” toward equilibriumwhich, again, is characterized by a boundary→ higher gradienthigher gradienttheoriestheories required !
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Constitutive relations for the diffusion flux (w/o thermo-diffusion)
w/o gradients with density gradients(w/o strain gradients)
TxBJ
jiji
1
( : mobility)
F̂Def.
chemical potential
Helmholtzfree energy
density
δ
ˆδDef. TF
T
kl
klk
k
Def.
δ
δ
),,,,(ˆ ijiji cTF
ijB
functional derivative:
,,1 , ),,(ˆ ijcTF
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Constitutive relations for the stress tensor
without gradients
Castigliano’s
2nd theorem
specific Helmholtz free energy
with density gradients(no strain gradients)
/2
31
Def. Fp kk pressure
),,,( ijccTF
),,,,,,( ijijiiji CcccTF
ij
ij
C
Ft
02
1,,1 , ),,(
ijCcTF
),,,,,,,( ijijiiji ccccTF
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
18
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principle I: Historic remarks
Entropy principles:
a) Clausius & Duhem (18th century)
b) Coleman-Noll (1963)
c) Green-Nagdhi (1967)
d) Müller (1968), Liu (1972)
Shortcomings of the above principles:
a) Entropy flux relation, application to mixtures of solids ?
b) Entropy balance global / local / principle for every constituent (too strong) ?
c) Lagrange multipliers in order to consider the
balances as constraints (some class of materials requires
“additional” constraints, e.g., materials with gradients as variables)
Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles
19
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principle I: Historic remarks
Entropy principles:
a) Clausius & Duhem (18th century)
b) Coleman-Noll (1963)
c) Green-Nagdhi (1967)
d) Müller (1968), Liu (1972)
Shortcomings of the above principles:
a) Entropy flux relation, application to mixtures of solids ?
b) Entropy balance global / local / principle for every constituent (too strong) ?
c) Lagrange multipliers in order to consider the
balances as constraints (some class of materials requires
“additional” constraints, e.g., materials with gradients as variables)
Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles
R. Clausius
P. M. M. Duhem
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principles:
a) Clausius & Duhem (18th century)
b) Coleman-Noll (1963)
c) Green-Nagdhi (1967)
d) Müller (1968), Liu (1972)
Shortcomings of the above principles:
a) Entropy flux relation, application to mixtures of solids ?
b) Entropy balance global / local / principle for every constituent (too strong) ?
c) Lagrange multipliers in order to consider the
balances as constraints (some class of materials requires
“additional” constraints, e.g., materials with gradients as variables)
Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles
Entropy principle I: Historic remarks
B.D. Coleman
W. Noll
A.E. Green
P.M. Nagdhi
I. Müller
I-S. Liu
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principles:
a) Clausius & Duhem (18th century)
b) Coleman-Noll (1963)
c) Green-Nagdhi (1967)
d) Müller (1968), Liu (1972)
Shortcomings of the above principles:
a) Entropy flux relation, application to mixtures of solids ?
b) Entropy balance global / local / principle for every constituent (too strong) ?
c) Lagrange multipliers in order to consider the
balances as constraints (some class of materials requires
“additional” constraints, e.g., materials with gradients as variables)
Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles
Entropy principle I: Historic remarks
Duhem: Tq ii /
radiation: Tq ii /3
4ideal gas:
)( 32 ijp
ijTqi j
mixtures:
iTT
qi Ji
1
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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principle I: Historic remarks
Clausius: form?locald tS T
Q
Entropy principles:
a) Clausius & Duhem (18th century)
b) Coleman-Noll (1963)
c) Green-Nagdhi (1967)
d) Müller (1968), Liu (1972)
Shortcomings of the above principles:
a) Entropy flux relation, application to mixtures of solids ?
b) Entropy balance global / local / principle for every constituent (too strong) ?
