finite element modeling of fast - materialwissenschaft · 2020. 7. 7. · what you need to do in...
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FINITE ELEMENT MODELING OF FAST
Antonios ZavaliangosDrexel University
D t t f M t i l S i & E i iDepartment of Materials Science & Engineering, Philadelphia, PA 19104
Outline
• Introduction to FEM for usersIntroduction to FEM for users• Basic thermo‐electric modeling
ll l d d l• Fully coupled model• Discrete models• Open issues
THE NEXT FEW SLIDES CONTAINBLATANT
SELF PROMOTION AND MAY BE
ILLEGAL IN SOME STATES
4/11/2011 4www.materials.drexel.edu
11 311 3 F l M bF l M b11.3 11.3 Faculty MembersFaculty Members1 Lecturer1 Lecturer
116 Undergraduate116 UndergraduateStudentsStudents
Where We StandWhere We StandRANKED 11th OUT OF 88 MATERIALS
PROGRAMS
58 PhD Students58 PhD Students $5M$5M
IN THE 2010 NRC S‐RANKINGS
70% Domestic 70% Domestic 10 PhDs/year 10 PhDs/year awarded awarded
for 2004for 2004‐‐20102010
$5M $5M Annual Annual Research Research ExpendituresExpenditures
4/11/2011 5
for 2004for 2004 20102010
FACULTY
AREAS OF RESEARCH Nuclear Materials ScienceNanomaterials & Nanotechnology Materials for EnergyBiomaterials Polymers and Soft MaterialsElectronic and Photonic Materials Materials Processing
Faculty DistinctionsFaculty Distinctions
• 1 PECASE Awardee (Spanier)1 PECASE Awardee (Spanier)
• 3 NSF CAREER Awardees (Zavaliangos, Gogotsi, Li)
• 1 NIH 1 ARO and 1 Whitaker Foundation• 1 NIH, 1 ARO, and 1 Whitaker Foundation
Young Investigator (Marcolongo, Spanier)
• 2 x R&D 100 Awardees
(Gogotsi, Taheri)
• Collegiate Inventors Award
(Gogotsi)( g )
Faculty DistinctionsFaculty Distinctions• Fellows:
2 ACerS (Barsoum, Gogotsi)2 ASM (Lawley, Knight); 1 ECS, 1 AAAS, 1 MRS (Gogotsi)1 TMS (Lawley), 2 ΑΣΜ (Knight, Kalidindi)
• 1 NAE Member (Lawley - emeritus)• 3 Alexander von Humboldt Awardees
(Barsoum, Li, Gogotsi)• 3 Journal Editors (Gogotsi,
Kalidindi, Lawley), and 3 on Journal Editorial Boards (Li, Marcolongo, Zavaliangos)
RESEARCH THRUSTS
Soft materials• soft materials for biomedical
Earth‐abundant electronic & catalytic materialssoft materials for biomedical
applications• interfacial phenomena• Soft materials synthesis• Soft materials for bio‐sensing
catalytic materials• complex oxide films and heterostructures• ferroic and multi‐ferroic materials• eco‐friendly synthesis of quantum dots• eco friendly growth and processingg
Materials at extremes
• eco‐friendly growth and processingmethods
Nanomaterials for energy & bi di l li iMaterials at extremes
• materials under irradiation and nuclear materials
• dynamic microscopy• MAX phases
biomedical applications• carbon and carbon‐derived based materials• supercapacitors• nanodiamondsp
• low‐temperature synthesis• coupled phenomena
• carbon nanotube‐based celluclar probes• nanotubes and nanowires
PhD Student Placement (2007‐10)• Defense
– Wright Patterson Air Force BaseArmy Research Laboratory (2)
• National Labs– Los Alamos (2), Lawrence Berkeley, NISTLos Alamos (2), Lawrence Berkeley, NIST
• Large companies– Honeywell, United Technologies, FujiFilm, Slumberger
• Small companies– TBT Group (2), Y‐Carbon [Drexel technology licencees] – Molecular Biometrics Ingeni SAMolecular Biometrics, Ingeni SA
• Faculty– University of Bath (UK), The College of New Jersey, Çankaya (Turkey)
• Postdocs– MIT, Yale, ETH, UPenn, Clemson, UConn, Rutgers
Shameless self promotion over
ContributorsContributors
Drexel PhD Students: Ji Zh (2004 f l U Al k )Jing Zhang (2004, now faculty at U. Alaska)Brandon McWilliams (2007, now with Army Research Lab)
P f J G ’ @ UCD iProf. Joanna Groza’s group @ UCDavis
Funding: NSF-DMII and ARO/ARL
Simulation of FAST
What is modeling?
==
4/11/2011 14
What is simulation?
4/11/2011 15
Why model & simulate?
• Access to “information” that cannot beAccess to information that cannot be obtained by experiments
• What‐if scenarios (scale up, tool design,What if scenarios (scale up, tool design, control algorithm testing)
• Cost minimization by reduction of experimentsCost minimization by reduction of experiments
Note 1: Models don’t need to be exact (they never will)( y )Note 2: “Don’t try this at home – I am a trained professional”
FINITE ELEMENTS fan overview for users
LAB CHARACTERIZATION INDUSTRIAL PROBLEM
Real parts haveReal parts have
• Complex geometries that can be the origin ofComplex geometries that can be the origin of variations in temperatures, stress, properties
• Complex interactions with the environment• Complex interactions with the environment, such as friction, radiation, convection etc. which also may be related to variations inwhich also may be related to variations in temperatures, stress, properties…
Non‐uniform everything
zz everything…
Analytical solutionimpossible
Uniform stress & Temperaturep
straightforward to analyze
(but you can be surprised with the difficulties with a simple test)
i l Si l iNumerical Simulationof Pressing and Sintering
in the Ceramic and Hard Metal Industryy
T. Kraft, O. Coube and H. RiedelFraunhofer-Institute for Materials Mechanics
Wöhlerstrasse 11, D-79108 Freiburg, Germany
Instead of …. We need to solve ……
klijklij E
0
xzxyxx
jj
E
0
zyx
0
zyxyzyyyx
0
zzzyzx
zyx
Instead of We need to solveInstead of …. We need to solve ……
d dxdkAq
t
rcqk pV
IRV J
THE GOOD NEWS
•You don’t have to do this on your own… y•Software exists that solves these complex
mathematical equations for you
THE BAD NEWS
• It is not a push button solution• You need to have some understanding in order toYou need to have some understanding in order to get meaningful results fast…
How do FEM work?
