Finite element modeling of electric current-activated sintering: The effect of coupled electrical potential, temperature and stress

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Acta Materialia 55 (2007) 3611–3622

Finite element modeling of electric current-activated sintering: Theeffect of coupled electrical potential, temperature and stress

X. Wang, S.R. Casolco, G. Xu, J.E. Garay *

Department of Mechanical Engineering,University of California, A303 Bourns Hall, Riverside, CA 92521, USA

Received 24 October 2006; received in revised form 9 February 2007; accepted 12 February 2007Available online 5 April 2007

Abstract

A finite element model is developed to simulate three-way coupling of thermal, electrical and mechanical behavior of electric current-activated sintering. In addition to the direct simulation of the evolution of electrical potential and temperature in the system, the model iscapable of simultaneously calculating the displacement fields and the stress distributions. Experimentally measured temperatures arefound to be in excellent agreement with simulations. The displacement and stress distributions were found to depend on the materialproperties. In the copper system, owing to its large thermal expansion, the steady-state vertical stress has a gradient of 33% acrossthe sample at a 1000 A current input (�700 �C); the radial and angular stresses are significant and have larger gradients. The maximumshear stress analysis indicates high stress concentrations at the surfaces at the ends of plungers that are in contact with the spacers.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Sintering; Finite element modeling; Stress distribution

1. Introduction

Current-activated sintering methods have demonstratednumerous benefits over conventional processing and havebeen used to produce materials with excellent propertiesoften unattainable with other densification methods [1–4].The benefits include dramatically reduced processing timesand temperatures for a variety of material classes, includ-ing metals [5], ceramics [6] and composites [7]. Recently,the technique has been shown to be a reliable method forattaining bulk nanostructured materials [8,9]. These andother developments have brought growing interest andwidespread use.

In the literature, current-activated sintering methodshave been given several other names, including sparkplasma sintering (SPS), field-assisted sintering (FAST)and pulsed electric current sintering (PECS). Althoughthe current delivery method varies among laboratories

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.02.022

* Corresponding author. Tel.: +1 951 8272449; fax: +1 951 8272899.E-mail address: [email protected] (J.E. Garay).

(DC, varying frequency pulsed-DC), the unifying themeis that large electric currents are used, and external loadsare applied during processing. The application of high-den-sity electric currents produces high heating rates and canpotentially significantly enhance diffusion and/or reactionprocesses [10,11]. The stress that is generated by the exter-nal loads also plays a significant role in the densificationprocess.

Despite the significant advantages over other densifica-tion methods, further advances could be hindered by stressor temperature heterogeneities, especially as applied to theproduction of very large or net-shaped parts. In order toovercome these challenges and ensure materials with uni-form properties, a precise understanding of the cause ofpotential temperature and stress heterogeneities is needed.For this reason, the parameters that effect the current/tem-perature/stress distributions and their time evolution areextremely useful for the continuing development of theprocess and materials produced.

Recently, there has been some work in simulating thetemperature [12–15] and current distributions [13–15] of

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3612 X. Wang et al. / Acta Materialia 55 (2007) 3611–3622

the process. These studies were primarily prompted by con-cerns over possible temperature inhomogeneities that couldadversely affect the materials produced or the accuracy ofthe temperatures being reported in experimental studies.Yucheng and Zhengi [12] used analytical solutions of theone-dimensional Fourier equation to calculate temperaturedifferences in the die and sample. Zavaliangos et al. [13]applied the commercial package ABAQUS and performeda coupled electrical–thermal analysis of the process payingspecial attention to contact resistances. Anselmi-Tamburiniet al. [14] applied the CFD-ACE+ code to examine theeffect of widely varying sample conductivities on the cur-rent and temperature distributions. They showed that thesample conductivities significantly effect both distributions.Vanmeensel et al. [16] also presented a coupled thermal–electrical analysis using the ANSYS code. They used theiranalysis to propose a measurement scheme that minimizesmeasured and actual temperatures. For the most part,these models have been qualitatively consistent and agreedwell with experiments.

Mechanical aspects such as stress distributions havebeen neglected thus far. This is a critical omission, becausethe applied load is one of the main control parameters, andexperimental studies have demonstrated that the processingpressure is of crucial importance to the materials obtained[9]. Thus, unforeseen stress gradients can play as importanta role as temperature gradients. The present study presentsthe development of a finite element (FE) model that can beused to simulate three-way coupling of thermal, electricaland mechanical behavior of the current-activated sinteringprocess. The focus is on studying the influence of variousmaterial and control parameters on the thermal and stressdistribution in the system. The results provide informationthat can be used for die design that could subsequently leadto the reliable production of large net-shaped parts withcontrolled properties.

2. Finite element modeling

The FE model is based on a set of governing equationsthat involve a three-way dynamic coupling of the chargeconservation law, Fourier’s law and the mechanical equa-tions of motion. The dynamic elastic behavior and electri-cal potential, however, reach the steady state in a muchshorter period compared with the heat transfer, so thatthe mechanical equations of motion and electrical potentialare treated as quasistatic, thus reducing the unnecessarycomplexity of modeling and simulation. The electricalpotential and temperature coupling is strong because ofthe relatively high dependence of thermal and electricalcoefficients on the temperature. However, while the tem-perature has a significant effect on the stress distributionthrough thermal expansion and moderate influence onthe elastic constants, the effect of stress on the thermaland electrical coefficients is assumed to be negligible. Asa consequence, in the linearization process of modeling,the temperature-dependent coefficients are treated as func-

tions of positions and are updated individually at eachnode at every time step.

