Materials Science & Engineering A 561 (2013) 183–190

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Materials Science & Engineering A

0921-50

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An analysis on necking effect and stress distribution in round cross-sectionspecimens of pure copper with different diameters

W.J. Yuan a, F. Zhou a, Z.L. Zhang b, Y.J. Su a,n, L.J. Qiao a, W.Y. Chu a

a Corrosion and Protection Center, Key Laboratory for Environmental Fracture (MOE), University of Science and Technology Beijing, Beijing 100083, Chinab Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), Richard Birkelands vei 1a, N-7491 Trondheim, Norway

a r t i c l e i n f o

Article history:

Received 3 September 2012

Received in revised form

22 October 2012

Accepted 25 October 2012Available online 7 November 2012

Keywords:

Tensile properties

Elongation

Finite element method (FEM)

Diameter influence

Pure copper

93/$ - see front matter Crown Copyright & 2

x.doi.org/10.1016/j.msea.2012.10.077

esponding author. Fax: þ86 10 62333884.

ail address: yjsu@ustb.edu.cn (Y.J. Su).

a b s t r a c t

The size dependence of the tensile properties and necking effect of pure copper exposed to various heat

treatments are investigated in this paper using round bar cross-section specimens through experi-

mental and 3D finite element methods (FEM). The results show that the total and post-necking

elongation increased dramatically as the specimen diameter increased. The size effects on necking and

stress distribution of different diameter specimens are analysed. These results show that the post-

necking deformation length is only dependent on the diameter of the specimens and is independent of

the specimen’s gauge length, which results in the size dependence of the post-necking elongation.

Moreover, FEM simulations show that the size dependence of the tensile properties can be eliminated

when the ratio of the gauge length to the diameter is larger than 10.

Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved.

1. Introduction

Tensile properties include strength parameters, such as ulti-mate tensile strength (UTS) and yield strength (YS), and ductilityparameters, such as elongation and cross-sectional area reduc-tion. These parameters are important index properties in materi-als research. Specimens with rectangular or round cross-sectionsare often used to determine tensile properties. In practice, roundbar specimens are used primarily out of convenience [1–5]. Theshape and geometric size of a round bar specimen can bedesigned according to the ASTM standards, which states thatthe ratio of the gauge length to the gauge diameter should be fouror five. Unless this ratio is maintained, the elongation values maynot be comparable with those obtained from a standard testspecimen [6]. However, in most circumstances, the dimensions ofthe specimens fail to fulfil the ASTM standard: for example, whensamples are cut from structures and when the ratio of the gaugelength to the diameter is smaller or larger than the value (four orfive) specified in the standard [7–10]. This discrepancy leads tothe question of how the results obtained from different tests canbe compared if they did not use the same specimen size and howthe results from one testing size can be properly compared toothers.

012 Published by Elsevier B.V. All

Recent studies have shown that the size of a rectangularspecimen influences its tensile properties [11–26], includingstrength [11–14,19], ductility [19,20,22,24], and the constitutiverelation [23]. The effect of the diameter of a round bar specimenon its tensile properties has also been investigated [27,28]. Maticet al. [27] studied the effects of L0/D0 (where L0 and D0 are thegauge length and the gauge diameter, respectively) on theengineering stress–strain curves of HY-100 steel specimens withdifferent diameters and gauge lengths. The results showed thatthe engineering stress–strain curves were dependent on thespecimen geometry, and the fracture strain was reduced asL0/D0 increased. Matic et al. also found through finite elementmethods FEM that the initial length of the neck on the deformedspecimen was equal to the value of the diameter of the specimen,thus the L0/D0 must be at least unity for a fully developed neck tooccur. Gupta et al. [28] investigated the tensile behaviour ofaluminium and copper specimens with different gauge lengthsand diameters. Their studies showed that when the load reachedits maximum, the onset of necking depends on the length anddiameter of specimen, and the deformation length after neckingincreased linearly with the diameter and was found to beindependent of the gauge length. Experiments and FEM resultshave shown that the tensile properties are strongly dependent onthe geometric size of the specimens [27,28]. However, themechanism of the size effect on specimens with round cross-sections in tensile tests remains unclear, and there are few reportsexamining the consistency between the finite element simula-tions and experimental results.

rights reserved.