The geometry is discritized into elements.
=
)(xfy Nyyy ,..,, 21UNKNOWNS:
4/11/2011 26
)(fy N
How do FEM work? (cont.)How do FEM work? (cont.)The underlying partial differential equation are written in an equivalent
t i fmatrix form
0)(
extQxTk + BOUNDARY CONDITIONS
=
kkkk
extQ
xx CONDITIONS
n
n
n
bb
TT
kkkkkkkkkkkk
2
1
2
1
3333231
2232221
1131211
nn
nnnnn
n
bTkkkk
321
3333231
27
What you need to do in preparation of la FEM analysis
• discretize the geometryd fi th i iti l diti• define the initial conditions
• define the boundary conditions• define the constitutive behavior of the material, e.g., for adefine the constitutive behavior of the material, e.g., for a
thermal analysis you need:heat conductivity, density, heat capacity
• select how fast the analysis will march in time (for a non-steady state analysis)
28
What the FEM software will do for you?What the FEM software will do for you?
• convert the PDE to a system of algebraic equations (possiblynon linear)non linear)
• (try to) solve the system of equations• plot the results
29
What you need to do to successfully l h lcomplete the analysis
• understand why the analysis fails (and it will…) (iterations divergence, severe element deformation,(iterations divergence, severe element deformation,
numerical instabilities)• minimize the errors but selecting a adequately fine mesh• validate convergence• evaluate the importance and the effect of the various
assumptionsp
30
A brief (and simple) exampleSteady State Heat Transfer
Θ1 Θ2
W d t lWe need to solve:
For this (very simple) case an analytical solution exists
Discretization and basis functions
0 1 2 3 4 5
h (x)hi(x) θi hi(x)
Approximation of solution and boundary conditionboundary condition
0 1 2 3 4 5
)()()( xhxx ii
??
)()()( xhxx ii
??
ResidualResidual
0)(1020)()1001()(100Residual 2
2
x
dxxdx
dxxd(x)
Residual(x)
MinimizationMinimization of Residual Residual(x)
f( ) )(R id l xf(x)
121
121
RESIDUALMINIMIZETO,..,,CHOOSE
),,..,,(Residual
Residual(x)2
d( )LS L
2R id li
n)Collocatio Square,Least (Galerkin,RESIDUAL MINIMIZETO
dx(x)LS 0
2Residualmin Optimum Residual(x)2
N‐2 equations (due to BC)
Analytical Solution
Residual
FEM solutionN=5
FEM solutionN=10
FEM solutionN=40
MACROSCOPIC FEMMACROSCOPIC FEM MODELING OF FAST
Sintering under electrical current is a strongly coupled problemstrongly coupled problem.
ELECTRICITY THERMAL
Temperature dependenceof resistivity
ELECTRICITY THERMAL
Temperature dependenceof resistivity
Density Thermal activation
TRANSPORT ANALYSISJoule heating
Electroplasticeffect
Density Thermal activation
TRANSPORT ANALYSISJoule heating
ElectroplasticeffectEffect of current on diffusion or plasticity
Density dependence of electrical properties
Thermal activationof sintering Density
dependence of thermal properties
Density dependence of electrical properties
Thermal activationof sintering Density
dependence of thermal properties
DIFFUSION-BASED SINTERING
DIFFUSION-BASED SINTERING
Early modeling efforts
• Yucheng, & Zhengyi, Mat. Sci. & Eng. B, 2002– 1D only compact and die thermal only ‐ not coupled with electric current– 1D, only compact and die, thermal only ‐ not coupled with electric current
• Matsugi, et al., J. Mat. Proc. Tech., 2003– Steady state finite difference method of thermal/electric problem was
dependent on material properties
Thermoelectric Coupling Models
• Zavaliangos, Zhang, Krammer, and Groza, Mater Sci Eng. 2004h l l l d h h f ll d b d l d– Thermoelectrical coupling, recognized that the full system needs to be modeled, contact resistance
• Vanmeensel , Laptev, Hennicke, van der Biest Acta Met. 2005h l i li i i h i f d d– Thermoelectric coupling, comparison with experiments for a conductor and an insulator
• Anselmi‐Tamburini, Gennari, Garay and Munir, Mater. Sci. Eng. A 2005Si il i V l– Similar in Vanmeensel
• McWilliams B, Zavaliangos A, Cho KC, et al., JOM 2006– Role of part die dimensions on gradients
• Wang, Casolco, Xu and Garay, Acta Mat. 2007– Includes elastic stress from thermal expansion
“Simple” Thermoelectric FEM Analysis
• Thermal‐Electrical• Thermal‐Electrical coupled analysis
• No displacement
Punch/die
No displacement degrees of freedom
• Material properties Specimen Temperature
monitoring position
p pfunction of temperature
• Radiation to the Conduction
chamber walls and between parts
C d i b• Conduction between parts is considered in the model
COUPLING SIMPLIFICATIONS
ELECTRICITY THERMAL
Temperature dependenceof resistivity
ELECTRICITY THERMAL
Temperature dependenceof resistivity
ELECTRICITY TRANSPORT
THERMAL ANALYSISJoule heating
ELECTRICITY TRANSPORT
THERMAL ANALYSIS
Effect of current on diffusion or plasticity
Joule heating
Density dependence of electrical properties
Thermal activationof sintering Density
dependence of thermal properties
Density dependence of electrical properties
Density dependence of thermal propertiespropertiesproperties
DIFFUSION-BASED SINTERING
DIFFUSION-BASED SINTERING
Modeling framework – theoryConservation of charges
d dVrdSdVd
d
VC
SV nJJ
x
Conservation of energy
dSqqqqdVqdVkdVt
c ecS rconvcV eVV P )(