In order to ensure good mechanical strength, the graph-ite in typical current-assisted sintering experiments is denseand polycrystalline, with a relatively small grain size. Themeasured properties of this type of graphite are quite uni-form, and thus it is treated as isotropic in the model. Inpractice, powder is always used in experiments. Inmodeling, however, the sample is treated as an isotropicsolid—the densification process is not treated. Despite thissimplification, the results serve to shed light on the impor-tant consequences of temperature/potential dependentstress distributions in the process. It should be noted thatall previous models [12–15] of the process have also madethis simplification. Moreover, the present model shouldapproximate reality relatively well in the latter stages ofthe densification process (final stage of sintering).

2.1. Modeling of temperature and electrical potential field

The electrical potential and temperature distributionsare governed by the following system of simultaneous par-tial differential equations (PDE)

r � ~J ¼ 0 ð1Þ

r � ~f þ qcpoTot¼ qe ð2Þ

where ~J ¼ r~E is the current density, ~E is the electric field,and r is the electrical conductivity. ~f ¼ �krT is the heatflux density, T is the temperature, and k is the thermal con-ductivity. q is the density, cp is the heat capacity, andqe ¼~J � ~E ¼ j~Jk~Ej ¼ JE is the heat generated by the flow-ing current per unit volume per unit time. Considering that~E ¼ �r/, where/ is the electric potential, the equationscan be written as

r � ~J ¼ r � ðr~EÞ ¼ r � ð�rr/Þ ¼ 0 ð3Þ

r � ð�krT Þ þ qcpoTot¼ JE ð4Þ

The initial and boundary conditions used for the solu-tions are: (a) the initial temperature is 300 K; (b) the pro-cess takes place in vacuum, so heat losses by conductionor convection through the gas are neglected. All the lateralsurfaces, however, have heat losses by radiation towardsthe chamber walls, which are held at room temperature.This heat loss is given by f ¼ mnðT 4

w � T 40Þ, where f is the

heat flux per unit area, m is the emissivity (which is assumedto be equal to 1 in this analysis), n is the Stefan–Boltzmannconstant, Tw is the temperature of the die surface, and T0 isthe temperature of the walls of the chamber; (c) the temper-ature of the two extreme upper and lower copper surfaces(copper electrodes, as shown in Fig. 1) is set at 300 K; (d)current flux can be considered for two operational modes,constant voltage or constant current. The constant currentmode was used in the present study because it more accu-rately represents the experimental control parameters ofthe apparatus acting as a constant current power supply.

Copper

Graphite

Sample (Cu or Al2O3)

Electrode

Electrode

Spacer

Y [m]

X [m]

Die

Plunger

Spacer

Sample

0.004 0.038

0.080

0.020

Point A

Point C

Point B

0.019

0.044

0.076

σz

σr

σθ

Fig. 1. Schematic of the current-activated sintering apparatus showingdimensions and coordinate system with origin at the middle of the sample.The inset in the top right-hand corner shows the relation of the three-dimensional stresses in the radial, angular and vertical directions.

X. Wang et al. / Acta Materialia 55 (2007) 3611–3622 3613

Taking advantage of the axisymmetrical configurationand converting Eq. (3) into a cylindrical system gives

r � ð�rr/Þ¼� r;r/;zþr/;rrþrr/;zþr;r/;zþr/;zz

� �¼ 0

ð5Þ

r;r/;rþr/;rrþrr/;rþr;z/;zþr/;zz¼ 0 ð6Þ

Similarly, Eq. (4) becomes

k;xT ;r þ kT ;rr þkx

T ;r þ k;zT ;z þ kT ;zz þ JE � qcpT ;t ¼ 0:

ð7ÞLet

T ðr; zÞ ¼ Naðr; zÞT a ð8Þ/ðr; zÞ ¼ Naðr; zÞ/a ð9Þrðr; zÞ ¼ N aðr; zÞra ð10Þkðr; zÞ ¼ N aðr; zÞka ð11Þ

where N(r, z) is the linear shape function for triangularelement.

Applying the Galerkin method, the following equationsare obtained:Z

XNbðr;r/aN a;r þ r/aN a;rr þ

rr/aNa;r þ r;z/aNa;z

þ r/aN a;zzÞdX ¼ 0 ð12ÞZX

Nbðk;rT aNa;r þ kT aNa;rrþ k

rT aN a;r þ k;zT aNa;z

þ kT aNa;zzþ JE � qcpT ;tÞdX ¼ 0 ð13Þ

Now we can adopt the linear interpolation between timesteps n and n + 1.