Fig. 1. Experimental (solid lines) and FEM (dashed lines) results showing the

dependence of the ultimate tensile strength (sb) of pure copper under different

heat treatments on the diameter of the specimen.

W.J. Yuan et al. / Materials Science & Engineering A 561 (2013) 183–190184

In a previous study [23], a failure criterion was proposed,wherein the true fracture stress was assumed to be equal forspecimens of various thicknesses with rectangular cross-sectionsto simulate the elongation of the specimen by finite elementmethods (FEM). The influence of the thickness of the specimen onthe tensile properties was then investigated through both experi-ments and FEM. However, because of the limitations of therectangular cross-section, the failure criterion proposed for theFEM was not experimentally verified. The objective of this paperis to experimentally verify the fracture criterion by simulating thetensile properties with FEM to determine the elongations ofspecimens of various geometric sizes. The effects of specimendimensions with round cross-sections on the tensile properties ofpure copper are investigated. In the combined experimental andcomputational results, the stress distribution after necking isstudied for specimens with different dimensions to reveal themechanism of the size effect on the tensile properties.

Fig. 2. Experimental (solid lines) and FEM (dashed lines) results showing the

dependence of the area reduction (c) of pure copper under different heat

treatments on the diameter of the specimen.

2. Experiment and simulation

In this study, experiments were performed to determine thetensile behaviour of round bar tensile specimens with variousdiameters and the same gauge length cut from cold-drawn purecopper bar. Tensile experiments were carried out using specimenswith a gauge length of 25 mm and cross-sectional diameters thatvaried from 4 to 9 mm. Two different heat treatments wereexamined: as-received and annealed at 473 K for 10 min., fol-lowed by cooling in a furnace, corresponding to low and highductility, respectively. The surfaces of the gauge sections of thespecimens were polished using 60] to 1000] SiC paper. Tensiletests were performed at a strain rate of 10�4 s�1. At least fourspecimens were tested under each of the conditions, and eachexperimental datum presented in this paper is the correspondingaverage value. The strain was calculated by measuring the cross-head displacement. Before the tensile test and after specimenfracture, the gauge length of the specimen was measured and theelongation was calculated.

ABAQUS was used to calculate the effects of the geometric sizeof the specimen on its mechanical properties. Because of geo-metric symmetry, a 1/8 symmetry model was used. The boundaryconditions were as follows:

x¼ l02 , ux ¼ uprescribed

x¼ 0, ux ¼ 0

y¼ 0, uy ¼ 0

z¼ 0, uz ¼ 0

ð1Þ

A Youngs modulus of 120 GPa and Poisson’s ratio of 0.3 forboth the as-received and the annealed copper were the inputparameters for the FEM. The load-displacement data measured byan extensometer were used to calculate the parameters K and n,using the least-squares method, according to the ISO standard[29]. The true stress–true strain curve was calculated from therelationship S¼ Ken, and the material properties with the follow-ing rate-independent strain hardening power law were given bysf ¼ s0 1þep=e0

� �n, where sf is the flow stress, ep is the equivalent

plastic strain, s0 is the yield stress, e0 ¼ s0=E is the yield strainand n is the plastic strain hardening exponent [24].

3. Results and discussion

Fig. 1 shows the variation in the UTS (sb) of the pure copperround bar specimens with various heat treatments and diameters.As shown in Fig. 1, the UTS from the experimental stress–straincurves change from 356.4 MPa to 356 MPa for the as-received

specimens as the diameter increases from 4 to 9 mm, and from258.8 MPa to 259.2 MPa for the annealed specimens as thediameter increases from 4 to 9 mm. It is clear that the UTS isindependent of the diameter, as shown in Fig. 1 by the solid lines.The FEM results corroborated this conclusion, as shown in Fig. 1by the dashed lines.

Fig. 2 shows the experimental results for the area reduction(c) of the pure copper round bar specimens with different heattreatments and diameters. As shown in Fig. 2, the c of as-receivedCu changes from 86.9 to 91% as the diameter increases from 4 to9 mm. The c of the annealed specimen changes from 87.5 to85.1% as the diameter increases from 4 to 9 mm.