Conservation of energy
45Temperature evolution
Heat conductionin component
Joule heatingin component
Heat conductionacross surface
RadiationJoule heatat interface
Convection
45evolution in component in component across surface
Coupling (Joule heating)
xσ
xJE
dd
ddqe
It does not look like it, butit is Ohm’s law
Constitutive equationsConstitutive equations
• Need to provideNeed to provide – Thermal propertiesElectrical properties– Electrical properties
• Can be temperature dependent• Can be anisotropic (e.g., graphite sheets)• Usually input in tabular form
Temperature dependent materials propertiesproperties
7080
ity,
2000
2500100000120000
vity
,
0102030405060
herm
al C
ondu
ctiv
iW
/m-o C
0
500
1000
1500
2000
Spec
ific
Hea
t,J/
kg/o C
0200004000060000
80000
ectri
cal C
ondu
ctiv
( -m
)-1
00 500 1000 1500 2000
Temperature, oC
Th
00 500 1000 1500 2000
Temperature, oC
00 500 1000 1500 2000
Temperature, oC
Ele
Graphite*
1 E+061.E+081.E+101.E+121.E+14
ivity
, -m
30
40
50
ondu
ctiv
ity,
m-o C
800100012001400
fic H
eat,
kg/o C
1.E+001.E+021.E+041.E+06
0 500 1000 1500 2000
Temperature, oC
Res
isti
0
10
20
0 500 1000 1500 2000
T t oC
Ther
mal
C W/m
0200400600
0 500 1000 1500 2000
T t oC
Spec
ifJ/
k
47
Temperature, CTemperature, oC Temperature, oCAl2O3**
* Tokai G540 Data Sheet, Tokai Carbon Co. Ltd.
** Baikalox Ultra‐Pure Alumina Data Sheet, Baikowski International Corporation.
Thermal & electric gap resistance
J1 2
SA
1 or 1
• A result of imperfect interfaces
• Electric gap resistance results or
2or 2
J
• Electric gap resistance resultsin local heat at the interface
• Thermal gap resistance is
H d fl
• Thermal gap resistance is effectively an insulator
)( 21 gc hq
Heat and current flux across the interface
g
)( 21 gJ
ELECTRICELECTRIC POTENTIAL
ELECTRIC CURRENT DENSITY
Note that although the neck is an equipotential surfaceHolm’s resistance
Note that although the neck is an equipotential surfaceThe current density is not uniform – there is a peak at the root of the neck a
R2
Holm’s contact resistanceHolm s contact resistance SA
R LR J1 2
jCj a
R2
A
R 1
α
a
RC
21
α
Contact resistance – size effect For large scales, the behavior of contacts is relatively well known (Maxwell, Holm), the solution of the electrostatic problem
id ti t f th t t i tprovides an estimate of the contact resistance.
When the size of the contact ‘neck’ a decreases, the scattering l h f h h b bl h h l h l
length LS of the charges becomes comparable with the length scale of the contact – Knudsen effect
aL
aK
aR S
34
2Maxwell‐Holm Knudsen‐Sharvin
The behavior becomes more complex at nanoscaleThe presence of contamination on the surface of the powders usually
means even higher contact resistance
Thermal and electric gap conductancesconductances
We group the contact resistances into two categories based on the orientation of interfaces:
VerticalcontactResistance Horizontal contact
conductances and vertical contact conductances
Resistance(red lines)
Horizontalcontact conductances.Resistance(green lines)
Why different horizontal and vertical contact conductances?
52
Why different horizontal and vertical contact conductances?
Calibration of thermal & electric gap conductancesconductances
The gap conductancesare determined by four i d d tindependent calibration experiments and a series of numerical
(a) (b)series of numerical simulation runs.
53(c) (d)
Difference between (a) and (b)• extra punch length (can be calculated) and • extra contact resistance (can be extracted from the difference (careful that temperature gradient should be small)
Difference between (b) and (c)• presence of the die• presence of the vertical contact resistance
Difference between (b) and (c)• vertical contact resistance is now over a smaller length
Flow chartRun single punch and double punch experiments. Log potential, current and temperature Run experiment with graphite
cylinder. Measure temperatures in
c
Flow chart Guess axial electric gap conductance and run single punch and double punch simulations and record potential, current and temperature.
y pthe specimen and die surface simultaneously. Guess thermal gap conductance in transverse direction. Run simulation with graphite cylinder.
temperature.
Predicted resistance fits experimental single and
NoPredicted temperature No
Guess transverse electric gap
double runs
Yes
difference fits experimental data of graphite cylinder runs
Yes
No
conductance and use axial gap electric gap conductance from previous step. Run dummy test simulations. Record potential, current and temperature.
Derive thermal gap conductance in axial direction based on
=1.25107(‐m2)‐1;hg
Predicted resistance fits experimental data
No
=7.5106 (‐m2)‐1;=2.2103W/m2‐K;=1.32103 W/m2‐K
vghghvgh
57
of dummy run
Yes
c
Typical values Note proportionality(assumes geometry effect only)
Close loop temperature control Voltage feedback
Temperature signal
Compare with Preset Temperature Profile,
T
U =-k T
T
time
DU =-k (q read –q preset )
58
A subroutine is developed to implement the control scheme. Simulation at preset heating profile can realized.