T nþ1 ¼ T n þ Dt ð1� bÞ dT n

dtþ b

dT nþ1

dt

� �ð14Þ

and define

Kabc ¼ N a;rN b;r

Z ZrN c dr dz ð15Þ

P abc ¼ N a;r

Z ZN bNc dr dz ð16Þ

Qabc ¼ N a;zNb;z

Z ZrN c dr dz ð17Þ

Mab ¼Z Z

rN aN b dr dz ð18Þ

Then the following recurrence equations are obtained:

� rc/nþ1a ðKabcþQabcÞ ¼ 0 ð19Þ

� kcT nþ1a ðKabcþQabcÞ �

qcqMab

bDtT nþ1

a

¼�rcðð/naN a;rÞ2þ ð/n

aN a;zÞ2ÞMbc þ qcqMab

� � 1

bDtT n

a�1� b

bdT n

a

dt

� �þZ

r¼Rmnð T n

w

� 4� T 40ÞrN b dy

ð20Þ

2.2. Modeling of mechanical displacements

The strain energy of the system can be expressed by

U ¼ 1

2

ZXðfegT ½C�feg � 2fegT ½C�fe0gÞdX ð21Þ

where

feg ¼

er

eh

ez

erz

8>>><>>>:

9>>>=>>>;¼

oor 01r 0

0 ooz

ooz

oor

26664

37775fdg

are the elastic strains. Here, {d} are the mechanicaldisplacements,

½C� ¼

l1 þ 2l2 l1 l1 0

l1 l1 þ 2l2 l1 0

l1 l1 l1 þ 2l2 0

0 0 0 l2

26664

37775

is the elastic matrix for the isotropic solid, and

fe0g ¼

atT

atT

atT

0

8>>><>>>:

9>>>=>>>;

are the thermal strains, where at is thermal expansioncoefficient.

The total mechanical potential energy is

3614 X. Wang et al. / Acta Materialia 55 (2007) 3611–3622

P ¼ U �Z

CfdgTf�T gdC ð22Þ

where f�T g is the vector of surface tractions. The bodyforces are ignored in the analysis.

Minimization of the total potential energy results inmatrix equations, which can be used to solve thedisplacement.

3. Device set-up and simulation scheme

3.1. Experimental procedure

The experiments were performed on a laboratory appa-ratus capable of delivering 4800 A and a 150 kN load. Inthis system, the load was delivered by a Universal testframe (Instron 5584, Instron Inc., USA) and the currentwas provided by programmable power supplies (XantrexInc., Canada). All experiments were performed using DCcurrent. The experiments were performed under vacuum(<4 · 10�5 Torr), in a custom fabricated water-cooled highvacuum chamber (MDC Corp., USA). The sample used inall experiments was a fully densified alumina piece (>99%theoretical density).

The temperature was measured using grounded andshielded type-K thermocouples (0.10-mm diameter wiresand 0.50-mm sheath diameter, Omega Engineering,USA). The thermocouples were inserted in holes drilledin the graphite die at a point halfway between the outerand inner diameter of the die (point A in Fig. 1). The datafrom the thermocouples were acquired using NationalInstruments data acquisition cards.

3.2. Simulations

A schematic of the current-assisted sintering apparatusused in the experiments and simulation is shown inFig. 1. The two extreme ends (electrodes) are water-cooledcopper. In the simulations, they were kept at a constanttemperature of 300 K. As previously mentioned, the spac-ers, die and plungers are graphite. Fig. 1 also shows pointsof particular interest for which results are presented anddiscussed in later sections.

The sample (copper and alumina) properties used in thecalculations are listed in the Appendix. Graphite propertiesare highly dependent on the type of graphite used. Thegraphite properties derived from fitting (with experimentalresults, see Section 4.2) and from a literature search arealso listed in the Appendix. Contact resistance wasneglected in the simulations, as it has been previouslyargued that its role is secondary [14].

In experiments, a constant load of 40 kN was applied tothe top electrode. In the simulations, a constant pressure of8.72 MPa was applied to the top electrode; the electrodehad a diameter of 76 mm, equivalent to applying a 40 kNload. This translates to �140 MPa of nominal applied pres-sure on the sample, which is typical of experiments. In the

simulation, the initial temperature of the whole apparatuswas set to be 300 K, the boundary temperature at the upperand lower ends was set to be a constant 300 K. Uniformpressure was applied on the top boundary; and zero verti-cal displacements were assumed for the lower boundaryand the middle point of the die outer surface. These elasticboundary conditions do not exactly reflect the actual exper-imental conditions and may to some extent cause an asym-metry to the displacement fields of the upper and lowerspacers. However, as these effects are negligible in the cru-cial plunger–die–sample portion, they are consideredappropriate approximations. In calculating the mechanicaldisplacements, friction is neglected at the contactinterfaces.

An adaptive mesh was used to model the section, withthe finest mesh section corresponding to the area near thesample. There are a total of 7980 elements. Under currentconstruction, on a computer with Pentium4 2.6 GHz CPUand 512 MB memory, the calculation at each time steptakes approximately 4 s to calculate the temperatures andelectrical potentials and one minute to calculate themechanical displacements for each iteration of the contactconditions.

4. Results

4.1. Experimental verifications

In Fig. 2, the simulation results of temperature evolutionat point A are compared with the experimental results. Thesample used in the experiments was fully dense alumina(>99% density). To demonstrate the accuracy of the simu-lation, results of two different current inputs (500 A and1000 A) are compared in the figure. For the 500 A input,the temperature reaches �290 �C and �700 �C for 1000 Ainput for both the experimentally measured values andthe simulation. Both pairs of curves show excellent matchbetween experiment and simulation.

