UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Determination of the dynamic strength of iron at low temperature Lockhart, Gary T. 1992

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1992_spring_lockhart_gary.pdf [ 3.67MB ]
Metadata
JSON: 831-1.0086493.json
JSON-LD: 831-1.0086493-ld.json
RDF/XML (Pretty): 831-1.0086493-rdf.xml
RDF/JSON: 831-1.0086493-rdf.json
Turtle: 831-1.0086493-turtle.txt
N-Triples: 831-1.0086493-rdf-ntriples.txt
Original Record: 831-1.0086493-source.json
Full Text
831-1.0086493-fulltext.txt
Citation
831-1.0086493.ris

Full Text

DETERMINATION OF THE DYNAMIC STRENGTH OF IRON AT LOWTEMPERATUREByGary T. LockhartB. A. Sc. University of British Columbia, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESMETALS AND MATERIALS ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992© Gary T. Lockhart, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureMetals and Materials EngineeringDepartment ofThe University of British ColumbiaVancouver, CanadaDate ^29 /qqZDE-6 (2/88)AbstractMeasurements of the stress sensitivity of the strain rate and of the strain vs stress in atitanium-doped, "interstitial free" (IF) iron were done at three temperatures, -20, -50and -75 °C. By application of a deformation model, based on a new theory of mobiledislocation density, these measurements permit the determination of (1) the constants ofthe power law describing the dislocation velocity as a function of the effective stress, and(2) the mean free dislocation path, which is descriptive of parabolic strain hardening.The theory is then used to calculate the microstructural parameters which determinethe inelastic strain rate and the deformation resistance as they evolve, over time, withstress, stress rate and temperature. These parameters include the mobile and networkdislocation densities, dislocation velocity, and effective stress. Theoretical predictionsof the dynamic properties show excellent agreement with experiment. These propertiesinclude: (1) the stress sensitivity of the strain rate, as influenced by stress, stress rateand the magnitude of the stress decrease; (2) the nature and recovery time of the strainrate following a stress decrease; (3) the relative level of hard machine stress vs straincurves as a function of crosshead speed; -and (4) the temperature dependence of both themicroyield and macroyield stresses.1 1Table of ContentsAbstract^ iiList of Tables vList of Figures^ viAcknowledgement ix1 Introduction 11.1 The Nature of Dynamic Strength ^ 21.2 Rate Equations ^ 51.3 Direct Measurement of the Dislocation Velocity ^ 71.4 Indirect Methods of Determining n ^ 91.5 Progress in Modelling n* Data 111.6 Equations of the Model ^ 121.6.1^The Mobile Density 121.6.2^The Dislocation Velocity ^ 131.6.3^Strain Hardening ^ 141.7 Applications of the Model 151.8 Stress Decrease Experiments in the Soft Tensile Machine ^ 152 Experimental Methods 172.1 Sample Preparation ^ 172.2 Stress Drop Tests 181112.3 Stress versus Strain Tests  ^202.4 Model Simulations ^  223 Results and Discussion 263.1 Strain versus Stress  ^263.2 Stress Sensitivity Measurements ^  293.2.1 Effect of Stress and Strain  ^323.2.2 Effect of the Magnitude of the Stress Drop ^ 343.2.3 Effects of Stress Rate and Temperature  363.3 Calculation of Dislocation Velocity Constants ^  383.4 Determination of Static Strength Constants  433.5 Additional Measured and Simulated Tests ^  513.5.1 Recovery of Strain Rate ^  513.5.2 Constant Crosshead Speed Tests  554 Further Discussion^ 694.1 Summary of Results  694.2 Comparison with Prior Study ^  715 Conclusions^ 74References 76Appendices^ 80A Hard Tensile Machine Calibrations^ 80A.1 Calibration of Initial Specimen Gauge Length ^  80A.2 Determination of Machine Stiffness Constant  81ivList of Tables2.1 Composition of ARMCO IF-iron. ^ 172.2 Material constants used in the theoretical calculations ^ 233.3 Summary of the fitted constants ^ 493.4 Calculated mechanical and microstructural variables. ^ 684.5 Calculated mechanical and microstructural variables using dislocation ve-locity constants determined by direct methods for edge dislocations. . . . 72List of Figures2.1 Schematic diagram showing the essential features of a soft tensile machine. 193.2 Soft tensile machine (a) strain vs. stress, and (b) slope vs. strain curvesat four temperatures, and a constant stress rate of 1.0 MPa/s ^273.3 Temperature dependence of the strain hardening rate^ 283.4 The original experimental extension and extension rate vs time curves froma typical stress decrease test.   313.5 Stress dependence of (a) the strain rate ratio, and (b) the stress sensitivity. 333.6 Strain rate ratio vs stress decrease at several stress rates and three tem-peratures  ^353.7 Stress sensitivity of the dislocation velocity, as defined by dlnvIda =n1cre,vs stress rate for three temperatures.  ^373.8 Strain rate ratio vs applied stress rate for various stress drops at -20, -50and -75 °C ^  393.9 The optimized Chi—squared fit of the dislocation velocity constants andtheir calculated confidence intervals for (a) -20 °C, (b) -50 °C and (c) -75 °C. 413.10 Temperature variation of the dislocation velocity constants.   423.11 Curve fitting to (a) the strain vs stress and (b) the slope vs strain curvesestablishes the initial static strength and mean free path for this material. 