Dummy run calibration
50006000700080009000
er, W No specimen
Calibration experiment
01000200030004000
Pow
e
ExperimentalNo contact resistanceWith horizontal contact resistanceWith vert+horiz contact resistance
Calibration experimentInput=Temperature q(t)
5
6ExperimentalNo contact ResistanceWith horizontal contact resistance
1200
1400
e, C
Time, sec
1
2
3
4
5si
stan
ce*1
000,
With horizontal contact resistanceWith vert.+horiz. contact resistance
200
400
600
800
1000
ace
Tem
pera
tur
ExperimentalNo contact Resistance
0
1
0 500 1000 1500 2000 2500
Time, sec
Re
0
200
0 500 1000 1500 2000 2500
Time, sec
Surf With Horizontal Contact Resistance
With vert+horiz contact resistance
Pressure dependence of contact resistanceExperimental resultsExperimental results
3
3.5
4
hm) LOW PRESSURE
0 5
1
1.5
2
2.5
Res
ista
nce
(mO
h
Linear (Die run (no sample, 32 MPa))
Linear (Die run (no sample, 15 MPa))
NO SAMPLE
HIGH PRESSURE
0
0.5
0 100 200 300 400 500 600 700Time (s)
200
250
Die surface 32 MPaDie surface 15 MPa
NO SAMPLE
100
150
200
Tem
pera
ture
(oC
)
Die surface 15 MPa
UpperPunch
Die
0
50
0 100 200 300 400 500 600 700Time (s)
NO SAMPLE
Total system resistance decrease (top) and die surface temperature decreases as applied pressure increases as a result of the better contactThis affect local heat generation (at contacts) and heating loss along the axis
Closing the parenthesis in contact resistance
• It is crucial as it affects current flow significantlyt s c uc a as t a ects cu e t o s g ca t y• It is also crucial in terms of temperature distribution because contact resistances cause local heating
• There are still issues because there is a complex dependence of thermal and electrical contact resistances on not just the material pair but
t t ( d ti )– pressure, temperature (and time – creep)• It is not straightforward to deal with this problem (especially in complex shapes)(especially in complex shapes)
Before pushing ahead with the FEM:b k f h l l lA back of the envelope calculation
Simplified lumped electric circuit modelelectric circuit model
Rsys= 0.5 m contains horizintal contact
Rsys= 0.5 m I0I0A
L1
horizintal contact resistance
Rdie = 0.5mRsp = infinite for
Rp = 1.5m
BCD
EF G
ALR
1
spalumina powderand 0.3m for graphite cylinder
LRR
R inout
2)/ln(1
L 2
63
Geometry, function and materials of each componentof each component
Component Resistance group
Function Material Outer Diam./ Inner Diam./ Thickness (mm)
A Rsys Water cooled Ram Graphite OD: 120, Th.: 200
B Large disc, current transfer
Graphite OD: 155, Th.: 20
C Small disc, spacer Graphite OD: 120, Th.: 20
D Spacer, thermal Graphite OD: 76.2, Th.: 40buffer
E Rp Punch Graphite OD: 19.1, Length: 25.4
F Rdie Die Graphite OD: 44.6, ID: 19.1, Length: 38.1Length: 38.1
G Rsp Specimen Alumina, Graphite
OD: 19.1, Th.: 6.0
64
Current distribution and Joule heating in specimenheating in specimen
0.3
eat,
4
ie
0.2ed
Jou
le h
ep/I 0
Rdi
e
2
3
t rat
io, I s
p/I d
220 )( spdie
spdie
die
sp
RRRR
RIP
0
0.1
Nor
mal
ize
P s
0
1
Cur
rent
die
sp
sp
die
RR
II
0 1 2 3 4
Rdie/Rsp
The amount of current flowing through specimen depends on the
Insulating specimen
Conducting specimen
The amount of current flowing through specimen depends on theresistance ratio of specimen to die.
Maximum Joule heating in specimen when resistance is same as
65
Maximum Joule heating in specimen when resistance is same asgraphite die. Metallic powders may experience significant Jouleheating during FAST, while ceramic powder will not.
Where does heating occur?System Rsys
Punch Rp Die Rdie Specimen Rsp
0I 0I 0IRR
R
spdie
sp
0IRR
R
spdie
die
Resistance ,m1.0 3.0 0.5 0.3/infinite
Current
sysRI 20 PRI 2
0202
2
)()(
IRRRR
spdie
diesp
202
2
)()(
IRRRR
spdie
spdie
Joule heat
Joule heat % (specimen: 0.24 0.71 0.03 0.03graphite)
Joule heat %(specimen: alumina)
0.22 0.67 0.11 <0.01 die
punch
The maximum Joule heating is generated in the punches (~ 70%) independently of the resistivity of the specimen, mainly because the
66
p y y p , ypunches offer the smallest cross section.
Voltage drop across the specimen
The voltage drop across the specimen is only a small fraction of the total voltage difference applied on the system, Vtot.
The voltage across the specimen increases monotonically with Rsp. Its maximum value, , occurs when Rsp approaches to infinity (insulating
i )
maxspV
specimen):
dieRVV max
This means that if the goal is to maximize the voltage across the specimen
syspdie
dietotsp RRR
VV
max
This means that if the goal is to maximize the voltage across the specimen for an insulating specimen, the resistance of the die should be maximized (Rdie >> Rp+Rsys).