4.2. Electrical potential and temperature distribution

In Fig. 3a and b, contour plots of steady-state tempera-ture in the sample/die regions for non-conductive alumina(Fig. 3a) and conductive copper (Fig. 3b) samples are plot-ted. The magnitude of the highest temperature (�700 �C) issimilar for both samples, although temperatures in the alu-mina system are �10 �C higher. Comparison shows thatthe conductivity of the sample has some impact on the tem-perature distribution in the die sample region. This is to beexpected, because the current is the only heat source in theprocess. In the non-conductive alumina sample, there is alarge current concentration in the die at regions close tothe sample. By contrast, for the conductive copper sample,the current concentration is in regions within the samplethat are close to the die wall. The consequence is that thereare more contour lines in the middle region of the appara-tus for conductive sample, and Fig. 3c shows that the tem-

Time(s)

Tem

per

atu

re( ˚

C)

0 500 1000

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

500A Exp1000A Exp1000A Sim500A Sim

Fig. 2. Experimental verification, temperature evolution at the middle ofthe die (point A).

X. Wang et al. / Acta Materialia 55 (2007) 3611–3622 3615

perature gradient inside the non-conductive sample is moreuniform than that inside the conductive sample.

4.3. Steady-state mechanical displacements

The mechanical displacements of the system at steadystate (with an applied current of 1000 A) are shown inFig. 4. As shown earlier, the temperature distribution ata given current level is complex. In order to facilitate dis-cussion, the steady-state temperature in the center of thesample will be given along with the applied currentthroughout this paper. As an example, the temperature atthis center point is �692 �C for copper and �705 �C foralumina at 1000 A. In Fig. 4a, the displacement in theradial direction is shown for a simulation using an aluminasample. In the plunger–die–sample region, the radial dis-placement increases fairly uniformly with increasing radius,and thus the highest displacement occurs near the surfaceof the die. These radial displacements are due to the com-bination of the thermal expansion and a Poisson-typeexpansion due to the pressure applied in the vertical direc-tion. Fig. 4b is a plot of radial displacements for the appa-ratus using a copper sample. Comparison of Fig. 4a and breveals different patterns of contour lines of radial displace-ments for alumina and copper samples. In contrast to thealumina case (Fig. 4a), the radial distribution of the coppercase (Fig. 4b) does not increase uniformly with radial posi-tion. Instead, the portion where the copper is in contactwith the die wall is more displaced than the portions wherethe plunger is in contact with the die wall (graphite ongraphite). The top portions have higher displacement dueto uniform pressure applied to the top electrode and theboundary condition that the vertical displacements of thelower end of the apparatus are set to be zero. Althoughnot explicitly shown here, similar differences between the

copper and alumina samples exist in the angular and verti-cal displacements. These displacement differences are man-ifested in the stress distributions shown in the next section.

4.4. Steady-state stresses

In Fig. 5, contour plots of stresses in the radial and ver-tical directions for both alumina (Fig. 5a and c) and copper(Fig. 5b and d) samples at 1000 A current input and steady-state temperature are presented. The stresses are calculatedat the center points of the triangular elements and then dis-tance weighted to nodes. The radial stress contours (Fig. 5aand b) reveal that the maximum radial stress gradientoccurs where the plunger comes into contact with thespacer. In this small region, the radial stress changes fromcompressive in the spacer to tensile in the plunger.Although the magnitudes are different, this gradient behav-ior is observed for both the copper and the alumina sam-ples. Also in both cases, there is a relatively lowcompressive radial stress in the majority of the plungerand a very low radial stress in the radial extremities ofthe die. Larger differences between copper and aluminasamples are observed close to the sample. In the case of alu-mina, the radial stress is uniform (zero) in the entire sam-ple, while the copper sample has an inhomogeneouscompressive stress distribution with a higher magnitudetowards the edge. Correspondingly, the radial stress is alsorelatively high in the die where the sample comes into con-tact with the die wall.

Fig. 5c and d are contour plots of the vertical stress inthe alumina and copper samples, respectively. In contrastto the angular and radial stresses, the vertical stresses arecompressive over the entire modeled section in both cases.The maximum stresses are found in the corners where theplunger comes into contact with the spacer. Once again,the stress is uniform across the sample for alumina andnon-uniform for copper. Although the contour plots arenot explicitly shown here, steady-state shear stresses rrz

were also calculated in the system. For the alumina samplesystem, the shear stresses are zero everywhere except at thecorners of the plunger–spacer interface. The situation atthe plunger–spacer corner is similar for the copper samplesystem, yet shear stresses are also observed in the sampleand in the die. The shear stresses occur at the corners ofthe die wall and sample interface.

The situation in the sample is more clearly shown inFig. 6, where stress distributions (radial, angular, verticaland rz shear) along the cross-section of the middle of sam-ple and die are plotted for 1000 A at steady state (Fig. 6aand b). This cross-section corresponds to the X-axis shownin Fig. 1. For comparison, the results for a 500 A currentinput are also plotted (Fig. 6c and d). At 1000 A, the tem-perature in the middle of the sample is 705 �C for the alu-mina sample and 692 �C for the copper sample while, at500 A, the temperature is 283 �C for alumina and 275 �Cfor copper. The stresses are qualitatively and quantitativelysimilar for both input levels in the alumina system. The

X [m]

Y[m

]

0 0.01 0.02 0.03 0.04-0.02

-0.01

0

0.01

0.02

T [0C]706705704703702701700699698697696695694693

X [m]

Y[m

]

0 0.01 0.02 0.03 0.04-0.02

-0.01

0

0.01

0.02

T [0C]695694693692691690689688687686685684683682681680679678

a b

Radial Position (m)

Tem

per

atu

re (

°C)

0 0.005 0.01 0.015 0.02

680

685

690

695

700

705

710

715

1000A Copper1000A Alumina

c

Fig. 3. Steady-state plots at 1000 A current input of: (a) temperature contour plot of sample–die area of alumina sample; (b) temperature contour plot ofsample–die area of copper sample; (c) temperature distribution across the radial middle (X-axis of Fig. 1) of the sample and the die. Note: In (a) and (b),the temperature below the least number shown in the legends are shown in the same color as the least number.