463.12 Comparison of the measured and calculated 0.2 % yield stresses^ 48vi3.13 Theoretical predictions of (a) microstructural parameters of the inelasticstrain rate, and (b) static and dynamic components of the deformationresistance, for a stress vs strain test at -50 °C and 1.0 MPa/s rate of stressincrease  503.14 Recovery of strain rate after the abrupt stress decrease. The theory pre-dicts both (a) the initial slow recovery rate and (b) the recovery time. . .^533.15 Theoretical prediction of the microstructural parameters governing therecovery of the strain rate, following a stress decrease.  ^543.16 Measured hard tensile machine stress vs strain curves for four crossheadspeeds, ranging from 8.5 x 10 to 8.5 x 10-i mm/s, and at four temper-atures, spanning -75 °C to 25 °C  573.17 Theoretical predictions of (a) microstructural parameters of the inelasticstrain rate, and (b) static and dynamic components of the deformationresistance, for a stress vs strain test at -50 °C and a nominal strain rateof 3.0 x 10-4 s-1  593.18 Comparison of measured and calculated hard tensile machine stress vsstrain curves for four crosshead speeds differing by a factor of 10, for fourtemperatures  613.19 Temperature and strain rate dependences of the measured and theoreticalflow stresses, at 4.0 % strain.  ^623.20 Comparison of the estimated and calculated values of the effective stressas they vary with temperature and strain rate.  ^664.21 Dislocation velocity vs effective (shear) stress as determined by experimentand theory.  ^70vii4.22 Comparison of the calculated effective stresses and the yield stresses asso-ciated with the long range motion of edge and screw dislocations. . . 73A.23 Determination of the hard machine stiffness: (a) the total and inelasticelongations are determined from the load vs elongation curve for a speci-men prestrained to 8.8 % uniform strain; and (b) the elastic loading rateis determined from the time derivative of the reloading curve vs specimenextension.   82viiiAcknowledgementThe author is grateful to Dr. T. H. Alden for his advice and direction given throughoutthe course of this study.He is also grateful to fellow students for their friendship and helpful discussion, andto members of the technical and support staff - in particular, P. R. Musil for an exquisitehigh vacuum furnace and much improved soft tensile machine.Further, I thank my wife, Christine Kiyomi, for her encouragement and devotion.ixIn presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Metals and Materials EngineeringThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date:2qChapter 1IntroductionThe dynamic nature of the deformation properties of metals, as indicated by the effectsof temperature and deformation rate on their strength, is an important problem in thetheory and practice of physical metallurgy. One problem that has received much attentionover the past several decades is the strong temperature and rate dependence of the yieldand flow stresses of a-iron at low temperature. For instance [1], the yield strength of acommercially available interstitial-free (IF) iron, pulled at a conventionally slow testingmachine rate, increases 6-fold when tested at -100 °C rather than at ambient temperature.A similar, if smaller, increase in strength is also encountered at ambient temperature byincreasing the strain rate to about 103 s-1 [2]. Considering that most cold formingoperations are done within the range of strain rates from 100 to 103 s-1, over whichthe yield stress for this material doubles, a prediction or interpretation of the material'sstrength in response to either the operating parameters of a deformation process or theexternal deformation variables of a mechanical test is of practical importance.This thesis is an extension of an earlier ambient temperature investigation [3] andis an attempt to measure and to predict the dynamic properties of IF iron at threetemperatures, -20, -50 and -75 °C. Although the technique in this investigation has beenapplied to IF iron, it is believed to have wider application and may significantly advanceour quantitative understanding of inelastic deformation of other metals and alloys. Inkeeping with the earlier study, no attempt is made to involve a thermally-activated flowanalysis, which emphasizes a precise mechanistic description of dislocation dynamics;1Chapter 1. Introduction^ 2instead, following Johnston and Gilman [4], we obtain an empirical power law function todescribe the stress dependence of the dislocation velocity. In this present work, however,this relationship is not obtained by direct observation of individual moving dislocations[4], but rather by the application, to stress sensitivity data, of a recently proposed semi-empirical model [3,5,6] purporting to describe the evolution of the mobile dislocationdensity.The formulation of a quantitative theory of mobile density alone represents a signif-icant advance in the study of dislocation dynamics, and the application of this theorydifferentiates this technique from all others. In principle, the equations of the theorycan be used to predict various deformation behaviours since the mobile density and thedislocation velocity link to the external deformation rate through Orowan's equationpmbv. In the present study, these equations are applied to the results of an experi-mental study of IF iron, in which the stress is abruptly reduced by a