67
Comparison ofComparison of a conductive
versus non‐conductivenon‐conductive
sample
FEM model
Electric potential gradient (Field)
El t i t ti l di t
NOTICE THE VALUE OF THE FIELDElectric potential gradient magnitude (V/m)
Conductive material:~ 5V/m
alumina powders graphite cylinder
Non‐conductive materialO[100 V/m]
Compare with valuesalumina powders graphite cylinder
Local potential difference is smaller than the overall one
Compare with values required for arcing
(usually > 10,000V/m)
Local potential difference is smaller than the overall one.
Alumina: Significant potential gradient exists across the specimen. About ~10% of the total V. The condition is often claimed to cause micro spark/plasma at particle contactb h fi ld i ll d h i d d f i / l– but the field is very small compared to what is needed for arcing/plasma etc.
Graphite: almost no potential gradient within sample. (~1‐2% of the total V)
Electric t current
density Electric current density magnitude (A/m2)y g ( / )
alumina powders graphite cylinder
NOTICE THE VALUE OF THE CURRENT DENSITY
Non conductive material
Electric current density is maximum at the end of punch end due to smallest cross section area
Non‐conductive material~Zero
Conductive material:smallest cross section area.
Alumina: the majority of current is diverted towards the die due to the high resistivity of specimen.
~ 100 A/cm2
Compare with values electromigration
70
Graphite: Approximately. 20% of total current goes through specimenelectromigration
Joule heating
Joule heating energy (J/m3)
alumina powders graphite cylinder
l h h d f h d d ll Joule heating is maximum at the end of punch end due to smallest cross section area in the system. Joule heating is almost zero for both alumina and graphite cylinder. Alumina: because all current passes through die
71
Alumina: because all current passes through die.
Graphite: because of low resistivity of specimen.
Joule heating in specimen
0.01
0.012
m
0.004
0.006
0.008sp
/Wsy
ste
0
0.002
0.004
1E 10 1E 06 1E 02 1E+02 1E+06 1E+10
Ws
1E-10 1E-06 1E-02 1E+02 1E+06 1E+10
/graphite
Joule heating is maximum with graphite specimen. The observation is consistent with lumped resistance model.
72
Joule heating in specimen is small(<1%) compared with punches. In terms of specimen heating, heat conduction is more important than Joule heating.
Validation studyValidation study
Temperature evolution1000
1200
C
0
200
400
600
800
Tem
pera
ture
, o C
ExperimentSimulation
(oC)
00 200 400 600 800 1000
Time, s
(oC)
Highest temperature
Generated heat is partially diffused into the specimen and partially conducted into
The pattern of heat flow results in temperature of specimen is higher than the control temperature on the
74
Highest temperature develops in the punches during the early stages.
partially conducted into the machine and radiated from die surface.
control temperature on the surface of the die by at least 100oC.
Evolution of temperature on outer surface of the die outer surface of the die
1000
1200
C600
800
1000
erat
ure,
o CB
0
200
400
Tem
pe
ExperimentSimulationA
00 200 400 600 800 1000
Time, s
Evolution of temperature on the surface of the die – comparison of prediction and experiment. A and B represent two instances, just after the beginning of heating and at the late heating stage, respectively.
75
g g g g g , p y
Evolution of resistance of the systemthe system
7
8
E i 7
8
4
56
7
ance
, m
ExperimentSimulation
4
5
6
7
ance
, m
ExperimentSimulation
0
12
3
Res
ista
0
1
2
3
Res
ista
Al2O3 (0.1 m) Graphite (fully dense) 0
0 200 400 600 800 1000
Time, s
00 200 400 600 800 1000
Time, s
h h f h l l d With the incorporation of contact resistance, the calculated resistance of system matches experimental data in both cases.
The difference in FAST of Al2O3 at initial stage is caused by (1)
76
2 3 g y ( )presence of nano particles at interface between punch and die during filling, and (2) inaccuracy of electric conductivity of graphite at lower temperature (<300oC).
Temperature difference
300
400nc
e,
o C ExperimentWith thermal contact resistanceNo thermal contact resistance
200
atur
e di
ffere
n
center
surface
0
100
400 600 800 1000 1200 1400
Tem
pera
•Experimental data shows the temperature excess of interior temperatures th di f t t
400 600 800 1000 1200 1400
Surface temperature Tsurface, oC
center over the die surface temperature surface.
•The incorporation of thermal contact resistance enables the simulated results to match experimental data.
77
•To know the true specimen temperature, a calibration-based correction with respect to surface temperature is necessary.
Temperature difference
200
250
cent
er,
SimulationE i t
100
150g
rate
at c
o C/m
inExperiment
0
50
0 50 100 150Hea
ting
Heating rate at die surface, oC/min
The actual heating rate is linearly proportional to the heating h d f d h h h h d f rate on the die surface and it is higher than the die surface
temperature of 70%. The linear relation between the heating rate in the specimen and on the die surface indicates that a single calibration for the specific die/punch configuration
78
single calibration for the specific die/punch configuration suffices to cover a large range of heating rate.
Temperature differenceModel with max emissivity
Model with calibrated thermalresistance
Model - base case
Model - half thermal diffusivity
0 20 40 60 80 100 120 140 160
Experiment
(oC) at Tsuface=900oC
Overestimation of the thermal diffusivity. For a 50% lower thermal diffusivity the corresponding temperature difference increases by about 80%. (doubtful)
Underestimation of the emissivity of the die surface. Using emissivity of e=1.0 instead of the value =0.8 results in a 10% increase of the temperature difference. (insignificant)
79
The presence of thermal contact resistance on the inner vertical surface of the die. By scaling the contact resistance it is possible to match exactly the experimental data. (most probable explanation)
Heat loss mechanism
100%d
CQ
dsQ
dtRQ 60%
80% Radiation loss ‐ die side
Radiation loss ‐punch side
Radiation loss die toppsRQ
dsRQ
RQ
0%
20%
40%
Heat loss through loading train
p
CQ
dtRQ
psRQ
0%350 550 750 950 1150 1350
Temperature on die surface ,Tsurface,oC
At low temperature heat conduction through the loading train is dominant.