The single factor analysis method (SFAM) [23] was used toanalyse the correlation between the c and the diameter of thespecimens. The rejection region, F, was calculated by

F ¼SA

f A

=SB

f B

, ð2Þ

where SA and SB are the sums of the squares of the standarddeviations of the factor A and the error B, respectively. Thevariables fA and fB are the number of degrees of freedom for thefactor A and the error B, respectively. From the single factoranalysis, the F factor for the c of the as-received specimens was2.27 and 2.40 for the annealed specimens, which was less than2.77 (F0.05). It was concluded that the c of Cu with different heattreatments is independent of the specimen diameter within thediameter ranges tested.

Fig. 3. Experimental (solid points) and FEM (open points) results of the variation

of the total elongation (dtotal), post-necking elongation (dpe) and uniform elonga-

tion (due) of the as-received pure copper (a) and the annealed copper (b) with the

diameter of the specimen.

Fig. 4. The tensile stress S11 distribution in the length direction for different

diameter specimens of the annealed copper before (a) and after the onset necking

(c); (b) is the tensile stress vs. the true strain before the onset of necking.

W.J. Yuan et al. / Materials Science & Engineering A 561 (2013) 183–190 185

However, increasing the specimen diameter clearly influencedthe elongation of the specimens. Fig. 3(a) and (b) shows that thetotal elongation (dtotal) increased as the diameter of the specimenincreased. As the diameter increased from 4 to 9 mm, the totalelongation of the as-received Cu increased significantly, from15.7% to 34.8%. The total elongation of the annealed specimensincreased from 27.5% to 41.3% as the diameter of the specimenincreased from 4 to 9 mm.

The total elongation of the tensile specimens dtotal occurred asa two-part process, consisting of uniform elongation due, followedby post-necking elongation, dpe, i.e., dtotal ¼ dueþdpe. The due canbe determined using Considere’s criterion [30] to describe theonset of the localised deformation

@s@e ¼ s: ð3Þ

The post-necking elongation can be calculated usingdpe ¼ dtotal�due. Fig. 3(a) and (b) also shows the dependence ofdue and dpe vs. the diameter of the as-received and annealedspecimens, respectively. The SFAM results show that the F factorof the due was 1.27 for the as-received specimens and 1.35 for theannealed specimens, which are both less than 2.77 (F0.05). How-ever, the F factors of the dpe were 35 and 43, respectively, for theas-received and the annealed specimens, which were larger than2.77 (F0.05). These results indicate that due is independent and thatdpe is markedly dependent on the specimen diameter. Therelationship of both the experimental and the simulated dpe anddtotal to the diameter of the specimen can be well fit linearly, asshown by the dashed lines in Fig. 3(a) and (b). The R-squaredvalue for dpe is 95% and 99% for the experimental and simulated

results of the as-received specimens, respectively, and 94% and99% for the annealed specimens, respectively.

The true fracture strain is defined as e¼ ln A0=A� �

¼�ln 1�cð Þ

[24]. Fig. 2 clearly shows that c is independent of the diameter ofthe specimens. The true fracture strain and stress is found to beindependent of the diameter of the specimens. Therefore, thefailure criterion, which assumed the true fracture stress to beequal for the specimens with various diameters, was used tosimulate the dependence of the elongation on the diameter ofspecimens using FEM. Interpolation and the least-squares methodwere used to fit the experimental elongation vs. diameter curve.A similar method was mentioned in our previous work [23].

The FEM simulation results are shown in Fig. 3(a) and (b) asopen points and are in agreement with the experimental results.The critical simulated engineering fracture stress was found to be60 MPa for the as-received specimens and 77 MPa for the annealedspecimens, which correspond to 595 MPa and 526 MPa of true

Fig. 5. The distribution of tensile stresses (S11) at the centre section of the gauge after necking for the annealed copper with 4 mm (a) and 9 mm (b) diameters; the

distribution of the true stress for a true strain of 0.5 in the length direction greater than the yield stress of 350 MPa for the as-received specimens (c) and 250 MPa for the

annealed specimens (d); (e) shows the dependence on length of the region (dc), in which S11 is greater than the yield stress vs. diameter of the specimens for a true strain

of 0.5.