small amount andthe resulting decrease of strain rate is measured. If these tests are repeated for differentapplied stress rates, then stress sensitivity profiles are defined for various values of stressdecrease. From a theoretical fit of such profiles, it is possible to obtain the constantsof the dislocation velocity equation. When these constants have been determined, thetheory permits the calculation of the dynamic strength of this material for a specifieddeformation rate and temperature.1.1 The Nature of Dynamic StrengthIt is an essential feature of our understanding of low temperature deformation that thestrength of a solid is partly static, i.e. temperature and time independent, and partlydynamic, i.e. temperature and time dependent. The physical basis for this understand-ing is that the magnitude of the flow stress is determined by the interaction of mobileChapter 1. Introduction^ 3dislocations with two general kinds of obstacles: (1) those possessing short range stressfields, i.e. of less than about 10 atomic diameters; or (2) those possessing long rangestress fields, i.e. of the order of 10 atomic diameters or greater.Short range obstacles are termed thermal obstacles because the external stress re-quired to overcome them can be reduced by thermal fluctuations. The conventional viewis that thermal fluctuations can assist the applied stress even at temperatures approach-ing 0 K. At 0 K, thermal activation is not possible, and the applied stress alone mustrelease dislocations from short range obstacles. At a sufficiently high temperature, theavailable thermal energy may be sufficient to release dislocations with a very small (in-cremental) increase of the external stress above the long range (static) resistance of thesolid. In this case, the obstacle is said to have become thermally transparent. Apart fromthese two limiting cases, the thermal release of dislocations from short range obstacleswill be stress-assisted.Besides being inherently temperature dependent, the release of dislocations from shortrange obstacles is a time dependent process which limits the strain rate. However, therelease rate maybe enhanced by raising the external stress. Thus, in order for deformationto proceed at some imposed rate, the external stress must be increased. Conversely,it follows that the resistance conferred by short range obstacles is dynamic; i.e. thisresistance will be partly determined by the external deformation variables of temperatureand imposed deformation rate.Long range obstacles are termed athermal obstacles because the dislocation-obstacleinteraction occurs over too great a distance for thermal activation; thus mobile disloca-tions must overcome them by the application of stress alone. The resulting release ofdislocations is said to be mechanically-activated or time independent. However, timeindependent release is never admitted except perhaps at at 0 K for which thermal acti-vation is not possible, or at a sufficiently high enough temperature for which short rangeChapter 1. Introduction^ 4obstacles become thermally transparent. Consequently, low temperature deformation isviewed as being entirely dynamic, but the flow stress is only partly so.It is expected that the stress field experienced by a moving dislocation at any pointin a crystal will be the algebraic sum of all stresses at that point [7]; i.e. the short rangestresses arising from thermal obstacles will simply be superimposed onto the long rangestresses arising from athermal obstacles. The simplest assumption is that the flow stress(o-) can be split into two additive parts:a = ae + C78  (1.1)where a, is the effective stress which exactly matches the dynamic resistance conferred byshort range obstacles, and a., the residual, is by definition the long-range static resistanceof the solid. A literal interpretation of Eq. [1.1], if the effective stress is the only ratesensitive component of the strength, is that deformation occurs only when the appliedstress is in excess of the static resistance of the material. In principle then, Eq. [1.1]predicts that if, during a strain vs stress tensile test for example, the flow stress isreduced abruptly by an amount equal to ae, the deformation ceases; i.e. the strain rategoes to zero. To illustrate this point, an experiment of this type was done in the earlierstudy [3] of IF-iron at room temperture. For a stress rate of 0.2 MPa/s, a stress decreaseof 4.6 MPa (the value of the calculated effective stress) reduced the strain rate to zero.The result is remarkable, considering that the static strength for this material not onlydominates the strength at large strains because of strain hardening, but also at smallerstrains near the yield stress, which is about 70 MPa (0.2% offset). Of course in order toperform such a test, without trial and error, the effective stress must be known a prioriChapter 1. Introduction^ 51.2 Rate EquationsThe treatment of low-temperature deformation is by the theory of thermally activatedflow, in which barriers to elementary units of flow are overcome by applied stress andthermal fluctuations. The units of flow are modelled as dislocation segments which areheld up at short—range obstacles, waiting for thermal activation. It is usually true that theaverage dwell time at an obstacle is much longer than the average transit time betweenobstacles. Thus, the release