c) Lagrange multipliers in order to consider the
balances as constraints (some class of materials requires
“additional” constraints, e.g., materials with gradients as variables)
Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles
23
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principle I: Historic remarks
Entropy principles:
a) Clausius & Duhem (18th century)
b) Coleman-Noll (1963)
c) Green-Nagdhi (1967)
d) Müller (1968), Liu (1972)
Shortcomings of the above principles:
a) Entropy flux relation, application to mixtures of solids ?
b) Entropy balance global / local / principle for every constituent (too strong) ?
c) Lagrange multipliers in order to consider the
balances as constraints (some class of materials requires
“additional” constraints, e.g., materials with gradients as variables)
Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles
24
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principle I: Historic remarks
Entropy principles:
a) Clausius & Duhem (18th century)
b) Coleman-Noll (1963)
c) Green-Nagdhi (1967)
d) Müller (1968), Liu (1972)
Shortcomings of the above principles:
a) Entropy flux relation, application to mixtures of solids ?
b) Entropy balance global / local / principle for every constituent (too strong) ?
c) Lagrange multipliers in order to consider the
balances as constraints (some class of materials requires
“additional” constraints, e.g., materials with gradients as variables)
Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles
25
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
0 z
zz DF
Entropy principle II: Formulation
Two constitutive quantities: entropy density and entropy flux .Constitutive relation of the entropy density:
Local balance for entropy density:
Entropy production positive definite & of the form fluxes x driving forces:
(Absolute) temperature defined as follows:
Stress tensor decomposed into an elastic and a dissipative part:
)( riablesprimary va theofsderivativeriables,primary vaS
i
ii
ixt
2nd law
ijijijdisselast
0eq
eT
Def.1
TTD ijii
z
1,,
26
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Entropy principle III: The evaluation procedure
H.W. Alt, I. Pawlow: On the entropy principle of phase transition models with a conserved order parameter. Advances in Mathematical Science and Applications, 6(1), pp. 291–376, 1996:
Balances interpreted as “evolution equations” for variables:
Balances viewed as a system of algebraic equations, i.e.,
choose right hand sides arbitrarily & calculate left hand sides.
Right hand sides of the balances + chain rule list of arbitrary terms:
Construct constitutive relations such that 2nd law is identically satisfied.
... t
ij
i
i
ij
ij
i
i
j
ii
i
i
i
i
ii
x
q
x
ee
xxxx
J
xx
,,,,,,,,,,,,
27
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
28
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Mixtures w/o higher gradients I: Choice of variables
Simple -component mixture without reactions / viscous effects:
Different representations of entropy density:
with a constant number of variables, e.g.:
Different representations useful under different circumstances, e.g.:
},{ ijij cC
ijijelast
ijtS
pS
S
S
stressesKirchhoffPiola2ndtheofDefinition:
pressuretheofDefinition:
,...,1,potentialchemicaltheofDefinition:ˆ
law2ndtheofonExploitati:~
1,,1,,,1
),,(),,,(),,(ˆ),,(~
ijijijij CcTSccTScTScS
