10STWJ_Arora Cooling Rate From Dim Analyses

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Cooling rate in 800 to 500uC range fromdimensional analysis

A. Arora*1, G. G. Roy2 and T. DebRoy1

Structure and properties of steel welds are affected by their cooling rates in the 800 to 500 uC

range. The available cooling rate correlations are based mostly on heat conduction equation that

ignores convective heat transfer and are inaccurate or empirical in nature and valid only for a

limited range of welding conditions. Numerical heat and fluid flow models can accurately

calculate cooling rates, but they are not widely available. In the present paper, the authors

propose and test a generalised correlation for cooling rate developed through dimensional

analysis, valid for a wide range of welding processes, operating parameters and materialproperties.

Keywords: Cooling rate, Dimensionless correlation, Arc welding, Laser welding, Steels

Introduction

The properties of steel welds13 are affected by theircooling rates in the 800 to 500uC range where the phase

transformations important for the evolution of final

weld microstructure occur.4 The cooling rates of welded

samples are often calculated using analytical solutions of

heat conduction equations,5

because they are widelyavailable and easy to use. For three-dimensional heat

flow, the cooling rate R is commonly obtained from the

heat conduction equation as follows5

R~2pk T{Ti

2

Qg(1)

where kis the thermal conductivity of the weld material,T is the temperature at which the cooling rate is

calculated, Ti is the interpass temperature, Q is the heat

input per unit length of weld deposit and g is the arc

efficiency. However, Svensson et al.4 have observed that

such calculations based on heat conduction model areinadequate in representing experimental cooling curves

for a wide range of welding conditions. It is also worth

noting that other heat flow models such as that of

Rosenthal6 have also been found inappropriate for

application to the fusion zone.4

Calculations that ignore convective heat transfer withinthe weld pool overestimate the spatial variation of

temperature because they do not consider the mixing of

relatively hot and cold liquid metal. As a result, these

calculations also overestimate the cooling rate, which is

the product of the spatial gradient of temperature and the

solidification velocity. It will be shown subsequently in

this paper that the errors resulting from the use of heatconduction models can be unacceptably high becauseconvection is often the dominant mechanism of heattransfer within the weld pool. The significant discrepan-cies between the cooling rates computed from thecorrelation based on heat conduction model5 and theexperimentally measured cooling rate values available in

reference 4 are shown in Fig. 1. The cooling rates fromthe heat conduction equation required values of boththermal conductivity and arc efficiency. The values of arcefficiency were taken from the values provided inSvensson et al.4 However, Ref. 4 did not include anyvalue of thermal conductivity. In the calculations withheat conduction equation the authors used a value of27 W m21 K21 for simplicity. It should be noted that theuncertainty in the thermal conductivity values is insignif-icant when compared with the errors in the computedvalues of the cooling rates in Fig. 1. In fact, thermalconductivity value used is near the lower end of thetemperature dependent thermal conductivity of steels. A

higher value of thermal conductivity would result in evenhigher discrepancy between the experimentally measuredcooling rate4 and the values computed5 from the heatconduction equation. It can be observed that heatconduction model5 overestimates the experimental cool-ing rate4 by a factor of 25 in many cases.

Because of the difficulties in obtaining reliable coolingrates from heat conduction models, Svensson et al.4

suggested the following form of empirical correlationsfor computing cooling rate R

R~C1 DT

C2

Qg(2)

where DT is the temperature difference, and C1 and C2are the empirical constants whose values vary dependingon the welding processes and parameters. They pro-posed 27 sets of values for C1 and C2 for various weldingconditions. Each set of values of C1 and C2 is valid for aspecific set of welding conditions considered in their

1Department of Materials Science and Engineering, The PennsylvaniaState University, University Park, PA 16802, USA

*Corresponding author: email [email protected]

2Department of Metallurgical and Materials Engineering, Indian Institute ofTechnology, Kharagpur, India

2010 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 30 April 2010; accepted 8 June 2010DOI 10.1179/136217110X12720264008394 Science and Technology of Welding and Joining 2010 VOL 15 NO 5 42 3


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empirical correlations, and cannot be used for weldingconditions other than the ones used to derive these

correlations. Furthermore, the 27 correlations covered

cooling rates only up to 48 K s21. In other words,equation (2) is not applicable to many of the high energy

density processes that use a high welding speed andinvolve a high cooling rate.