rate of dislocation segments from obstacles, which is thereciprocal of the average dwell-time, accounts for the time and temperature dependencesof the deformation. The release rate is given by an Arrhenius rate equation, which is theproduct of an attempt frequency and the probability of a successful thermal fluctuation.The strain rate is then, simply, the release rate times the strain produced by a successfulthermal fluctuation. The usual form of the strain rate equation, for a single thermally-activated mechanism, is:= N Abv exp   — exp  —kTG — • ( —k TG (1.2)where N is the number of activation sites per unit volume, A the area swept out by adislocation segment per thermally-activated event, which is given as the product of thedislocation segment length (1*) and the distance of motion after activation (d*), b theBurgers' vector, 11 the attempt frequency, AG the Gibbs free energy of activation, andexp(—AG/kT) the probability of a successful thermal fluctuation. Additionally, AG isusually expressed as AG = AH — TLS, where AH is the activation enthalpy, and ASthe activation entropy, at constant temperature.Both and AG will depend upon stress, temperature and structure, but in theanalysis of temperature and strain rate effects on deformation, the dependence of isassumed weak in comparison to that of G. Of particular interest here is the stressdependence of AG which mainly controls the strain rate at a particular temperature.Chapter 1. Introduction^ 6The variation of AG with stress is given asAG = AG0 — f V* do-^(1.3)where V* = (aAG150-)T has the dimensions of volume and is referred to as the activationvolume. Within a limited range of stress and temperature, the stress dependence of theactivation energy may be linearised. Then, the amount of energy provided by the appliedstress (i.e. the work done) is equal to V*cre. Thus, an increase in the effective stress hasthe effect of reducing the activation energy and increasing the rate of thermal release ofdislocation segments. Following Eq. [1.2], changes in the external deformation variables,namely strain rate and temperature, may effect changes in the activation energy. Inparticular, separately raising the strain rate or reducing the deformation temperature(but still maintaining the same strain rate) raises the effective stress and reduces theactivation energy.While the ideas discussed above are sufficient to explain the temperature and ratedependence of deformation, experimental evaluation of all the activation parameters isnot usually possible. In particular, AS is usually assumed a priori to be either zero ortemperature and stress independent, conditions which Li [8] suggests may not be ap-propriate. As well, the utility of this method has been limited by at least the followingconsiderations: (1) a general failure of various theoretical models to distinguish betweenmechanisms [9,10]; and (2) the insufficient treatment of cases in which competing mech-anisms operate [11].Another form of Eq. [1.2] is used in the literature of dislocation dynamics,pmbv,^ (1.4)which follows from the equalities pm = Nl* and v = d* t, exp — (AG/ kr). This isOrowan's equation [12], which gives the deformation rate in terms of the density ofChapter 1. Introduction^ 7mobile dislocations, pm, and their average velocity, v. Implicit in this formulation is thatthe stress dependence of v is assumed to be much stronger than the stress variation ofPm. Although has a precise physical description (Eq. [1.2]), when it is calculated fromEq. [1.4], empirical relations are generally used to describe the stress dependence of vand evolution of the pm with increasing stress and strain.1.3 Direct Measurement of the Dislocation VelocityOne of the principal achievements in the experimental study of the deformation of ma-terials has been the direct measurement of the velocity of individual dislocations as afunction of stress. In general, these measurements involve applying a shear stress pulseof known duration, onto a slip plane of an oriented single crystal, and observing the po-sition of dislocations before and after the application of the stress. Such measurementsare useful because they potentially separate the effects of structure from the effects oftemperature and stress, and provide a basis for calculating the strain rate. Johnstonand Gilman [4] were the first to measure by direct observation, in lithium fluoride, thestress dependence of the dislocation velocity. In this seminal work, they found that thedislocation velocity, at low and intermediate stresses, is described by a power law in theapplied stress of the form:0. ) nV = v0( — .To(1.5)Here, To and n are material and temperature dependent constants. To is the stress requiredto move a dislocation at unit velocity, v0; in this paper we take vo = 1.0 m/s. n, thepower exponent, is descriptive of the stress sensitivity of the dislocation velocity. In LiF,the dislocation velocity was found to be a very sensitive function of the applied stress,increasing by the the twenty-fifth power.Johnston and Gilman found that the relation given by Eq. [1.5] was applicable for aChapter 1. Introduction^ 8range of velocities that spanned eight orders of magnitude from 10-9 m/s, which defineda minimum stress for dislocation motion, to 10-1 m/s, at high stress, which is about 4.5orders of magnitude below the limiting velocity of transverse