29
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Mixtures w/o higher gradients II: Entropy production
Local entropy balance:
Chain rule for
Kinematic relation
i
i
i
i
ii
xxS
x
S
t
S ~~~
t
c
c
S
t
S
t
S
t
S ij
ij
~~~~
right hand side of balances
(note: balance of momentum
not required for exploitation,
since not in )
),,(~ ijcS
i
kl
kliii x
c
c
S
x
S
x
S
x
S
~~~~
,
)(3
2
d
d
d
d32
32
32 ikjliljk
j
i
i
iklkl
kl
FFFFx
Jx
CJCJtt
c
chain rule
kinematic relation
ii S
~
30
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Mixtures w/o higher gradients III: Entropy production
Substitution and rearrangement: a) separate terms into flux / force:
b) separate terms linear in (cf., arbitrary terms)
Definition entropy-flux (first parenthesis): Residual inequality:
klklij
klikjliljk
ij
j
i
ii
iii
ii
i
c
SCJ
S
TS
c
SFFFFJ
Tx
x
Tq
S
xJ
SJ
T
q
x~
3
2~
~~
/1~~
32
32
1
elast
11
z
zzDFji x /
1
Def.~S
JT
q ii
i
0.../1
~
1
j
i
ii
ii
xx
Tq
S
xJ
Q P
ij
ij ,0andinlinear
0Q0P
terms linear in drop out
Galileian invariance
i
31
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Mixtures w/o higher gradients IV: Heat & diffusion flux
0/1
~
1
i
ii
i
x
Tq
S
xJ
Helmholtz free energy density ),,(ˆ),,(~ ijij cTFcF
experimentallyinconvenient quantity
T
F
T
S
Def.ˆ1
~(chemical potential) Legendre Transformation
1
111
0
Tx
JTx
JJi
ii
ii Constraint
jij
jiji
x
TT
x
Tq
)(/1 (Fourier’s law of heat conduction)
1
1
1
1
,
iij
iji JJTx
BJ ( : mobility)
No coupling, quadratic form
ijB
Q-bit
32
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Mixtures w/o higher gradients V : Selected results
Pressure
0~
3
2~
~~
0... 32
32
1
elast
klklij
klikjliljk
ij
j
i
c
SCJ
S
TS
c
SFFFFJ
Tx
Legendre transform applied to ),,,( ijccTF
/2 F
p
Legendre transform applied to
1~
~)
~(
STST
F
kkp elast3
1
)pressureofdefinitionfromdirectlyfollows(
),,( ijCcTF
ijijij
C
F
C
FJt
/
2/
2 0
1
1p
P-bit
Gibbs-Duhem relation
2nd Piola-Kirchhoff stress tensor
33
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
34
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Mixtures with higher gradients I: Functional representations
functional representation of the entropy density
Note:- Choice of Higher Gradients depends on the problem
- Present choice: Convenient for diffusion problems & for definition of the chemical potential
111
,,,,,
,,,,,,,,,,,,,
,,,,ˆ,,,,~
ijijiijiijijiiji
ijijiijiji
CccceScccceS
cTSceS
35
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Mixtures with higher grad. II: Specialization to binary alloy
Now: Binary alloy A-B:
Relating difference of chem. pot. to derivative of Helmholtz free energy density:
with Helmholtz free energy density
diffusion flux for a binary alloy:
1
1
1
1
,)(
iij
iji JJ
xT
BJ
T
BB
xBJ
xBJ
ijAAij
jABiji
BjABiji
A
,)(
,)(
(isothermal case)
BAB ccc
F ,
δ
δ1
),,,,,,,( ijijiiji ccccTF
c
F
xTBJJ
jiji
Bi
δ
δ1)(
Def.
36
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
lkij
kl
lkij
klklklijkl
ijij
x
c
x
cb
xx
caTcTC
F
2
local )Δ)(,(
Transformation to the reference configuration / re-definition of mobility
Problem with free energy density (more general case):
Decomposition & Taylor expansion
HGC can be identified & calculated by microscopic atomistic theories (e.g., EAM)
it follows (Böhme et al., 2007): (periodic arrangement of the lattice)
Approach for elastic contributions
ijijB 20
phase diagram ? ?
