October 28, 1997 This is a preprint of a paper intended for publication in a journal or proceedings. Since changes may be made before publication, this preprint is made available with the understanding that it will not be cited or reproduced without the permission of the author. Lawrence Livermore National Laboratory UCRL-JC-128674 PREPRINT This paper was prepared for submittal to the 1998 TMS Annual Meeting "Symposium on Superplasticity and Superplastic Forming" San Antonio, TX February 15-19, 1998 The Evolution of Grain Size Distribution During Deformation of Superplastic Materials D. R. Lesuer R. Glaser C. K. Syn
Transcript
October 28, 1997
This is a preprint of a paper intended for publication in a journal or proceedings. Sincechanges may be made before publication, this preprint is made available with theunderstanding that it will not be cited or reproduced without the permission of theauthor.
Lawre
nce
Liverm
ore
National
Labora
tory
UCRL-JC-128674PREPRINT
This paper was prepared for submittal to the1998 TMS Annual Meeting
"Symposium on Superplasticity and Superplastic Forming"San Antonio, TX
February 15-19, 1998
The Evolution of Grain Size DistributionDuring Deformation of Superplastic Materials
D. R. LesuerR. GlaserC. K. Syn
DISCLAIMER
This document was prepared as an account of work sponsored by an agency ofthe United States Government. Neither the United States Government nor theUniversity of California nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does not necessarily constitute orimply its endorsement, recommendation, or favoring by the United StatesGovernment or the University of California. The views and opinions of authorsexpressed herein do not necessarily state or reflect those of the United StatesGovernment or the University of California, and shall not be used for advertisingor product endorsement purposes.
THE EVOLUTION OF GRAIN SIZE DISTRIBUTION DURING DEFORMATION OF SUPERPLASTIC MATERIALS
Donald R. Lesuer, Ronald Glaser and Chol K. Syn
Lawrence Livermore National Laboratory, Livermore, CA 94551
Abstract
Grain size distribution and its evolution during superplastic deformation has been studied fortwo materials - ultrahigh carbon steel, which has a two phase microstructure, and a copperalloy, which has a quasi-single phase microstructure. For both materials the distribution ofinitial grain sizes is very accurately represented by a lognormal function. As the materials aredeformed the distributions retain their lognormality throughout the deformation history. Theevolution of the parameters characterizing the log normal distribution (mean and standarddeviation) have also been studied and found to vary in a systematic manner results. Resultscan be used to specify the grain size distribution as a function of strain during superplasticdeformation and thus should prove useful for computational studies in which grain sizedistribution is evaluated.
Introduction
It is generally recognized that the behavior of superplastic materials is strongly dependent ongrain size. Most studies of this dependence have used some average measure of grain size,such as mean linear intercept. Materials, however, possess a collection of different grain sizeswhich can have a strong influence on deformation behavior. In many annealed metals (e.g. Cu[1], α-brass [1], β-brass [1], Al [1], [2] and Fe [3]) this collection of grain sizes can berepresented by a lognormal distribution. In addition for systems in which grain sizedistribution has been studied during grain growth (Al [2] and Fe [3]) the distribution retains itslognormality during normal grain growth.
Superplastic deformation can be accompanied by static (normal) as well as strain-enhancedgrain growth [4] [5]. These two grain growth processes occur by different mechanisms and theinfluence of strain-enhanced grain growth on grain size distribution is unknown. However, aspreviously shown [6] grain size distribution and its evolution can have a significant influenceon the stress-strain-strain rate behavior of superplastic materials. As grain growth occurs, boththe average grain size and the dispersion (standard deviation) of the distribution increase.Thus, since the constitutive response for different deformation mechanisms is grain sizedependent, grain growth can result in a broader range of deformation mechanisms and flowbehavior in the deforming material. In this paper we report on an experimental study of grainsize distribution and its evolution during superplastic deformation. The results should beuseful for computational studies in which the evolution of grain size distribution is accountedfor in the calculations.