sound waves in the material.It may be noted here, that while the power law equation has no theoretical basis, it hasbeen shown to give a good representation of the data over the useful range of velocitiesnormally encountered in engineering applications [14]-[30].Combining the concept of the stress dependence of the dislocation velocity with adetermination of the strain dependence of the mobile dislocation density, Johnston andGilman [4] devised a model to calculate stress vs strain curves, in which n was a keyparameter. Later, Johnston [13], with the aid of a IBM-704 computer, calculated variousmechanical behaviours including stress vs strain curves, yield point phenomenon anddelay times in creep tests.The success and novelty of Johnston and Gilman's work, which demonstrated theimportance of the concept of the stress dependence of the dislocation velocity, generatedmuch interest in the study of dislocation dynamics and measurements of dislocation mo-bilities in other materials. However to date, some thirty years later, dislocation velocitieshave been measured in only a few materials: among the metals are W [14], Mo [15,16], Fe[17], Fe-Si [18], Nb [19], Zn [20], Ni [21], Cu [22,23], Cu-Ni [24,25], Cu-Al [26], Cu-30%Zn[27], Ti [28], Al [29], Pb and Pb-In [30]; some data are also available for a few materialshaving the diamond, sodium chloride and sphalerite structures. It is true that the use ofdirect methods to determine the stress dependence of the dislocation velocity is severelylimited, particularly by the difficulty in acquiring single crystals, reliable etchants, elab-orate and special mechanical devices and then the tedium inherent in making a sufficientnumber of measurements.Chapter 1. Introduction^ 91.4 Indirect Methods of Determining nAs a consequence, there has been interest in establishing techniques which are simplerand more convenient than the direct methods to determine the stress dependence of thedislocation velocity. To this end, Guard [31] suggested that if there is a simple stress-dependent velocity change on change of strain rate according to Orowan's equation,then the stress dependence of the dislocation velocity could be determined indirectly bymeasuring the logarithmic stress sensitivity of the strain rate, i.e.dingndln crHere, n* is usually determined as the reciprocal of the strain rate sensitivity of the flowstress, n* 1/m, measured by an "instantaneous" crosshead speed change test in theconventional hard machine. (In these tests, the stress change is usually measured inresponse to an imposed abrupt upward change in the crosshead speed.) Accordingly,Guard measured the stress sensitivity in Fe-3.25%Si and observed its value was of theorder of 60. This value is much larger than the intrinsic value, n = 35, measured by Steinand Low [18] using direct methods.The reasons for this discrepancy have been extensively discussed in the literature ofdislocation dynamics. One possibility is to suppose that not only the velocity but alsothe mobile dislocation density changes in response to the upward change in crossheadspeed. An increase in the mobile density would occur, according to Johnston and Stein[32], if the sudden increase of stress unpins some of the dislocations that have becomeimmobilized in the network by strain hardening. Alternatively, Alden [3] has suggestedthat sources may operate to increase the total dislocation density. Using the power lawequation for the dislocation velocity, the measured stress sensitivity becomesdinTi^72+ ^dln a- (1.7)(1.6)Chapter 1. Introduction^ 10where n is the stress sensitivity associated with velocity change alone. Evidently, n* = nwhen the increase of stress neither generates many new dislocations nor unpins thosepreviously stuck.Other analyses [33,34] have emphasized that it is the effective stress, not the totalapplied stress, which drives the dislocation velocity, and that the effective stress is usuallyless than the applied stress, Eq. [1.1]. (Note that when measurements of dislocationvelocities are done at small strains in single crystals having exceptionally high purity andperfection, the static component of the flow stress can be assumed to be relatively smallin comparison to ae; i.e. a, a.) This will be true even in the annealed material if itpossesses a microstructure that contributes an initial static strength; then the effectivestress will be less than the flow stress even at very small strain. As well, at larger strainsthe static strength will increase because of strain hardening, which is assumed to bemainly or entirely rate insensitive. Thus it can be argued that the dislocation velocity,as defined by Eq. [1.5], must be given as a function of ae, rather than a. The first termof Eq. [1.7] becomesdlnvd in = n  cr,din a dln aIf the effective stress is the only rate sensitive component of the flow stress, then onchange of crosshead speed, do-, = do. Thusdln v^o-^= n—dln a^a,(1.9)and, in most instances, n* will be larger than n, even at the yield stress (and increas-ingly so at larger strains). Accordingly, a simple linear extrapolation of the strain ratesensitivity data to zero strain, as suggested by some investigators [32,35], does not given* = n.In general, the differences between n* and n may be attributed to both changes in themobile density and an effective stress to flow stress ratio less than unity. Unless, however.