3616 X. Wang et al. / Acta Materialia 55 (2007) 3611–3622

radial, angular and shear stresses are uniform (zero) acrossthe entire alumina sample for both current input levels(Fig. 6a and c). There are small angular stresses and verti-cal stresses in the die (12 MPa and 8 MPa, respectively) atthe interface. The vertical stress quickly decreases to 0 MPaand is constant in the majority of the die, while the angularstress also decreases but remains tensile at a level of 4 MPaeven at the end of the die. The largest magnitude of stress isby far the compressive vertical stress that is nearly uniformat �140 MPa for both 500 and 1000 A. The similarity ofFig. 6a and c suggests that the stresses in the alumina sam-ple system are not temperature dependent.

In contrast, comparison of Fig. 6b and d reveals thatamplitudes and distributions of stresses inside the copper

sample system depend on the magnitude of the current(in reality the temperature caused by the current). In thiscase, only the shear stress is near zero over the plottedcross-section. It should be recalled that shear stresses doexist at the sample corner regions. At both current levels,the radial and angular stresses are similar and compressiveover the entire sample, although they decrease toward theedge of the sample. In the 1000 A case (Fig. 6b), they bothhave a maximum value of 72 MPa in the middle of the sam-ple and decrease to �45 MPa at the edge. In the die, theangular stress becomes tensile (24 MPa) and decreases to12 MPa, while the radial stress remains compressive anddecreases gradually to 0 MPa. The vertical stress is positiondependent as well. In the middle of the sample it is

X [m]

Y[m

]

0 0.025 0.05 0.075

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Ur[m]

0.0001266560.0001181940.0001097320.000101279.28075E-058.43455E-057.58834E-056.74213E-055.89592E-055.04971E-054.2035E-053.35729E-052.51108E-051.66487E-058.18666E-06

-2.75433E-07-8.73752E-06-1.71996E-05

X [m]

Y[m

]

0 0.025 0.05 0.075

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Ur[m]

0.0001340350.0001251460.0001162580.000107379.84812E-058.95929E-058.07045E-057.18161E-056.29278E-055.40394E-054.51511E-053.62627E-052.73743E-051.8486E-059.59763E-067.09274E-07

-8.17908E-06-1.70674E-05

a b

Fig. 4. Steady-state contour plots at 1000 A current input of: (a) radial displacements of sample–die area of alumina sample (�705 �C at center of sample);(b) radial displacements of sample–die area of copper sample (�692 �C at center of sample).

X. Wang et al. / Acta Materialia 55 (2007) 3611–3622 3617

144 MPa and increases radially, reaching a maximum of152 MPa near the sample edge. After this maximum, thevertical stress quickly decreases to 100 MPa at the edge.This represents a difference of �50 MPa across the sampleat this current level. The trends are similar for the 500 Ainput (lower temperature), although the magnitudes arelower.

The stress distributions along the vertical center (Y-axis)of the modeled section are shown in Fig. 7 for the aluminasample system (Fig. 7a) and copper (Fig. 7b). As previouslynoted, both systems have large gradients in radial andangular stress (compressive peaks and tensile peaks) atthe plunger–spacer interface. The main difference is in theradial and angular stresses sample region. These two stres-ses have a much larger magnitude in the case of copper. Inaddition, a larger plateau of the vertical stresses is observedfor the alumina sample.

Fig. 8 shows the effect of applied current and corre-sponding temperature on the steady-state stresses at vari-ous points in a copper sample as shown in Fig. 1. Thecurrent level is varied from 500 to 1000 A, resulting in tem-perature varying from 275 to 692 �C. The stresses of themiddle of the sample (point C) and the middle point ofthe edge of the sample are shown (point B). At both loca-tions, the shear stress remains unaffected by current overthe entire current range calculated. However, the otherstresses are affected by applied current (resultant tempera-ture), and the magnitude of change depends on the positionin the sample (different points B and C). At both locations,the radial and angular stresses increase significantly as thecurrent increases. The radial and angular stresses are thesame at the middle of the sample, and slightly different atthe edge. In addition, the stress increase is sharper in themiddle of the sample than at the edge. The vertical stressin the middle of the sample remains almost unaffected by

changes in applied current (temperature). By contrast, analmost linear drop in the vertical stress at the edge of thesample is shown.

A contour plot of the maximum shear stress at theregion where the top plunger comes in contact with thespacer is shown in Fig. 9. Note that these are not therz-shear stresses shown in previous figures, but the maxi-mum shears smax. The figure reveals very high shear stressesat the ends of the plungers that are in contact with the spac-ers. A picture of a plunger that failed at the point of calcu-lated maximum shear is shown in Fig. 10.