Materials, Experimental Procedures and Results
Materials and experiments
Two superplastic materials with significantly different microstructures were evaluated in thisstudy - ultrahigh carbon steel (UHCS) which has a microduplex structure and Coronze 638which has quasi-single phase structure. The UHCS had been processed to produce amicrostructure consisting of equiaxed ferrite grains and spheroidized carbide particles. TheCoronze 638 consisted of essentially pure copper with a sub-micron sized dispersion of CoSiand CoSi2 particles. Further details on material processing and characteristics have been
previously reported [6], [7].
Tensile tests were done at constant true strain rate in a computer-controlled test machine.Some specimens were strained to failure, whereas other specimens were deformed to apredetermined strain and then unloaded. All samples were then sectioned to their center andthe microstructures photographed. These microstructures were evaluated using a Quantametimage analysis system, which measured the area of each of the grains in the polished cross-section. The area for each of these grains was then converted to a linear intercept measure ofgrain size (according to ASTM procedure [8]) using the following expression.
D= πA4
(1)
where D is the grain size and A is the cross-sectional area of the grain. For UHCS the size ofthe ferrite grains are reported whereas for Coronze the size of the copper grains are reported.A histogram was produced in which the number of grains within a given grain size range wasplotted against grain size. A plot of relative frequency versus grain size was then obtained byconstructing a smooth curve through the histogram and normalizing the relative frequency sothat the curve integrates to one.
Results
Fig. 1 shows the influence of superplastic deformation at 750°C and a constant true strain rateof .001 s-1 on the microstructure of UHCS. Fig. 1(a) was taken in the grip section of thesample, which was thus exposed to the testing temperature without plastic deformation. Figs1(b) and 1(c) were taken in the gage section of the samples deformed to true strains of 0.92 and1.42 respectively. A comparison of the figures shows strain-enhanced grain growth in that theferrite grains have grown. The carbide particles have also coarsened and the size of the ferritegrains appears to be determined by the intercarbide spacing. This suggests that the kinetics ofgain growth are determined by the kinetics of carbide coarsening. The stress-strain curve inFig. 1(d) shows the importance of this grain growth on the deformation behavior of UHCS -increasing the grain size from its initial size (mean grain size = .8 µm) to the size at a strain of1.42 (mean size = 1.48) has raised the flow stress from 35 MPa to over 62 MPa.
Fig. 1. Microstructure of UHCS before deformation (1a) and after superplastic deformation toa true strain of .92 (1b) and 1.42 (1c) respectively. Microstructure consists of ferrite grainsand iron carbide particles. The true stress-strain behavior for this material is shown in Fig.1(d). Strains at which the photomicrographs were taken are indicated in the plots
The experimentally derived grain size distributions are shown in Fig. 2 for UHCS (2a) andCoronze (2b) superplastically deformed to different true strains. The UHCS was deformed at750°C and the Coronze was deformed to 550°C. The strain rate in both cases was .001s-1.The number of grains measured to determine each of the curves is indicated in the figure. Itwas found that section to section variations could be minimized by having at least 200 grains inthe population. As expected the average grain size increases with strain for both materials.The grain size distribution also broadens. This increases the opportunity for multiplemechanisms (such as grain boundary sliding and power law creep) to contribute to thedeformation rate, since many of the mechanisms are grain size dependent.
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5 4
0.0 700.42 686.92 427
1.43 326
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
Strain
Numberof
grains
UHCS
(a)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7
0.0 330.590 399
1.128 309
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
Strain
Numberof
grains
Coronze
(b)
Fig. 2. Grain size distributions for superplastically deformed UHCS (1a) and Coronze (1b) forseveral different strain levels. Measurements are based on samples that were deformed to theindicated strains at temperatures of 750°C for UHCS and 550°C for Coronze. The strain ratein both cases was .001s-1.