(1.8)Chapter 1. Introduction^ 11we can know the relative importance of these effects, strain rate or stress sensitivitydata do not permit a determination of n. The theoretical resolution of these problemsis particularly relevant to this thesis, in which stress sensitivity data will be used todetermine the dislocation velocity constants, n and 70.1.5 Progress in Modelling n* DataRecent progress has been made in the modelling of the strain rate sensitivity of a selectgroup of materials for which a comprehensive set of rate sensitivity data is available. Thisprogress is primarily the result of attempts to model the mobile dislocation density. Ina study by Pharr and Nix [34], the total dislocation density is given by a linear functionof the strain, following Johnston and Gilman, and the mobile density is determined bythe fraction of links of the dislocation network longer than a critical lengthaGbcre(1.10)This model was applied with considerable success to Fe-3.25%Si and copper, materialsthat are both rate insensitive (i.e. large n*), but show major differences in the evolutionof structure and effective stress. However, the predicted values of n* for iron, the onlyrate sensitive material they studied, were much larger than the values determined fromexperiment. This discrepancy resulted largely from the model's apparent over estimationof the stress sensitivity of the mobile dislocation density.In a parallel study, in which the strain rate sensitivity of iron was studied by ap-plication of a model of mobile dislocation density having a different physical basis fromEq. [1.10], Alden [36] was able to make successful predictions of the experimental results.In addition, the model was shown to have good predictive capabilities for other materials,LiF and Fe-3.25%Si. (This model will be described in the following section.)Chapter 1. Introduction^ 12Alden's model differs from the model of Pharr and Nix [34] in two ways: (1) thetotal dislocation density is determined by the stress, not the strain; and (2) the mobiledislocation density is determined from a competition, over time and strain, between stressrate (mechanically-activated) dependent generation and velocity dependent trapping.Despite these theoretical differences, a similar result of both models is that an upwardchange in the crosshead speed is followed by an abrupt increase in the mobile dislocationdensity. In Pharr and Nix's model, this increase results because the critial link lengthdecreases in response to a higher effective stress. In Alden's model, new dislocationsare injected into the material in response to the high stress rate transient following thecrosshead speed change. In light of these studies, it appears that the evaluation of thestress sensitivity of the dislocation velocity by way of an upward change of crossheadspeed is complicated by a structural change.1.6 Equations of the Model1.6.1 The Mobile DensityThe theory has been discussed in two earlier papers [5,38] and its essential equations willonly be briefly described here. The basis for the theory is an empirical relation linkingthe total dislocation density [39,40,41] to the applied stress, namelya = a* + aGbpt112. (1.11)a* is an experimental constant which is attributed to sources of hardness other thandislocations. In the theory, a* is a frictional stress which defines the threshold stressat which moving dislocations begin to multiply. Following the observations of Johnstonand Gilman, we take a* to be the stress required to move dislocations at the velocity of10-9m/s.Chapter 1. Introduction^ 13Time independence is implied by Eq. [1.11]; i.e. the total dislocation density dependson the level of stress, not the rate of stress increase. Thus, we assume that dislocationsources, which generate additional dislocations, are activated mechanically by the mono-tonic increase of stress. Consequently, in taking the time derivative of Eq. [1.11] andmaking the assumption that all newly generated dislocations are mobile, we obtain theresult that the generation rate is linked to the rising stress;.+^2pr2Pm —^ cr.aGb(1.12)Once dislocations are generated, they move at a characteristic velocity v to producestrain. After moving over a distance ro, the mean free path, they are trapped (i.e.immobilized) in the dislocation network. The attrition rate for the mobile population isPm = Pm^(1.13)where r^roil) is the statistical lifetime. Thus, the net change in the mobile density isa consequence of a competition between stress rate dependent generation and velocity-dependent trapping,=2p 2.aGtl/b^0" pm vro •PmPm1.6.2 The Dislocation VelocityThe dislocation velocity is given by a power law in the effective stress:V = Vo --Cre " .To(1.14)(1.15)1-0 is a constant descriptive of the frictional resistance to dislocation glide, and is definedas the stress required to move a dislocation at a velocity of v, = 1 m/s. a, is given by:o-,^— aGbp1,12,^ (1.16)Chapter 1. Introduction^ 14which contains the assumption that network dislocations contribute to strain hardeningby raising the static strength of the material. In a recrystallized material, an initialnetwork dislocation density is assumed in order to account for the contribution to theinitial static strength made by residual dislocations.1.6.3 Strain HardeningThe strain hardening coefficient is defined as the derivative of the static