5. Discussion

Pulsed electric current is used in most current-activatedsintering experiments. However, DC current was used in allthe simulations in this study. This is believed to be a goodapproximation because, as shown previously [14], most ofthe energy in the process is delivered at low frequencieswith a large portion at 0 Hz, i.e., DC. Moreover, there issome experimental evidence that the overall energy deliv-ered is the controlling factor, with pulse frequency havinglittle if any effect [15]. DC currents were used in all the ver-ification experiments in this work.

As mentioned previously, the samples in the simulationare treated as isotropic solids, so densification is nottreated. In the initial stages, when the sample is porous,the real stresses are most likely very different from thecalculations. However, it is well known that densificationrates decrease significantly with increasing density [18], sothat a large portion of the sintering process occurs atdensities where the sample is more appropriately modeledas a solid rather than a powder. This is because, at this time(frequently referred to as the final stage of sintering), theremaining pores have become isolated (undergone Rayleigh

X [m]

Y[m

]

0 0.025 0.05 0.075

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04σz[Pa]

-5.99341E+06-1.98528E+07-3.37123E+07-4.75717E+07-6.14311E+07-7.52905E+07-8.91499E+07-1.03009E+08-1.16869E+08-1.30728E+08-1.44588E+08-1.58447E+08-1.72306E+08-1.86166E+08-2.00025E+08-2.13885E+08-2.27744E+08-2.41604E+08

X [m]

Y[m

]

0 0.025 0.05 0.075

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04σr[Pa]

2.83692E+072.14949E+071.46206E+077.74636E+06872078

-6.00221E+06-1.28765E+07-1.97508E+07-2.66251E+07-3.34993E+07-4.03736E+07-4.72479E+07-5.41222E+07-6.09965E+07-6.78708E+07-7.47451E+07-8.16193E+07-8.84936E+07

X [m]

Y[m

]

0 0.025 0.05 0.075

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04σr[Pa]

2.1783E+078.47538E+06

-4.83223E+06-1.81398E+07-3.14475E+07-4.47551E+07-5.80627E+07-7.13703E+07-8.46779E+07-9.79855E+07-1.11293E+08-1.24601E+08-1.37908E+08-1.51216E+08-1.64524E+08-1.77831E+08-1.91139E+08-2.04446E+08

X [m]

Y[m

]

0 0.025 0.05 0.075

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04σz[Pa]

-6.42678E+06-2.02713E+07-3.41158E+07-4.79603E+07-6.18048E+07-7.56493E+07-8.94939E+07-1.03338E+08-1.17183E+08-1.31027E+08-1.44872E+08-1.58716E+08-1.72561E+08-1.86405E+08-2.0025E+08-2.14094E+08-2.27939E+08-2.41783E+08

a b

c d

Fig. 5. Steady-state contour plots at 1000 A current input of: (a) radial stress of sample–die area of alumina sample (�705 �C at center of sample);(b) radial stress of sample-die area of copper sample (�692 �C at center of sample); (c) vertical stress of sample–die area of alumina sample (�705 �C atcenter of sample); (d) vertical stress of sample–die area of copper sample (�692 �C at center of sample).

3618 X. Wang et al. / Acta Materialia 55 (2007) 3611–3622

break-up), and the material behaves much more like a solidthan a powder. The results are therefore applicable for a sig-nificant portion of the processing time. Further densifica-tion occurs by diffusion along grain boundaries and/orthrough the lattice—both are highly dependent on stress.Knowledge of the stress distribution during these stages isvaluable for understanding the process.

Fig. 3a–c shows that the temperature is not uniformover the modeled section, and that the temperature profilesare dependent on the conductivity of the samples. Theseresults are consistent with previous models that have firmlyestablished that the temperatures are location dependent[12–15] and highly sensitive to the electrical and thermalconductivity of the sample [14].

These findings have a significant bearing on the interpre-tation of the experimental results, because they suggest that

measurement location is a crucial component of the exper-imental process. In other words, if care is not taken, themeasured (reported) temperature is not necessarily thesample temperature.

It has been shown previously that the geometry anddimensions of the apparatus can significantly affect thetemperature uniformity and magnitude [16]. However,this is not much of a concern for the modeled geometryand dimensions. Despite the large differences in the con-ductivity of the two samples considered here, the steady-state temperatures are <15 �C different at 1000 A(Fig. 3(c)). Moreover, the temperature gradients in theactual samples are not large: temperature differences are�2 �C in the copper sample. This represents a differenceof 1.5%. The temperature gradient in the alumina sampleis even less.

Radial Position (m)

Stre

ss (M

Pa)

0 0.005 0.01 0.015 0.02-140

-120

-100

-80

-60

-40

-20

0

σr

σθσz

σrz

Radial Position (m)

Stre

ss (M

Pa)

0 0.005 0.01 0.015 0.02-160

-140

-120

-100

-80

-60

-40

-20

0

20

σr

σθσz

σrz

Radial Position (m)

Stre

ss (M

Pa)

0 0.005 0.01 0.015 0.02

-140

-120

-100

-80

-60

-40

-20

0

σr

σθσz

σrz

Radial Position (m)

Stre

ss (M

Pa)

0 0.005 0.01 0.015 0.02

-140

-120

-100

-80

-60

-40

-20

0

20

σr

σθσz

σrz

a b

c d

Fig. 6. Steady-state stresses distributions along the cross-section of the middle (X-axis from Fig. 1) of sample and die (�705 �C at center of sample):(a) alumina sample at 1000 A current input; (b) alumina sample at 500 A current input (�283 �C at center of sample); (c) copper sample at 1000 A currentinput (�692 �C at center of sample); (d) copper sample at 500 A current input (�275 �C at center of sample).