W.J. Yuan et al. / Materials Science & Engineering A 561 (2013) 183–190186

fracture stress, respectively. The experimental and simulationresults showed that the effect of specimen diameter on elongationcan be generally attributed to its effect on the post-neckingelongation component. As specimen diameter increased from4 to 9 mm, the dpe simulated by the FEM increased from 17.8% to30.8% for the as-received specimens and from 14.8% to 27.6% forthe annealed specimens when the diameter increased from 4 to9 mm. These results are in agreement with the experiments, whoserange was 14.2–29.8% for the as-received specimens and 12.8–28.8% for the annealed specimens, respectively.

To understand the mechanism by which the diameter affectedthe necking behaviour and the post-necking elongation of tensilespecimens, special attention was paid to the analysis of the stressdistribution in the specimens with different diameters.Fig. 4(a) shows the stress distribution in the length direction(S11) at various true strains for specimens with diameters of 4,6 and 9 mm. The stress is uniform along the length direction overthe centre gauge region of approximately 12 mm for a specimenof 25 mm gauge length. Beyond the region in which the stress isuniform, the stress changes more variable, and the length of thestress fluctuation region increased with the diameter due to theeffects of the radius of the fillet coming off of the gauge section.Fig. 4(b) shows the stress value in the centre region of the gaugevs. the true strain for the specimens with various diameters.This finding shows that the tensile stress in the length direction at

a given strain is independent of the specimen diameter. Thus, theplastic deformation occurred over the entire gauge section untilthe onset of necking, resulting in the same uniform elongation due

for all specimens, regardless of diameter. When the tensile stressalong the gauge length reached the critical level, approximately390 MPa for the as-received specimens and 290 MPa for theannealed specimens, diffused necking occurred. The critical stres-ses for necking were independent of the diameter of the speci-mens. After the onset of necking, the plastic deformation waslocalised to the necking region. The stress in the gauge lengthdirection, except in the necking region, decreased from the yieldstress, as shown in Fig. 4(c). The uniform deformation length didnot continue to increase with the increasing tensile strain afternecking. This result indicates that uniform elongation did notcontinue to accelerate with the increase of tensile strain afternecking.

The distribution of the tensile stress (S11) in the annealedspecimens with diameters of 4 and 9 mm at the true fracturestrain are shown in Fig. 5(a) and (b), respectively. These plotsshow that the greatest stress is located in the centre part of thegauge section, and the zone of maximum stress increases with thediameter. Fig. 5(c) shows the distribution of the tensile stress (S11)above the true yield strength to be 350 MPa for the as-receivedspecimens. Fig. 5(d) shows the tensile stress (S11) above the trueyield strength to be 250 MPa for the annealed specimens at

Fig. 6. The longitudinal section of the fractured specimens with a diameter of

4 mm (a) and 9 mm (b); the magnifications of the A, B and C positions (marked in

(a) and (b)) for the as-received specimens of 4 mm (c) and 9 mm (d) in diameter,

and for the annealed specimens of 4 mm (e) and 9 mm (f) in diameter,

respectively.

W.J. Yuan et al. / Materials Science & Engineering A 561 (2013) 183–190 187

various diameters from 4 to 9 mm for a true stain of 0.5. Thelength (dc) of the region in which the tensile stress is greater thanthe true yield stress (where plastic deformation occurs) increasedwith the diameter from 4 to 9 mm for a true strain of 0.5.Fig. 5(e) shows the relationship between the length (dc) of theplastic deformation zone for a true strain of 0.5 and the diameterof the specimen. It is observed that dc increases with the diameterof the specimen for different heat treatments. That is, the plasticdeformation area of the specimen along the length directionincreases with specimen diameter, which induces larger post-necking elongation in the specimen.

The plastic deformation in the grains of the as-received andannealed specimens was examined by SEM, as shown in Fig. 6.Magnified graphs of the different positions of A, B and C inFig. 6(a) and (b) were shown in Fig. 6(c–f). Fig. 6(c) and (d) shows

that for the annealed pure copper specimens, the grains in thelongitudinal section near the fracture surface were elongated indifferent diameter specimens; i.e., large plastic deformationoccurred in the necking region. The grains became larger andcoarser along the length direction from the fracture surface. Atthe same position from the fracture surface, the grains of the purecopper specimen with different heat treatments were consider-ably more elongated for the 9 mm diameter specimen than for the4 mm diameter specimen. This finding indicates that the stress ishigher in the thick specimen than it is in the slender specimen atthe same position, producing greater plastic deformation. Theincreased plastic deformation leads to extensive necking in thelength direction near the centre of the specimen and a relativelysmall post-necking elongation in the slender specimen. This resultis in agreement with the FEM results. The results of the as-receivedpure copper specimens were similar to that of the annealedspecimen, as shown in Fig. 6(e) and (f).