The values of Peclet number for heat transfer withinthe liquid pool, which represents the ratio of heat

transfer by convection to conduction, have been found

to be much greater than one in all welding processes,especially for low thermal conductivity materials such as

steels.712 Although several numerical models of con-vective heat transfer and fluid flow have been reported in

the literature that can accurately calculate cooling rates,these models are complex and not always widely

available. Therefore, there is a need for a generalisedcorrelation that can calculate reliable cooling rates for a

wide range of welding processes, workpiece materials

and welding parameters.

The nature of the fluid flow and the resulting

convective heat transfer affects the temperature profilesin the weld pool and its geometry. The weld geometry can

be characterised by its aspect ratio which is the ratio of

the width to depth of the weld pool. The aspect ratioaffects the cooling rates for various welding processes.

The aspect ratio depends on the type of material, thenature of the heat sources for various welding processes,

the power density of the heat source and the nature andamount of the surface active elements. The presence of a

surface active element in steel reverses the fluid motionwithin the weld pool and makes it narrow and deep, with

a low aspect ratio. In many cases, a high energy density

welding process such as a keyhole mode laser or electronbeam welding process produces a deep and narrow weld

pool. A weld pool with low aspect ratio tends to dissipateheat primarily in the width and length directions. Thesedirections are often larger than the thickness of the

workpiece, and a low aspect ratio may result in a low

cooling rate for a given welding speed. On the contrary, asmall hemispherical pool may dissipate heat in all three

dimensions yielding a high cooling rate.13

Here the authors propose and test a non-dimensional

correlation of cooling rate. The proposed correlation

considers convective heat transfer in the weld pool

indirectly by considering the aspect ratio of the weld

pool as an independent variable. It is shown that the

proposed correlation can overcome the overestimations

of cooling rates by the heat conduction equations.

Furthermore, the proposed correlation is shown to be

fairly accurate for a wide range of cooling rates resultingfrom various welding processes, welding variables and

workpiece material properties.

Dimensional analysis

Buckingham-p theorem was used to determine the

appropriate dimensionless numbers that affect the cool-

ing rate. This theorem states that an equation with n

number of variables each expressed by k fundamental

units, such as mass M, length L, time Tand temperature

h can also be expressed by n2k dimensionless num-

bers.

14

In this study, the following eight variables thataffect the cooling rate are defined with symbols and

dimensions in terms of fundamental units within

parenthesis. These are density of the material r

(ML23), thermal conductivity of the solid k

(MLT23h21), specific heat of solid CP (L2T22h21),

temperature at which cooling rate is calculated minusthe preheat temperature T2T0 (h) and absorbed heat

input per unit length q5gQ (MLT22). The other

variables are depth D (L), width W (L) of the weld

pool and the cooling rate R (hT21). Considering four

fundamental units M, T, L and h, the authors can derive

the four dimensionless numbers, as defined below

p1~W

D(3)

p2~k

rWC1:5

P T{To 0:5

(4)

p3~q

W2rCP T{To (5)

p4~RW

C0:5

P T{To 1:5

(6)

Two meaningful dimensionless numbers p1 and p2 canbe derived from these four dimensionless numbers,

where p1~W=D, which is the aspect ratio of the pool,

and p2~Rq= k T{To 2

h i. Combining the relevant

dimensionless numbers, namely, p1,p2, the authors get

an expression of reference cooling rate at any tempera-

ture T as follows

Rref~k T{To

2

q

W

D

(7)

Average time for cooling from T1 to T2 may be

calculated by integrating equation (7) with respect totime and may be expressed as follows

DtT1=T2~2 1= T2{To {1= T1{To

k=q W=D (8)

The average cooling rate between T1 to T2 may be

expressed as

1 Comparison of average experimental cooling rates4 in

temperature range of 800500uC with estimated cooling

rates using heat conduction model correlation5

Aro ra et al. Cooling rate in 800 to 500uC range from dimensional analysis

Science and Technology of Welding and Joining 2010 VOL 15 NO 5 42 4


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Raverage~ak T1{T2

2q 1= T2{T0 {1= T1{T0

W

D

b(9)

The empirical constants a and b are introduced forfitting equation (9) with the experimental data.