Data Analysis and Discussion
Grain size distribution and its evolution
The grain size data shown in Fig. 2 was analyzed for agreement with a lognormal distributionfunction. A random variable D (in this case grain size) is lognormally distributed if lnD isnormally distributed. If µD and σD denote the mean and standard deviation of D, and µlnD and
σlnD the mean and standard deviation of lnD, then these parameters are related by the
expressions [9]
(2)
. (3)
The probability density for D is given as
, (4)
where
. (5)
To verify lognormality of grain size D it is equivalent to verify normality of lnD. Normalpaper plots of lnD show, for each combination of material and strain level, strong evidence of amixture of two normal distributions. The mixture in each case consists of a minor component(representing from 0.2% to 1% of the population for UHCS and from 0.5% to 5% of thepopulation for Coronze) of very small grain sizes and a major component of regular grainsizes. These tiny proportions of very small grain sizes were viewed as outliers or experimentalanomalies, and were not used in the analysis. Maximum likelihood estimates of µlnD and σlnD
as a function of strain were computed from the remaining data and are shown in Table I. Thecorresponding maximum likelihood estimates of µD and σD, also shown in Table I wereobtained from Eqns (2) and (3).
Table IEstimated Lognormal and Normal Parameters for Grain Size Distribution
The relative frequency as a function of grain size calculated from Eqns. (4) and (5) (using theestimated values of µlnD and σlnD reported in Table I) is shown in Fig. 3(a) for UHCS and Fig.
4(a) for Coronze. Calculations were done for four different strain levels in the case of UHCSand five different strain levels in the case of Coronze. These calculated relative frequencyplots are compared with experimental data in Figs 3(b) through 3(e) for UHCS and Figs. 4(b)through 4(f) for Coronze. The agreement between calculation and experimental data isexcellent. Thus the lognormal distribution function provides an accurate description of thegrain size distribution in fine-grained, unstrained UHCS and Coronze (Figs 3(b) and 4(b)). Inaddition, for both materials the grain size distribution retains it lognormality as strain-enhancedgrain growth occurs during superplastic deformation. These findings are particularlysignificant, since the lognormal distribution function presented in Equation (2), has only twoindependent parameters (µlnD and σlnD). Thus, if the evolution of these parameters with the
deformation history is understood, then the entire grain size distribution can be determined andused in analysis and subsequent calculations. The evolution of these parameters is consideredin the next section.
Evolution of the mean and standard deviation
As shown in Table I, the estimated values of µlnD increase with strain; however the values of
σlnD are roughly constant. As a consequence, both µD and σD increase with strain according to
Eqns (2) and (3). Increasing mean grain size and the accompanying distribution broadening isapparent in Fig. 2.
The mean values of the normal distribution and the log normal distribution functions areplotted as a function of strain in Figs 5(a) and 5(b) respectively. The mean values for boththese distributions vary in a linear manner with strain. For the log normal distribution, theestimated slopes for the two materials are also identical, suggesting that the rate of increase ofµlnD with strain does not depend on the material type.
Since the estimated values of µlnD increase approximately linearly with strain, while the
values of σlnD remain approximately constant, the regression model
lnD = a + bε + E, (6)
becomes appropriate, where ε denotes strain, a and b are constants with µlnD = a + bε, and E is
a normally distributed random variable with mean 0 and standard deviation σlnD. Least
squares estimates of these regression parameters are given in Table II.
Table IIParameters for Eqn. (6)
Material a b σlnD
UHCS -0.245 0.455 0.326Coronze 0.643 0.456 0.282
The observation that the rate of increase of µlnD with strain appears to be insensitive to
material is consistent with mechanisms of strain-enhanced grain growth as proposed byWilkinson and Caceres [5]. Two mechanisms were proposed. In one mechanism, migration ofgrain boundaries is induced by grain boundary sliding. In the other mechanism, grainswitching and rotation enhance particle coarsening rates, which in turn increase the rate ofgrain boundary migration. In both these mechanisms, grain boundary migration is induced bythe deformation process itself and is independent of material.
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
0
.42
.92
1.43
(a)
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5
DataCalculations
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = 0
(b)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = .42
(c)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3 3.5
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = .92
(d)
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = 1.43
(e)
Figure 3. Grain size distributions for UHCS. Calculated distributions for four strain levelsare shown in 3(a). Comparisons between calculated distributions and experimental data areshown in 3(b), 3(c), 3(d) and 3(e).