80
Radiation becomes more important at higher temperatures.
FAST of high melting point materials requires radiation shielding.
Energy efficiency
1400
1600
put
600
800
1000
1200
1400
zed
ener
gy in
p
The total energy input into the system (normalized by mCpΔT , temperature ramp from 25 oC to
0
200
400
0 2 4 6 8 10 12
Nor
mal
iz p p1100 oC) as a function of electric gap conductance.
(log) gap electrical conductance (m2)-1
There is a transition of energy efficiency in the range of electric gap conductance of 104(Wm2)‐1. Above this level, the conversion of electric energy is not efficient
81
The effect of punch length
McWilliams B, Zavaliangos A, Cho KC, et al., JOM 2006
Effect of die diameter on sample center to die f disurface gradient
Smaller die reduces the temperature difference between the specimen center and the die surface
Important design consideration for process control
Tradeoff between mechanical strength and gradient
Summery of early FEM resultsA coupled thermal-electric-densification with temperature close-
loop controlling finite element analysis has been implemented.
A detailed evaluation of the accuracy of the model has been conducted for electrically conductive and insulator materials. The evaluation compared with experimental measurement indicatesevaluation, compared with experimental measurement, indicates that the model provides a reasonably accurate prediction of thermal and electric response.
Specimen is mainly heated up by heat conduction from punch irrespective of its resistivity. The most critical location for such temperatures is in the punches – high resistance (low cross p p g (section area=high Joule heating).
These temperatures may reach 200-600oC above the specimen
84
temperature – graphite may creep. Longer punches are particularly prone to this problem.
Some more results: 3D case studySome more results: 3D case study• Gradients become more complex when processing complex shapes and/or larger specimens
• Simulation as a design tool becomes critical
~150oC difference of compact edges!
3D simulations• Can we reduce the gradients in the sample in a “complex” shaped part to get uniform temperature p p p g pdistribution?
• One possibility: Electrically insulate punches to p y y pcontrol current flow through sample
Current can only pass through red areas
Temperature distribution in compactTemperature distribution in compact
1919oC
1692oC
1753oC
1627oC
2004oC
1919oC
1840oC1902oC
1844oC1684oC
2004 C
2043oC
1902 C
2019oC 2002oC
How to insulate?
• Punch insertsmaterial possibilities?
How to insulate?
Punch inserts material possibilities?– Zirconia: Tm=2700oC , ρ=10^11 Ohm*m k=1.15 W/mKW/mK
– BN: Tm=3000oC, ρ=7e‐5 Ohm*m, k=20 W/mK
• External heating of die• External heating of die– Thermally insulate sample
Potential issues & IdeasPotential issues & Ideas
• Current now only passes through parts ofCurrent now only passes through parts of sample rather than the entire volume– If there is an effect of current it will be seen here– If there is an effect of current it will be seen hereas temperature is (roughly) the same( g y)but the current is different
SINTERING + THERMOELECTIC MODEL
• McWilliams & Zavaliangos, J. Mat. Sci. 2008 g ,– Coupled sintering and thermoelectriic
Motivation
Titanium hip socketwith plastic liner
Complex shapes that evolve
Ball
substantial during sintering
Simple thermoelectrical model does not suffice Titaniumdoes not suffice
Also the simple stepwise approximation of sintering in a thermoelectrical
Titanium hip stem
model works for cylindrical sample but not for complex shape
Total Hip Joint Replacement (THR)
91
Prior work was mainly on the thermal+electrical problem butthermal+electrical problem but...
ELECTRICITY THERMAL
Temperature dependenceof resistivity
ELECTRICITY THERMAL
Temperature dependenceof resistivity
Density Thermal activation
TRANSPORT ANALYSISJoule heating
Electroplasticeffect
Density Thermal activation
TRANSPORT ANALYSISJoule heating
ElectroplasticeffectEffect of current on diffusion or plasticity
Density dependence of electrical properties
Thermal activationof sintering Density
dependence of thermal properties
Density dependence of electrical properties
Thermal activationof sintering Density
dependence of thermal properties
DIFFUSION-BASED SINTERING
DIFFUSION-BASED SINTERING
With the introduction of a sintering model the way opens for full coupling (i.e. for incorporation of current effects on diffusion or plasticity)
Introduction of a sintering model(e g Bouvard et al)(e.g., Bouvard et al)
= sintering strain rate
ij = viscoplastic strain rate
Experimentsto fit expression
IMPLEMENTED IN USER CREEP SUBROUTINE IN ABAQUS
Kim, Gillia, and Bouvard, J. Eur. Cer. Soc., 2003
Creep subroutine detailsCompactABAQUS estimates elastic
Creep subroutine details
Prescribed BCs and
and viscoplastic strains
σloads σt, dtSubroutine solves for strain
increment based on constitutive eqns.