X. Wang et al. / Acta Materialia 55 (2007) 3611–3622 3619

Much larger differences than those observed for temper-ature are found in the deformation and stress results.Fig. 4(a) and (b) reveals that the sample type plays a signif-icant role on the deformation of the system. The differencesin displacements between the alumina and copper samplescan be explained by differences in the coefficients of thermalexpansion (CTE) and modulus of the three materials con-cerned (graphite, copper and alumina). Graphite has amoderate CTE and a relatively low modulus. Copper hasa high CTE and a moderate modulus. Alumina has aboutthe same CTE as graphite but has a high modulus.

The result is that, although copper is quite stiff, its highthermal expansion causes it to deform significantly morethan graphite does. The copper sample therefore pushesthe die wall, causing deformation. By contrast, the thermalexpansion of the alumina sample is not greater than that of

graphite, so that it does not cause increased deformation ofthe die.

The particularly interesting effect of the inhomogeneousdeformation is the consequence on stress distribution. Thestress on the sample (usually referred to as pressure) that isquoted in experimental work is taken as the applied loaddivided by the cross-sectional area of the plunger. Thiscan referred to as a nominal stress rn given by

rn ¼F A

Acð23Þ

where FA is the externally applied uniaxial force, and Ac isthe cross-sectional area of the plunger. Using this scheme,the stress is assumed to be constant in the whole sample,regardless of the sample properties or temperature of thesystem. The direction dependence of stress is also ignored.

Vertical Position (m)

Stre

sses

(MPa

)

-0.1 0 0.1-140

-120

-100

-80

-60

-40

-20

0

20

40

Vertical Position (m)

Stre

sses

(MPa

)

-0.1 0 0.1

-140

-120

-100

-80

-60

-40

-20

0

20

40a b

σr

σθσz

σrz

σr

σθσz

σrz

Fig. 7. Steady-state stresses distributions across the vertical center (Y-axis) of the apparatus at 1000 A current input: (a) alumina sample (�705 �C atcenter of sample); (b) copper sample (�692 �C at center of sample).

-150-140-130-120-110-100-90-80-70-60-50-40-30-20-10

010

500 600 700 800 900 1000Current (A)

Stre

ss (M

Pa) hhhhhhhhhhhhh

Series3

Series4

Series2

thet

z

σr & θ Point C

σ z Point Cσ rz Point C&B

σ θ Point B

σr, Point B

σ z Point B

Fig. 8. Steady-state stress versus input current for copper samplecalculated at two different points: middle of sample (point C) and atsample–die interface (point B). The effect of current on stress is related tothe temperature that the current level causes. For reference, the temper-ature at the mid-point of the sample in this current range varies from275 �C (500 A input) to 692 �C (1000 A input).

X [m]

Y[m

]

0.007 0.008 0.009 0.01 0.011 0.012 0.013

0.025

0.0275

τ [Pa]1.37456E+081.2982E+081.22183E+081.14547E+081.06911E+089.92749E+079.16387E+078.40026E+077.63664E+076.87302E+076.1094E+075.34578E+074.58216E+073.81855E+073.05493E+072.29131E+071.52769E+077.64074E+06

Fig. 9. Contour plots of calculated maximum shear stress smax in theregion where the top plunger makes contact with the top spacer. The stresswas calculated at steady-state temperature.

3620 X. Wang et al. / Acta Materialia 55 (2007) 3611–3622

Yet the results indicate that all these factors are impor-tant, and the stress experienced by the sample is dependenton both the sample properties and the current level appliedfor the radial, angular and vertical directions.

In the case of alumina, only the vertical stress is signifi-cant in the sample: it is almost constant at �140 MPa forboth current input levels (Fig. 6a and c). The similaritybetween the two current levels suggests that the stressesin the alumina sample system are not temperature depen-dent. It should be noted that the vertical stress calculatedis very close to the nominal stress given using the tradi-

tional method (Eq. (23)). With this in mind, one can con-clude that, for samples with similar thermal expansion tothe die material (graphite), the stress reported in experi-mental studies is in reasonable agreement.

The situation is quite different in the copper sample sys-tem, where the stresses in all directions are both positionand current level dependent (Fig. 8). In this case, the stres-ses in the radial and angular directions are comparablewith those in the vertical direction. For example, undertypical conditions of 1000 A (a steady-state value of�700 �C), the stresses in the angular and radial directionsreach about half the nominal stress on a copper sample(Fig. 6b). More surprisingly, even the vertical stress differssignificantly. As seen in Fig. 6b, the difference in vertical

Fig. 10. Photo of a broken graphite plunger that failed under compressionat the location of calculated maximum shear stress.

X. Wang et al. / Acta Materialia 55 (2007) 3611–3622 3621

stress across a copper sample is �50 MPa, representing adifference of �33%.