The macro fractures of the as-received and annealed speci-mens with diameters of 4, 6 and 9 mm are shown in Fig. 7(a–c),and Fig. 7(g–i), respectively. The corresponding stress distributionin the cross-section obtained from the FEM is shown in Fig. 7(d–f)and Fig. 7(j–l), respectively. In the centre zone, there exists a fibreregion with fracture damage that was exposed to the maximumtrue stress, based on the FEM. The centre fibre region of the zoneof the maximum stress is found to increase with the diameter ofthe specimen. Outside of the fibre region, moving towards theedge, is the shear lip. The variation of the true stress on the cross-section with the distance away from the centre of the specimenswith different diameters is shown in Fig. 8. The maximum tensilestress is located in the centre of the cross-section. The fractureoccurred first in the central area, which formed the fibre fracturesurface. The size of the central fibre zone increased as thespecimen diameter increased from 4 to 9 mm. After the fractureoccurred at the centre of the cross section during necking, a shearstress occurs that formed a shear lip fracture surface. From thestress distribution in the length direction, shown in Fig. 8,combined with Fig. 7, the fibre region formed when the stressreached approximately 580 MPa for the as-received specimensand approximately 560 MPa for the annealed pure copper speci-mens (shown by dashed lines in Fig. 8), regardless of thediameter.

The elongation of the tensile specimen can be written as

dtotal ¼DLue

L0þDLpe

L0, ð4Þ

where L0 is the gauge length of the specimen, DLue and DLpe arethe uniform and post-necking deformation gauge lengths. Asmentioned above, the uniform deformation length did not con-tinue to increase with the increasing tensile strain after necking.It is reasonable to deduce that the uniform deformation length ofthe specimen is proportional to the gauge length of the specimen,that is, due ¼DLue=L0 is a constant, regardless of diameter andgauge length.

Fig. 9(a) shows the distribution of tensile stresses (S11) in thenecking region in the length direction for a 5 mm diameterspecimen with different gauge lengths L0; and Fig. 9(b) showsthe distribution for different diameter specimens with gaugelengths L0¼25 mm and 30 mm. From Fig. 9(a), it is shown thatthe tensile stress distribution (S11) in the necking region in thelength direction is constant for the specimens with the samediameter at a certain true strain, regardless of the gauge L0.However, the distribution is dependent upon the diameter of thespecimens with same length, as shown in Fig. 9(b). From the FEMresults, it is reasonable to conclude that S11 is dependent on D0 ofthe specimen but independent of L0.

Fig. 7. SEM macro fractures and the FEM stress distributions in the cross-sections of specimens with different diameters, (a) and (d) 4 mm, (b) and (e) 6 mm, (c) and

(f) 9 mm for the as-received pure copper as well as (g) and (j) 4 mm, (h) and (k) 6 mm, (i) and (l) 9 mm for the annealed pure copper.

W.J. Yuan et al. / Materials Science & Engineering A 561 (2013) 183–190188

The S11 distribution can be written as

S11 ¼ gD0ðxÞ, ð5Þ

where x is the distance from centre in the gauge direction of thespecimens. At a certain true strain, the tensile stress S11 willinduce a deformation length dL from the length of dx. The strain is

e11 ¼dL

dx¼ f S11ð Þ: ð6Þ

Combining Eqs. (5) and (6), the deformation length is defined as

dL¼ e11dx¼ f ½gD0ðxÞ�dx¼ FD0

ðxÞdx: ð7Þ

Because the plastic deformation occurred at the region oftensile stress greater than the yield stress, the deformation lengthof necking region DLi

pe at a given strain can be calculated as

DLipe ¼

Z dc

0FD0ðxÞdx: ð8Þ

The total post-necking deformation length of gauge during thenecking process was

DLpe ¼Xef

eu

DLipe, ð9Þ

where eu and ef are the true uniform strain and the true fracturestrain, respectively.