Results and discussion

The values of the constants a and b were determined byplotting the experimental average cooling rate4,1524

between 800 and 500uC (R8/5) against the expression inthe parenthesis on right hand side in equation (9). The

data used for the calculation are provided in Table 1. It

is assumed that the cooling rates in the fusion zone donot vary spatially. This means that cooling rate at one

location in the weld pool can be assumed to be same as

that at a location away in the same weld pool. A powerlaw fit, represented by the solid line in Fig. 2, yieldeda514?491 and b50?477. The final expression for theaverage cooling rate R8/5 is as follows

R8=5~14:491

300 k

2q 1= 1073{T0 {1= 773{T

W

D

0:477(10)

where T0 is the preheat temperature in Kelvin.

Table 1 Data used for calculation of cooling rate for experimental cases considered

No.Heat input per unitlength q, J m21

Preheat temperatureT0, K

Aspect ratioW/D

Thermal conductivityk, W m21 K21 Reference

1 2?796106 298 0?6 27 42 2?796106 373 0?6 27 43 2?796106 473 0?6 27 4

4 2?

776

10

6

523 0?

6 27 45 1?746106 298 0?8 27 46 1?756106 373 0?6 27 47 1?756106 523 0?6 27 48 8?516105 473 0?6 27 49 9?306105 523 0?6 27 410 2?526106 298 0?8 26?8 1511 3?946106 298 1?45 27?1 1612 3?156106 298 1?5 27?1 1613 2?406106 298 1?6 27?1 1614 4?646105 298 1?5 30 1715 1?446106 298 1?2 21 1816 2?256106 298 0?9 21 1917 3?736106 298 1?3 27 2018 3?516106 298 1?3 27 20

19 3?076106

298 1?3 27 2020 2?136106 298 1?3 27 2021 2?006106 298 1?3 27 2022 1?756106 298 1?3 27 2023 1?616106 298 1?3 27 2024 1?51610

6 298 1?3 27 2025 1?326106 298 1?3 27 2026 1?306106 298 1?3 27 2027 1?226106 298 1?3 27 2028 1?066106 298 1?3 27 2029 2?896106 298 1?60 36?4 2130 6?986105 298 1?5 40 2231 3?786105 298 1?2 40 2232 6?686105 298 1?20 40 2233 1?316106 298 1?15 40 22

34 1?686106 473 1?10 40 2235 7?046105 298 1?50 40 2236 1?356106 298 0?8 40 2237 1?556106 298 1?50 21 2338 1?666106 298 1?45 21 2339 1?806106 298 1?40 21 2340 1?926106 298 1?40 21 2341 2?056106 298 1?35 21 2342 2?17610

6 298 1?25 21 2343 2?306106 298 1?25 21 2344 1?55610

6 298 1?50 21 2345 1?666106 298 1?45 21 2346 1?806106 298 1?40 21 2347 1?926106 298 1?35 21 2348 2?056106 298 1?30 21 2349 2?756104 298 0?45 27 2450 3?676104 298 0?37 27 2451 4?596104 298 0?36 27 2452 5?526104 298 0?32 27 2453 6?446104 298 0?30 27 2454 7?366104 298 0?33 27 2455 8?206104 298 0?33 27 24