Equilibrium
Updated density Ω, ρ, and n evaluated at Tt
p y
Coupling algorithm schematic
t0,θ0 t1,θ1
Thermal ‐electric simulationRD 0 t0,θ0 t1,θ1
Thermal ‐electric simulationRD 0
•PROPERTIES FUNCTION OFLOCAL DENSITY
•TEMPERATURE EVOLVES•MESH DOES NOT EVOLVE
Si t iSi t i
t0,θ0 t1,θ1
Sintering simulation RD 1t0,θ0 t1,θ1
Sintering simulation RD 1
•THERMAL HISTORYFROM TH‐EL STEP
• MESH EVOLVES DUE TO SINTERING
θThermal ‐electric
i l ti θThermal ‐electric
i l tit1,θ1 t2,θ2simulationRD 1 t1,θ1 t2,θ2simulationRD 1
Solution requires convergence studySolution requires convergence study
1.2
0.8
1
sity
0.4
0.6
Rel
ativ
e de
ns
A RD 1 d
0
0.2
R Average RD - 1 secondAverage RD - 5 secondsAverage RD - 10 secondsAverage RD - 30 seconds
0850 870 890 910 930 950 970 990
Time (s)
Typically 50 100 steps within the range of densification are more than adequateTypically 50‐100 steps within the range of densification are more than adequate
Testing the algorithmTesting the algorithm
• 2D case studycase study– Rectangular compact– Free sintering (no applied load)
(l d b d b i i i h di i(load can be done but interaction with dies requires delicate handling in this simulation)
• Purposep– Verify sintering algorithm– Confirm that the evolution of density plays a role – Evaluate effect of initial density distributions and material
properties on overall sintering kinetics under coupled thermal‐electric conditions
McWilliams & Zavaliangos, J. Mat. Sci. 2008
“Paper” StudyHIGH DENSITY
• 2 RD0 configurations of conductive t i l ΔV
HIGH DENSITY
material– 80 % low /20% high density material
arranged in series
LOW DENSITY
– 80 % low /20% high density material arranged in parallel
R t ith th l diff i it ( ) d dΔV
• Repeat with thermal diffusivity (α) reduced by order of magnitude
• NO DIE, TO VISUALIZE RESULTS IN A DIE THERE WOULD BE INTERNAL STRESSESIN A DIE THERE WOULD BE INTERNAL STRESSES
TDE model ‐ Density evolutionSERIES PARALLEL
1
1.2
PARALLEL - Avg RDPARALLEL - Avg LD layerPARALLEL - Avg HD layer
1
1.2
SERIES Avg RD
SERIES PARALLEL
0.4
0.6
0.8
Rel
ativ
e de
nsity
0.4
0.6
0.8
Rel
ativ
e de
nsity
SERIES - Avg RDSERIES - Avg LD layerSERIES - Avg HD layer
0
0.2
700 750 800 850 900Time (s)
0
0.2
700 750 800 850 900Time (s)
High alpha High alpha
( )
0.8
1
1.2
nsity
SERIES - Avg RDSERIES - Avg LD layer
SERIES - Avg HD layer
0.8
1
1.2
nsity
0.2
0.4
0.6
Rel
ativ
e de
Low alpha 0.2
0.4
0.6
Rel
ativ
e de
PARALLEL - Avg RD
PARALLEL - Avg LD layerLow alpha0750 800 850 900 950 1000 1050 1100 1150
Time (s)
Low alpha0750 800 850 900 950 1000 1050 1100 1150
Time (s)
PARALLEL Avg LD layer
PARALLEL - Avg HD layerLow alpha
High thermal conductivityT l i iTwo layers in series
Current
Low
Highdensity
CurrentDensity contours
Lowdensity
Time
•Most of the Joule heating I2R occurs in the low density layer due to higher
Temperature contours
Most of the Joule heating I R occurs in the low density layer due to higher resistivity
•Thermal diffusivity is high enough to homogenize the temperature minimal intermediate distortion
Low thermal conductivityT l i iTwo layers in series
Current
Low
High
CurrentDensity contours
Lowdensity
Time
Temperature contours
•Joule heating I2R in the low density layer create temperature gradient • Strong intermediate distortion as low density layer densifies first!
High thermal conductivityT l i ll lTwo layers in parallel
High initial density Density contours
Current
Low initial density Time
Temperature contours
•Most of the Joule heating V2/R occurs in the high density layer (!)•Minimal intermediate distortion because thermal diffusivity is high enough to homogenize the temperature
Low thermal conductivityT l i ll lTwo layers in parallel
Density contoursHigh initial density
Current
TimeLow initial density
Temperature contours
•Most of the Joule heating V2/R occurs in the high density layer• Strong intermediate distortion as high density layer densifies first!
Low thermal conductivityS i P ll l i di iSeries vs Parallel at intermediate time
Current D it t
Low
High
Current Density contours
Current
Lowdensity
Lowdensity
High
Temperature contours
density
• Note that strong intermediate distortion has opposite curvatures
MessageMessage
UnderstandingUnderstanding the interplay between current temperature & sintering cancurrent, temperature & sintering can help us optimizeoptimize complex sintering operations under electrical currentelectrical current.