Eq. (9) clearly shows that the post-necking deformation lengthis only dependent on the diameter D0 and is independent of thespecimen’s gauge length L0. According to Fig. 3, the post-neckingelongation increased linearly with the diameter for the specimenswith a gauge length of 25 mm, i.e.,

dpe ¼DLpe

L0¼ kD0, ð10Þ

where k is a constant. Thus, the post-necking deformation length(DLpe¼L0dpe¼25kD0) increased linearly with the diameter of thespecimen. For the specimens with a given gauge length L0, thepost-necking deformation length can be written as

DLpe ¼ aD0, ð11Þ

where a¼ 25k is a constant coefficient, regardless of the gaugelength L0. Combining Eqs. (4) and (11) gives

dtotal ¼DLue

L0þDLpe

L0¼ bþaD0

L0: ð12Þ

Eq. (12) clearly shows that the total and post-necking elonga-tion are dependent on the ratio L0/D0. However, the value of L0/D0

Fig. 8. The variation of the true stress on the cross-section with the distance from

the centre of the specimens in the radial direction from the FEM results of the

fracture surface with different diameters for the as-received (a) and annealed

(b) copper specimens.

Fig. 9. Distribution of the tensile stresses (S11) in the necking region in the length

direction for a diameter of 5 mm with different L0 (a) and for different diameter

specimens with L0¼25 mm and 30 mm (b) for a true strain of 0.5 in the annealed

copper.

Fig. 10. Experimental (solid lines) and FEM (dashed lines) results showing the

dependence of the total elongation (dtotal), post-necking elongation (dpe) and

uniform elongation (due) of the annealed copper on the ratio between gauge length

and diameter of the specimen.

W.J. Yuan et al. / Materials Science & Engineering A 561 (2013) 183–190 189

must be at least unity to ensure a fully developed neck [27].In this paper, the gauge length L0 is maintained at a constant25 mm; therefore, a different diameter D0 is used to change theL0/D0 ratio. As observed in Fig. 10, the experiments showed thatthe elongation decreased from 41.3% to 27.5% as the L0/D0 ratioincreased from 2.78 to 6.25, a finding in agreement withMatic [27]. Fig. 10 also showed that the measured elongationdecreased with increasing L0/D0, up to a value of 20. However, themeasurements tend to be stable when the ratio increased above10. Therefore, for a certain range of L0/D0410, the elongationmeasurements may be comparable.

The results presented here suggest that uniform elongationmay be a better parameter choice than total elongation for thecharacterisation of the ductility of metallic materials for reliablecomparison in experimental studies, either with rectangular [23]or round cross-sections. According to Fig. 4(a), the length of thereduced section should be much more than gauge length forproper measurement of elongation because the radius of the filletmakes the stress distribution near the gauge end more variable.This shift induces inhomogeneous plastic deformation andreduces the accuracy and reliability of the measurement of theelongation.

4. Conclusions

The size dependence of the tensile properties of pure copperwith different heat treatments is investigated using round barspecimens with a gauge length of 25 mm and various diameters.It was found that the ultimate tensile strength, area reduction anduniform elongation are independent of the diameter of the

specimen. However, the total elongation and post-necking elon-gation were found to increase with the diameter of the specimen.

A failure criterion is proposed wherein the true fracturestrength is equal for specimens of various diameters, based onthe experimental results that the area reduction is independent ofthe diameter of the specimen to simulate the dependence of theelongation on the diameter of specimen by FEM. The FEMsimulation results were in agreement with the experiments.

The size effects on necking and stress distribution of differentdiameter specimens are analysed. The FEM simulation resultsshowed that after necking onset the size of the region where thestress is greater than the yield stress required for plastic

W.J. Yuan et al. / Materials Science & Engineering A 561 (2013) 183–190190

deformation along the length direction increased with the speci-men diameter. The post-necking deformation length increaseswith the specimen diameter and is independent of the specimen’sgauge length, which results in the size dependence of the post-necking elongation. However, the dependence of the total andpost-necking elongation on the specimen diameter can be elimi-nated when L0/D0 is above 10.

Acknowledgements

This project was supported by National High TechnologyResearch and Development Program of China under the GrantNo. 2012AA040104 and the National Nature Science Foundationof China under Grant No. 51071028.

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