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Figure 3 shows the computed cooling rates forvarious aspect ratios of the weld pool. Here it is

observed that the ratio of cooling rate estimation by

equation (10) to the experimental cooling rate is close to

1 for all values of the aspect ratio. This means that the

proposed correlation in equation (10) estimates thecooling rates that are in good agreement with

the experimentally measured cooling rates. In contrast,the ratio of cooling rate estimated by the heat

conduction model to the experimental cooling rate is

much greater than 1 for many cases. Especially, the

cooling rates estimated from the heat conduction model

are 1520 times greater than the experimental cooling

rates for aspect ratios ,0?5, i.e. for the case of keyhole

mode of welding. In situations where the keyhole forms,the conduction equation can be modified by consideringa line heat source and the cooling rates will be somewhatdifferent from those shown in Fig. 3. However, thecooling rates calculated by equation (1) for high aspectratios also do not produce satisfactory cooling rates asobserved from Fig. 3. In contrast, cooling rates com-puted from equation (10) agree well with the experi-

mental values of cooling rates both for the keyhole modewelding with aspect ratio ,0?5 and for other welds withmuch higher aspect ratios. The shape of the weld poolsometimes deviates considerably from a simple hemi-sphere. For such unusual shapes,12,25 where W/D maynot fully represent the weld pool shape, equation (10)may not be strictly valid.

The calculated average cooling rates R8/5 fromequation (10) are plotted against the reported experi-mental cooling rates in Fig. 4. The computed valuesrepresent heat input values in the range of 36104 to46106 J m21 and cooling rates from 5 to y120 K s21.It is observed that satisfactory agreement is achieved

between the cooling rates calculated from equation (10)and the experimental cooling rates. A linear fit with unitslope is shown in Fig. 4. The correlation coefficient R2

for the data shown in Fig. 3 is 0?965, where a R2 value of1 would mean an exact match between the estimated andthe experimental values.

Conclusions

A correlation based on dimensional analysis is proposedfor the calculation of cooling rates in the 800 to 500uCrange for steel welds. The correlation is tested for a widerange of welding processes, operating parameters and

material properties by comparing the computed coolingrates with the corresponding experimental results. Thecorrelation has been tested for heat input range between36104 and 46106 J m21, and cooling rate values fromy5 to 120 K s21. The fit between the experimentalcooling rates and the values computed from the proposedequation showed a correlation with R2 value of 0?965.

2 Power law correlation between experimental average

cooling rate4,1524 from 800 to 500uC and expression in

x axis: all symbols are defined in text

3 Ratio of average cooling rate between 800 and 500uC

estimations by heat conduction model5 to experimental

cooling rate4,1524 and ratio of estimation by equa-

tion (10) to experimental cooling rates,4,1524 with

respect to weld pool aspect ratio W/D

4 Comparison of estimations of average cooling rates

between 800 and 500uC by heat conduction model and

proposed correlation (equation (10)) with respect to

experimental cooling rates: experimental cooling rates

are for various welding processes from literature4,1524




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References1. H. K. D. H. Bhadeshia, L. E. Svensson and B. Gretoft: A model

for the development of microstructure in low-alloy steel (FeMn

SiC) weld deposits, Acta Metall., 1985, 33, 12711283.

2. H. Fredriksson and J. Stjerndahl: Solidification of iron-base

alloys, Met. Sci., 1982, 16, 575585.

3. N. S. Boulton and H. F. L. Martin: Residual stresses in arc welded

plates, Proc. Inst. Mech. Eng., 1936, 133, 295339.

4. L. E. Svensson, B. Gretoft and H. K. D. H. Bhaseshia: An analysisof cooling curves from the fusion zone of steel weld deposits,

Scand. J. Metall., 1986, 15, 97103.

5. C. L. Jenney and A. OBrien (eds.): in Welding handbook, 9th

edn, Vol. 1, Welding science and technology, 100; 2001, Miami,

FL, American Welding Society.

6. D. Rosenthal: The theory of moving sources of heat and its

application to metal treatment, Trans. ASME, 1946, 68, 819866.

7. W. Zhang, C. H. Kim and T. DebRoy: Heat and fluid flow in

complex joints during gas-metal arc welding Part I: Numerical

model of fillet welding, J. Appl. Phys., 2004, 95, (9), 52105219.

8. W. Zhang, C. H. Kim and T. DebRoy: Heat and fluid flow in

complex joints during gas-metal arc welding Part II: Application to

fillet welding of mild steel, J. Appl. Phys., 2004, 95, (9), 52205229.