strength withrespect to strain:dr„ ceGbg^9 —  ^ (1.17)^de^2ip1I2This equation can be rewritten recognizing that the rate of increase of • the networkdislocation density 16+„ is exactly equal to the attrition rate of the mobile dislocationpopulation —j); = pniv/ro:a2G2b^ •2r0 r8(1.18)If the trapping mechanism is the formation of an attractive junction with a second, inter-secting dislocation (which may be either mobile or already trapped in the network), thenthe mean free path should vary with the mean dislocation spacing (pt-1 /2 ), factored bysome statistical likelihood that such an encounter will form a stable attractive junction.Thus we write ro = TVpt1/2, where T* is the statistical constant. The mean free pathfalls in proportion to 4/2 as a result of the refinement of the dislocation network throughdislocation trapping. This leads to linear hardening. However, it is known from exper-iment [39] that iron does not harden linearly. Instead, like other body-centered cubicmetals, iron exhibits parabolic hardening. This result implies that a constant mean freepath determines the strain hardening coefficient. (It is not known whether some otherfeature of the microstructure controls the strain hardening rate, such as large athermalprecipitates or cell walls, or some competitive softening process, e.g. dynamic recovery.Chapter 1. Introduction^ 15which lowers the strain hardening rate by releasing network dislocations.) Despite thisuncertainity, the mean free path is estimated by fitting the slope of the strain vs stresscurve at intermediate strain.1.7 Applications of the ModelThere is a growing body of literature suggesting that this theory is both accurate anduseful. In particular, quantitative predictions have been made of (1) the strain ratesensitivity of the flow stress of LiF, Fe-3.25creep in 304 stainless steel [5], (3) yieldstresses of a wide variety of materials [37], and (4) more recently the dynamic propertiesof iron at room temperature.Among the various applications, the determination of the constants of the velocityequation, n and 70, is our primary effort, since these material constants control thedynamic properties of dislocations. In this present study, these constants are derivedfrom a theoretical fit to experimental measurements of the stress sensitivity of the strainrate (Eq. [1.6]).1.8 Stress Decrease Experiments in the Soft Tensile MachineThe usual method used to obtain stress sensitivity values is to measure the change offlow stress in response to an "instantaneous" crosshead speed change in a conventionalhard machine. However, this method can be severely limited if the material is rateinsensitive, which happens either because n is large, as it is for Fe-3.25%Si, or 70 is smallas for copper. In this case, a large change in the strain rate is accompanied by a smallchange in the flow stress, which may not be accurately measured. On the other hand, amaterial that is rate insensitive is, by definition, stress sensitive. Thus, a small changein the applied stress will produce a large change in the strain rate. Therefore, a betterChapter 1. Introduction^ 16method of measurement is to apply this small stress change. Such an experiment caneasily be done in the soft tensile machine, in which the stress and rate of stress increaseare controlled and the extension and extension rate are measured.We choose to employ a stress drop because the theory (Eq. [1.11]) indicates that anincrease of stress will result in a burst of dislocation generation. However, if a stressdrop is imposed, the mobile density is unchanged in the instant of the stress drop, andafterwards, only slowly declines with time and strain. (Later, it begins to increase again;see below Section 3.5.1). Consequently, the measurement of the stress sensitivity willnot be complicated by a structure change. Instead, the decrease in strain rate will beproportional to the decrease in the dislocation velocity; i.e.and from Eq. [1.9].^dln^dln vn =^— ^dln a dln aa.Ti = n— .cre(1.19)(1.20)Chapter 2Experimental Methods2.1 Sample PreparationThe test metal is IF iron, supplied by ARMCO, of composition shown in Table 2.1. Thisis a pure iron doped with titanium to remove residual interstitial carbon and nitrogen; itis nominally interstitial free. Slender tensile specimens are stamped from sheet having anas-received thickness of 0.09 cm. These specimens have a parallel gauge length of about2.54 cm with a typical cross-sectional area of 2.25 mm 2 . Prior to annealing, specimensurfaces are given a 600 grit polish, followed by a brief ultrasonic agitation in ethanol.Annealing is done in high vacuum at 820 °C for 64 h. The recrystallized grain size, as amean linear intercept, is 15 pm. Some supplimentary specimens were also prepared fromcold rolled sheet, having a reduced thickness of 0.06 cm; after recrystallization, the grainsize in these specimens was 35 pm. Prior to testing, specimens are chemically polishedfor 10 to 20 s in a 3 % hydrofluoric acid, hydrogen peroxide solution.Table 2.1: Composition of ARMCO IF-iron.C Mn N^Ti^Nb^P^S^Si^Al0.004 0.31 0.018 0.056 0.071 0.008 0.022 0.011 0.03617Chapter 2. Experimental Methods^ 182.2 Stress Drop TestsTests are performed in a soft tensile machine of the Andrade-Chalmers (cam) type inwhich the rate of increase of stress is controlled and the specimen extension is mea-sured. The load is applied