The nominal stress rn calculated with Eq. (23) for thecopper sample is identical to the alumina sample,140 MPa. The results shown here indicate that the stressesare significantly different from the nominal stress at allpositions. In the middle of the sample, the calculated verti-cal stress is �150 MPa or 7% different from the nominalstress, while in the edge region it is �100 MPa or 29% dif-ferent from the nominal stress. It is worth emphasizing thatthe nominal stress is that reported in experimental studies.

The significant differences in actual and nominal stressesare especially interesting in light of the temperature gradi-ents, which are only 1.5% different across the sample forthe same conditions. In addition, examination of Fig. 8indicates that, for copper, the sample stress distributionas well as magnitude is highly temperature dependent.Because in a typical experiment the temperature is timedependent, the stress that the sample experiences is timedependent as well.

The impact of stress in current-assisted sintering is thatit contributes significantly to the densification process.Stress is known to add to the driving force for sintering[17] and aid in other densification mechanisms such ashigh-temperature plastic deformation [18]. Thus the densi-fication rates or mechanisms in every stage of the densifica-tion process (initial intermediate and final) can be affectedby the inhomogeneous stress distribution observed here.Inhomogeneous densification in the samples can, of course,lead to materials with significantly varying properties.

In the simulations, constant thermal expansion coeffi-cients are used while, in reality, the CTE can vary with tem-

perature. Considering that the CTE of copper (and manyother materials) increases significantly with temperature,the stresses calculated are most likely a conservative esti-mate. It is probable that, at high temperatures, the radialand angular stresses of a high-CTE sample exceed the ver-tical stress whose amplitude is largely determined by theapplied load. Thus these thermal expansion-caused stressescan become the dominant contributor to the stress-relateddriving force in the densification process.

It is very likely that the geometry and dimensions playa role in stress distribution as well. Studies are under wayto examine the effects of dimensions and geometry onstress gradients, with particular attention being placedon the design of net-shaped part production dies. Adetailed parametric study of the effect of CTE and otherproperties on stress in the process using the model is alsounder way.

From the point of view of the materials produced, thestresses in the sample are the most important. Yet the stres-ses elsewhere can determine the performance and limita-tions of the overall system. For example, it is the strengthof the graphite die and plunger that largely determinesthe maximum load that can be applied in the process.Knowledge of the maximum shear stresses can shed lighton die/plunger failure. The plunger shown in Fig. 10 failedowing to the large shear stresses depicted in Fig. 9.

It is worth emphasizing that the stress heterogeneitiesobserved here can cause heterogeneities in the materialsproduced using this method. As previously discussed, thestress gradients in the sample are greater (33% difference)than the temperature gradients (1%), suggesting thatmicrostructural and property heterogeneities that are usu-ally attributed solely to temperature differences might alsobe caused by stress gradients.

6. Conclusion

The main conclusions of the present work are summa-rized as follows.

6.1. Temperature distribution

(1) The temperature distribution is dependent on theconductivity of the sample.

(2) Thermal gradients are not very significant in themodeled geometry and dimensions for either the cop-per or alumina system (<1.5% difference across thesample for a 1000 A input).

6.2. Stress distribution

(1) Displacements and stress distributions are dependenton sample properties (CTE and modulus).

(2) The stress gradients in the sample are larger than thethermal gradients.

(3) Observation for the alumina sample system:(a) Only the vertical stress is significant.

3622 X. Wang et al. / Acta Materialia 55 (2007) 3611–3622

(b) The vertical stress calculated is comparable withthe nominal stress typically reported in experimen-tal work.

(c) Steady-state stresses are not temperaturedependent.

(4) Observations for the copper sample system:(a) Angular radial and vertical steady-state stresses

are significant. Radial and angular stress is abouthalf the magnitude of the vertical stress for a1000 A input.

(b) At 1000 A current input (�700 �C) the steady-state stress across a copper sample varies by�33%.

(c) Stresses in all directions except the shear stress aretemperature dependent.

(d) The modeled stresses (including vertical stress) aresubstantially different from the stress calculated bysimple division of force divided by area. This sug-gests that experimentally reported stresses can besignificantly wrong for materials with high CTE.

Acknowledgments

Support for this work by the Army Research Office isgratefully acknowledged. One of us (S.R.C.) would alsolike to thank UC Mexus for support through a post-doc-toral fellowship.

Appendix

Graphite properties

q (kg m�3) = 1850Cp (J kg�1 K�1) = 310.5 + 1.7 · T

k (W m�1 K�3) = 65 � 0.017 · T

qe (10�6 X m) = 26 � 3 · 10�2T + 2 · 10�5T2 �6.4 · 10�9T3 + 7.8 · 10�13T 4

l1 (GPa) = 71.0l2 (GPa) = 78.0aT = 8.0 · 10�6

Alumina properties

q (kg m�3) = 3970Cp (J kg�1 K�1) = aþT

bþcT

where

a = 65181330.4, b = �669628.8 and c = �8175.8k (W m�1 K�1) = aT

bþT þ cTdþT þ eT

where

a = 770.25, b = 249.4, c = �790.15, d = 249 ande = 0.008qe (X m) = 1 · 108

l1 (GPa) = 143.0l2 (GPa) = 163.0aT = 8.0 · 10�6

Copper properties

Cp (J kg�1 K�1) = 355.3 + 0.1T

k (W m�1 K�1) = 420.66 + 0.07T

qe (X m) = (5.5 + 0.038T) · 10�9

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