finite element modeling of fast - … · finite element modeling of fast ... – steady state...

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


Density Thermal activation

TRANSPORT ANALYSISJoule heating

Electroplasticeffect



ElectroplasticeffectEffect of current on diffusion or plasticity

Density dependence of electrical properties

Thermal activationof sintering Density

dependence of thermal properties




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


ELECTRICITY THERMAL


ELECTRICITY TRANSPORT

THERMAL ANALYSISJoule heating

ELECTRICITY TRANSPORT

THERMAL ANALYSIS

Effect of current on diffusion or plasticity

Joule heating





Density dependence of thermal propertiespropertiesproperties



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 ( / )


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)


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


4

5

6

7

ance

, m


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


ELECTRICITY THERMAL




Electroplasticeffect



ElectroplasticeffectEffect of current on diffusion or plasticity









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


•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


•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


•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


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


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

finite element modeling of fast - … · finite element modeling of fast ... – steady state...

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