9. X. He, J. W. Elmer and T. DebRoy: Heat transfer and fluid flow in

laser microwelding, J. Appl. Phys., 2005, 97, (8), 084909.

10. S. Mishra and T. DebRoy: A heat transfer and fluid flow based

model to obtain a specific weld geometry through multiple paths,

J. Appl. Phys., 2005, 98, (4), 044902.

11. A. De and T. DebRoy: Reliable calculations of heat and fluid flow

during conduction mode laser welding through optimization of

uncertain parameters, Weld. J., 2005, 84, (7), 101S112S.

12. A. Arora, G. G. Roy and T. DebRoy: Unusual wavy weld pool

boundary, Scr. Mater., 2009, 60, 6871.

13. O. Grong: Metallurgical modeling of welding, 5; 1994, London,

The Institute of Materials.

14. V. Streeter and E. Wylie: Fluid mechanics, 8th edn, 164172;

1985, New York, McGraw-Hill.

15. H. Cerjak and E. Letofsky: Proc. 5th Int. Conf. on Trends in

welding research, (ed. M. Vitek et al.), 143148; 1999, Pine

Mountain, GA, ASM International.

16. Z. Yang and T. DebRoy: Modeling of macro- and microstructures

of gas-metal-arc welded HSLA-100 steel, Metall. Mater. Trans. B,

1999, 30B, 483493.

17. H. G. Svoboda, N. M. R. de Rissone, L. F. G. de Souza, D. B.

Santos and I. de S. Bott: Proc. 8th Int. Conf. on Trends in welding

research, (ed. S. A. David et al.), 154158; 2009, Pine Mountain,GA, ASM International.

18. A. Kumar, W. Zhang, C. H. Kim and T. DebRoy: A Smart Bi-

directional Model of Heat Transfer and Free Surface Flow in Gas

Metal Arc Fillet Welding for Practising Engineers, Weld. World,

2005, 49, (9), 3248.

19. L. Depradeux and J. F. Jullien: in Mathematical modeling of

welding of weld phenomena 7, (ed. H. Cerjak), 269294; 2005,

London, Institute of Materials, Minerals and Mining.

20. J. Amanie, I. N. A. Oguocha and S. Yannacopoulos: Proc. 8th Int.

Conf. on Trends in welding research, (ed. S. A. David et al.), 491

501; 2009; Pine Mountain, GA, ASM International.

21. W. Zhang, J. W. Elmer and T. DebRoy: Modeling and real time

mapping of phases during GTA welding of 1005 steel, Mater. Sci.

Eng. A, 2002, A333, (12), 320335.

22. G. T. Krause: Heat flow and cooling rates in submerged arcwelding, MSc thesis, The Ohio State University, Columbus, OH,

USA, 1978.

23. A. Kumar and T. DebRoy: Heat transfer and fluid flow during

gas-metal-arc fillet welding for joint configurations and welding

positions, Metall. Mater. Trans. A, 2007, 38A, 506519.

24. R. Rai, J. W. Elmer, T. A. Palmer and T. DebRoy: Heat transfer and

fluid flowduring keyhole mode laser weldingof tantalum, Ti6Al4V,

stainless steel and vanadium, J. Phys. D, 2007, 40D, 57535766.

25. A. Robert and T. DebRoy: Geometry of laser spot welds from

dimensionless numbers, Metall. Mater. Trans. B, 2001, 32B, 941947.