COMPLEX SHAPE FAST
SYSTEM + COMPACTSINTERING + THERMOELECTRICALSINTERING THERMOELECTRICAL
SIMULATION
Application of fully coupled model to complex h i ishape sintering
• New I(t)– Applied load
– Uniform initial RD0– More complex heat transfer
C t t ith h d di
I(t)
• Contact with punches and die• Conduction to/from heat sources/sinks
– Axisymmetric
• Thermal electric part:– Conduction, convection, and
di iradiation– Stepwise current profile
Top and bottom T fixed at 15oC
Fully coupled (TDE) vs thermoelectric (TE)ΔT f l t t di f ( l d l )ΔT from sample center to die surface (cylindrical geometry)
250
300
- die
sur
face
TDE
TE
150
200er
tem
pera
ture
m
pera
ture
(oC
)
0
50
100
Com
pact
cen
te tem
00 200 400 600 800 1000
Time (s)
C
TDE thermal + sintering + electricalTDE = thermal + sintering + electricalTE = thermal + electrical
TDE vs. TE ‐ ΔT within compact (cylindrical )geometry)
120
pact
TDE
80
100
erat
ure
- com
pat
ure
(oC
)
TE
40
60ct
cen
ter t
emp
edge
tem
pera
0
20
0 200 400 600 800 1000
Com
pac
Time (s)
Material in center of compact heats up first due to heat from punches and heat loss to die this material begins to densify first and leads to a situation i il t ll l fi ti f i t dsimilar to parallel configuration of previous case study
Fully dense material properties used for TE simulation
Explanation
Temperature gradient
Densification gradient
Current gradient (current prefers to go through the more densified area)
Pressure gradient (more densified area supportshigher stress)
EVEN IF AT THE END OF THE DAY THE DENSITY IS SIMILAR (TENDS TO 100%)DIFFERENT PARTS OF THE SAMPLE WILL UNDERGO DIFFERENTTEMPERATURE/DENSIFICATION HISTORY
ELECTRIC CURRENT DENSITYTHROUGH THE SPECIMENTHROUGH THE SPECIMENINCREASES DUE TO DENSIFICATION
relative density and temperature
t=645st 645s
t=676s
t=840s
Half models
Applied stress = 2.8 MPa
Effect of load on sintering (cylinder)Effect of load on sintering (cylinder)1.2
0.8
1
sity
0.6
elat
ive
dens
0.2
0.4Re
2.8 MPa4.2 MPa
0400 500 600 700 800 900 1000 1100
Time (s)( )
A “fully” coupled model has been l dimplemented in FEM
d d h d • Predictions are as good as the data we put in • A “good” sintering model is needed• If there is coupling of current and sintering then a sintering model with E effects should be gimplemented (no additional computational difficulty)y)
The big challengeThe big challenge
• Coupling of sintering and electrical effectsCoupling of sintering and electrical effectsAlthough the presented work provides the framework for FEM modeling of FAST the keyframework for FEM modeling of FAST the key problem appears to be delineating the coupling of sintering and currentcoupling of sintering and current
MICROSCOPIC CONSIDERATIONS
Some (not so) random thoughts
DOES SIZE MATTER?
If the same contact go through these two arrangements of particleswhich contacts see higher current density?
α1 α2
ifd1/α1 = d2/α2
d1 d2
1 1 2 2
I I
B
ALL
A
THE TWO ARRANGEMENTS SEE THE SAME CURRENT DENSITY
THE CURRENT DENSITY IS AMPLIFIED AT THE NECK BY > (d/α)2
α1 α2
d1 d2
I I
B
ALL
A
Contact resistance – size effect For large scales, the behavior of contacts is relatively well known (Maxwell, Holm), the solution of the electrostatic problem
id ti t f th t t i tprovides an estimate of the contact resistance.
When the size of the contact ‘neck’ a decreases, the scattering l h f h h b bl h h l h l
length L of the charges becomes comparable with the length scale of the contact – Knudsen effect
aL
aK
aR
34
2Maxwell‐Holm Knudsen‐Sharvin
Still this does not imply that the current density on nanoscale contacts will be much higher.
ELECTRICELECTRIC POTENTIAL
ELECTRIC CURRENT DENSITY
Note that although the neck is an equipotential surfaceThe current density is not uniform – there is a peak at the root of the neck
At low densities
th l l t d itthe local current density
is x2 3 times higheris x2‐3 times higherdue to the isolated paths
1. Zhang, J., Numerical Simulation of Transient Thermoelectric Phenomena in Field Activated Sintering. 2004, Drexel University: Philadelphia, PA.
Electric t current
density Electric current density magnitude (A/m2)y g ( / )
REMEMBER THE
alumina powders graphite cylinder
VALUE OF CURRENT DENSITY?
Conductive material:Conductive material:~ 100 A/cm2
multipled This is electromigration maybe even by (d/α) 2 or more
~10000 A/cm 2
for small neck
electroplasticity territory
HOWEVER THIS AMPLIFiCATION IS SIGNIFICANT ONLY AT
129
for small neck IS SIGNIFICANT ONLY ATTHE VERY EARLY STAGE OF DENSIFICATIONS
Electroplasticity (?)
Conrad MSE A322 (2002) 100–107
ElectromigrationElectromigration
Cu CuCu‐.8w/oSb Under
electric field10,000 A/cm2
Power input patternsPower input patterns
Same power input: V2t=const.
DC
V2 V2
p p
DC Double‐pulse
t(F)0 4 0 1 4 t(F)( ) ( )
F: Fourier number
Temperature evolution: DC vs. pulse current
Top: DC
Right: Double-pulse
Temperature evolution: DC vs. pulseTemperature evolution: DC vs. pulse
DC Double-pulse
t(F)0 4 0 1 4 t(F)
Temperature distributionTemperature distribution
25
15
20
25ge
, % DCDouble-pulse
Fourier number = 4
5
10
15
erce
ntag Double pulse
0
5
0 200 400 600
Pe
Temperature,oC
Pulse current offers more uniform temperature distribution than DC at the end of heating.
Is the contact much hotter than the bulk of the d i l ?powder particle?
For a 10 micron typical conductive (λ=1E‐5 m2/s) powder yp ( / ) pA temperature difference between the contact and the center will homogenize within 5‐10 x R2/λ = 12.5‐25 μs
Almost instantaneo s homogeni ation of temperat re differencesAlmost instantaneous homogenization of temperature differences.
Unless the powder particle is very large (>100 micron) the temperaturebetween contact and particle bulk is almost the same…
Food for thoughtFood for thought
• Experimental determination/delineation ofExperimental determination/delineation of current vs non‐current effects
• Proper model calibration• Proper model calibration• Models for small size (non‐continuum)• Understanding of local chemistry
Managing expectationsWHAT MODELING IS NOT
WHAT INDUSTRY WANTS WHAT ACADEMIA OFFERS
MODELING IS A TOOLMODELING IS A TOOL
1980s 1990s 2000s1980s 1990s 2000s