...))(()()(2
1),,(),(
),(),(
2
elast
cccc
Fc
c
Fc
c
FCcTFcTFF lk
kl
lkkl
kl
klk
k
k
ij
ijCcbijCcaL
0iL
,)Δ( Hooke
elast TF ijijij
leading term
2
1
Mixtures with higher grad. III: Specialization to binary alloy
c
F
XBJ
jiji
δ
δ0
),,,,,,,( ijijiiji ccccTFF
37
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Partial mass balance & mass concentration :
Bjij
i
i
ccc
F
XB
X
J
t
c
,
δ
δ
d
d00
/BBc
c
F
δ
δ
lk
mn
mn
kl
l
mn
k
op
mnop
kl
l
mn
kmn
kl
lk
kl
lkkl
XX
a
XX
a
XX
cA
X
c
X
c
c
A
XX
cA
c
FF
22
2elastdiagr.phase 22
)(
klkl
kl bc
aA
Material parameters:
c
baA
B
ij
ijijij
ij
, offunctionsas method
atom embeddedthebycalculatedbecanHGCs,:,,
t coefficiendiffusionthetolinkedbecanmobility,:
Mixtures with higher grad. IV: Specialization to binary alloy
38
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
39
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Application: Spinodal decomposition in AgCu I:
Simulations in 1D (no external stress)
Study concentration development along a line
40
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Application: Spinodal decomposition in AgCu II:
Simulations in 1D (no external stress)
Fortran 95, FFTPack, explicit Euler scheme
s103Δ 7t
29.00 c
K1000T
256N
41
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Application: Spinodal decomposition in Ag-Cu II:
Simulations in 1D (tensile stress: 103 MPa)
Fortran 95, FFTPack, explicit Euler scheme
s103 7t
29.00 c
K1000T
256N
stresses
accelerate
coarsening
42
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Application: Spinodal Decomposition in Ag-Cu III:
2D (no external stress)
initial 3000 time loops 6000 loops 60 000 loopssec10Δ 8t
29.00 cK1000T
Cu-rich phase
Ag-rich phase
43
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
44
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Experiment and simulation: Coarsening rates
By image analysis of experiments and computer generated microstructural evolution
3D simulations required ?
45
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
46
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Average elastic properties by homogenization I
Homogenization performed by S.V. Sheshenin, M. Savenkova (FE-program “Elast”)
Boundary value problem (plane strain analysis) for a given micrograph (= RVE):
0ij ij
011 0
012 0
022 0
Loading sequences applied, e.g.:
etc.(0) 0i ij ju x
, ,
(0)
0, , , , , 1, 2
ijkl s k l j
i s i
C x ui j k l s
u x u
1( )dij ij s
V
x VV
1
( )dij ij s
V
x VV
effij ijkl klC eff 0
11 11/ , 1, 2ij ijC ij
eff 012 12/ , 1, 2ij ijC ij
47
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Average elastic properties by homogenization II
AgCu28 simulated temporal development of microstucture
1 2 3 4 5 6
Conclusion
material is cubic, just like its constituents Ag and Cu
changing microstructure leads to no change in elastic coefficients
48
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Average elastic properties by homogenization III
SnPb37 effective elastic moduli of simulated microstucture
Conclusions
composite material is less “tetragonal” due to the slight difference between C1111 and C2222 for Sn and the presence of the cubic Pb
laminate theory gives similar results:
Tin (Sn) 29.751111 C 00.441122 C 52.952222 C 93.211212 C
Lead (Pb) 66.491111 C 31.421122 C 66.492222 C 98.141212 C
Attention
lead (Pb) is cubic = 3 elastic constants, tin (Sn) is tetragonal = 6 elastic constants
2D excerpt:
nkllmijmnkqqppllmijm
ijnkijnk
CCCCCCCC
CC
21
22221
22
1122
1222
)2(2
)1(1 ijklijklijkl CvCvC
49
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Outline Introduction and motivation: Three types of microstructural change
An experimental investigation of spinodal decomposition and coarsening
Constitutive equations for diffusion flux and stress
Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)
Reduction to the case of binary mixtures
Numerical simulation of spinodal decomposition and coarsening
Comparison with the experiment
Homogenization and effective properties
Conclusions and outlook
50
Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik
Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller
Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008
Conclusions and Outlook
Phase field / higher gradient models to be used for microstructural changes in multi- component alloys should not simply be postulated but rather based on balance laws for physical quantities.
Material theory should and can be used to derive the corresponding field equations.
Material parameters in these relations should not simply be “guessed,” rather they should be obtained from experiments that are independent of the to-be-described phenomenon and, eventually, also be obtained from atomic methods (e.g., embedded atom methchnique).
The spinodal decomposition observed in some solder/welding materials as well as the subsequent process of coarsening can be modeled quantitatively using such a strategy.
Homogenized elastic properties for experimentally observed as well as predicted micrographs showing microstructural change have been obtained.
Non-linear homogenized material properties were not obtained yet.