http://www.ingentaconnect.com/content/external-references?article=0001-6160(1985)33L.1271[aid=5128033]http://www.ingentaconnect.com/content/external-references?article=0001-6160(1985)33L.1271[aid=5128033]http://www.ingentaconnect.com/content/external-references?article=0001-6160(1985)33L.1271[aid=5128033]http://www.ingentaconnect.com/content/external-references?article=0001-6160(1985)33L.1271[aid=5128033]http://www.ingentaconnect.com/content/external-references?article=0306-3453(1982)16L.575[aid=4734416]http://www.ingentaconnect.com/content/external-references?article=0306-3453(1982)16L.575[aid=4734416]http://www.ingentaconnect.com/content/external-references?article=0306-3453(1982)16L.575[aid=4734416]http://www.ingentaconnect.com/content/external-references?article=0306-3453(1982)16L.575[aid=4734416]http://www.ingentaconnect.com/content/external-references?article=0020-3483(1936)133L.295[aid=9032209]http://www.ingentaconnect.com/content/external-references?article=0020-3483(1936)133L.295[aid=9032209]http://www.ingentaconnect.com/content/external-references?article=0020-3483(1936)133L.295[aid=9032209]http://www.ingentaconnect.com/content/external-references?article=0020-3483(1936)133L.295[aid=9032209]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5210[aid=6894736]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5210[aid=6894736]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5210[aid=6894736]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5210[aid=6894736]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5220[aid=6894735]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5220[aid=6894735]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5220[aid=6894735]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5220[aid=6894735]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)98L.044902[aid=7243122]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)98L.044902[aid=7243122]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)98L.044902[aid=7243122]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)98L.044902[aid=7243122]http://www.ingentaconnect.com/content/external-references?article=1359-6462(2009)60L.68[aid=8958257]http://www.ingentaconnect.com/content/external-references?article=1359-6462(2009)60L.68[aid=8958257]http://www.ingentaconnect.com/content/external-references?article=1359-6462(2009)60L.68[aid=8958257]http://www.ingentaconnect.com/content/external-references?article=1359-6462(2009)60L.68[aid=8958257]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=0022-3727(2007)40L.5753[aid=8741258]http://www.ingentaconnect.com/content/external-references?article=0022-3727(2007)40L.5753[aid=8741258]http://www.ingentaconnect.com/content/external-references?article=0022-3727(2007)40L.5753[aid=8741258]http://www.ingentaconnect.com/content/external-references?article=0022-3727(2007)40L.5753[aid=8741258]http://www.ingentaconnect.com/content/external-references?article=1073-5615(2001)32L.941[aid=7335405]http://www.ingentaconnect.com/content/external-references?article=1073-5615(2001)32L.941[aid=7335405]http://www.ingentaconnect.com/content/external-references?article=1073-5615(2001)32L.941[aid=7335405]http://www.ingentaconnect.com/content/external-references?article=1073-5615(2001)32L.941[aid=7335405]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5210[aid=6894736]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5220[aid=6894735]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)98L.044902[aid=7243122]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=0022-3727(2007)40L.5753[aid=8741258]http://www.ingentaconnect.com/content/external-references?article=1073-5615(2001)32L.941[aid=7335405]http://www.ingentaconnect.com/content/external-references?article=0001-6160(1985)33L.1271[aid=5128033]http://www.ingentaconnect.com/content/external-references?article=0306-3453(1982)16L.575[aid=4734416]http://www.ingentaconnect.com/content/external-references?article=0020-3483(1936)133L.295[aid=9032209]http://www.ingentaconnect.com/content/external-references?article=1359-6462(2009)60L.68[aid=8958257]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5210[aid=6894736]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2004)95L.5220[aid=6894735]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)97L.084909[aid=9267919]http://www.ingentaconnect.com/content/external-references?article=0021-8979(2005)98L.044902[aid=7243122]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5615(1999)30L.483[aid=5053587]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=1073-5623(2007)38L.506[aid=9267917]http://www.ingentaconnect.com/content/external-references?article=0022-3727(2007)40L.5753[aid=8741258]http://www.ingentaconnect.com/content/external-references?article=1073-5615(2001)32L.941[aid=7335405]http://www.ingentaconnect.com/content/external-references?article=0001-6160(1985)33L.1271[aid=5128033]http://www.ingentaconnect.com/content/external-references?article=0306-3453(1982)16L.575[aid=4734416]http://www.ingentaconnect.com/content/external-references?article=0020-3483(1936)133L.295[aid=9032209]http://www.ingentaconnect.com/content/external-references?article=1359-6462(2009)60L.68[aid=8958257]

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