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
0
.42
.92
1.43
(a)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7
CalculationsData
Rel
ativ
e F
requ
ency
Grain Size ( µm)
ε = 0
(b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5 6 7
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = .264
(c)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = .590
(d)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7 8
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = .827
(e)
0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5 6 7
Rel
ativ
e fr
eque
ncy
Grain size ( µm)
ε = 1.128
(f)
Fig. 4. Grain size distributions for Coronze. Calculated distributions for five strain levels areshown in 4(a). Comparisons between calculated distributions and experimental data areshown in 4(b), 4(c), 4(d), 4(e) and 4(f).
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5
µD (
µm)
Strain
Coronze
UHCS
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5
µ lnD
Strain
Coronze
UHCS
.46
Fig. 5 mean values as a function of strain for the normal distribution (a) and the log normaldistribution (b).
Conclusions
Grain size distribution and its evolution during superplastic deformation have been evaluatedfor UHCS, which has a two phase microstructure, and Coronze, which is quasi-single phase.In both materials strain-enhanced grain growth is observed. The primary conclusions are asfollows.1. For both materials the distribution of initial grain sizes is very accurately represented by alognormal function.2. As the materials are superplastically deformed, the distributions retain their lognormalitythroughout the deformation history.3. The log normal distribution has two independent parameters (µlnD and σlnD) that were
found to evolve in a systematic manner with superplastic deformation. The mean value of thelog normal distribution (µlnD) was found to increase in a linear manner with strain while the
standard deviation (σlnD) was roughly constant.
4. The slope of µlnD versus strain was the same for both materials studied suggesting that the
evolution of the grain size distribution is dependent on deformation alone and independent ofmaterial type. This observation is consistent with previously developed theories of strain-enhanced grain growth.5. The results provide the basis for determining the grain size distribution as a function ofstrain during superplastic deformation. The distribution can then be used in constitutiveequations and modeling studies that account for a range of grain sizes in the deformingmaterial.
Acknowledgments
The authors are indebted to Jack Crane (Olin Corporation) for providing the Coronze 638 andOleg Sherby (Stanford Univ.) for providing the ultrahigh-carbon steel. We also thank KerryCadwell and Scott Preuss for experimental assistance. This work was performed under theauspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratoryunder contract No. W-7405-ENG-48.
References
[1] F. Schuckher, in Quantitative Microscopy. New York: mcgraw Hill, 1968, 201.[2] P. Feltham, Acta Metallurgica, 5 (1957), 97.[3] C. S. Pande, Acta Metallurgica, 35 (1987), 2671.[4] D. S. Wilkinson and C. H. Caceres, An Evaluation of Available Data for strain- Enhanced grain Growth During Superplastic Flow , Journal of Materials Science Letters, 3(1984), 395 - 399.[5] D. S. Wilkinson and C. H. Caceres, On the Mechanism of Strain-Enhanced Grain Growth During Superplastic Deformation , Acta Metallurgica, 32 (9), (1984), 1335 - 1345.[6] D. R. Lesuer, C. K. Syn, K. L. Cadwell, and C. S. Preuss, “Modeling MicrostructuralEvolution and the Mechanical Response of Superplastic Materials,” in Advances inSuperplasticity and Superplastic Forming, N. Chandra, H. Garmestani, and R. E. Gogorth, Eds.Warrendale: TMS, 1993, 55-70.[7] D. R. Lesuer, C. K. Syn, K. L. Cadwell, and S. C. Mance, “Microstructural Change andIts Influence on Stress-Strain Behavior of Superplastic Materials,” , S. Hori, M. Tokizane, andN. Furushiro, Eds. Osaka, 1991, 139-144.[8] ASTM Specification E112-84 .[9] I. Miller, J. E. Freund, and R. A. Johnson, Probability and Statistics for Engineers.Englewood Cliffs, NJ: Prentice Hall, 1990.