by dead weight through a lever of a variable moment arm(Figure 2.1(a)). The arm is designed such that, for any instantaneous load P and inde-pendent of changes in specimen length, the stress a is given by the equation a = 4P/A0;A, is the initial specimen area. Various stress rates are achieved by varying the densityand flow rate of particulate matter into the loading container. In these tests, stress ratesranging from 6.5x 10-3 to 5.0 MPa/s are achieved by using lead shot and glass balls,sized for 2 mm diameters, and two fractions of Ottawa sand (ASTM C109), screened for-35 to +50 mesh and -50 mesh. The time and rate of flow of lead shot and glass balls arecontrolled by a motor driven shutter of variable diaphragm. Delivery of sand is throughpyrex funnels which are flamed to produce different openings.Specimen extension in the stress drop tests is measured by a clip-on extensometerMTS model 632.27C-21 of gauge length one inch (2.5 cm), calibrated 0 to 10 V for0.02 inch extension (0.051 cm). (Another extensometer, MTS model 632.11C-21 cali-brated for 0.15 inch extension (0.38 cm), is used to determine strain vs stress curves.) AHoneywell Accudata 218 bridge-amplifier also provides a low-pass filter of 100 Hz. Theoutput signal is digitized using a sampling interval which is scaled to be appropriateto the stress rate of each test. The digital recording instrument is a Bascom Turnermodel 5120, which writes sequential files of extensometer voltage to floppy disc. Thesedata may be either retrieved and analyzed using the built-in functions provided by theBascom Turner or transferred to a microcomputer. In most tests, because the loadingtimes are long, only a portion of the loading vs time curve that pertains to the stressdecrease is recorded by the Bascom Turner; about 5 MPa of loading is recorded prior toChapter 2. Experimental Methods^ 19Figure 2.1: Schematic diagram showing the essential features of a soft tensile machine.Chapter 2. Experimental Methods^ 20the stress drop and another 5 MPa afterwards. In total 2000 data points are acquired.In addition, specimen extension vs time is charted continuously with a Honeywell E196chart recorder.For low temperature testing, a steady stream of cooled nitrogen gas is admitted into aninsulated box constructed from 5 cm thick polystyrene foam. The stream of nitrogen gasis adjusted by a valve; in this way, temperature control better than ±0.5 °C is routinelyachieved. Test temperature is measured with a copper-constantin thermocouple attachedto the lower grip and monitored by a Fluke 8502A digital multimeter.A typical stress drop experiment involves several tests on a single specimen at constantstress rate but increasing stress. A "safe" prestress intended to be below the elastic limitis applied at the start of each test. After about 1.0 % inelastic strain, while the stresscontinues to rise, a small stress decrease is imposed by lifting a small weight, which hangsfrom a lever attached to the cam, and was a small addition to the total weight supportedby the specimen. The specimen extension rate is measured both before and after thestress drop. At the start of each test, the output voltage of the extensometer is zeroedby repositioning its knife edges on the specimen to re-establish the one inch initial gaugelength. For the low temperature experiments, the thermocouple voltage is monitoredcontinuously and the flow of cooled nitrogen gas adjusted to insure that testing proceedsat the targeted temperature. Initial cooling takes approximately 1 h compared to about15 min between consecutive tests. Once thermal equilibrium has been established, thetemperature is fairly stable and requires only slight adjustment.2.3 Stress versus Strain TestsStress vs strain tests are performed in a conventional Instron machine, in which the rateof specimen extension is controlled and the load is measured. In these tests, carefulChapter 2. Experimental Methods^ 21consideration has been given to the rigidity and alignment of the testing system. Toavoid sloppy linkages, pins and universal joints have not been used; instead the pull rod is screwed directly into the load cell and is held firmly in place by a wide-faced nut.Specimen grip sections are bolted against file inserts. The lower grip is bolted to the lowercage and the upper grip is held firmly by a nut on the threaded pull rod. Alignment ofthe grip faces is achieved by shimming the load cell at three points. Good alignment isdetermined visually. Easy mounting, to ensure the vertical alignment of the specimens,is achieved by mounts slotted to exactly accommodate the width of the specimen gripsections (about 4.8 mm). Near axial loading of the specimen is insured by offsetting thefile inserts of the grips by half of the specimen thickness.Specimen load is measured using a FR load cell calibrated 0 to 10 mV for 200 lbs(890 N). The output signal from the load cell amplifier is digitized using a samplinginterval which is appropriate to the crosshead speed of the test. In most tests, 10 000points are stored for 0.34 cm extension. An external low-pass filter of 10 Hz filters highfrequency noise. The cut-off frequency is sufficiently high to filter the output signalwithout loss of detail of the inelastic transient at yielding for the highest crosshead speedtests. The digital recording instrument is a Bascom Turner model 5120. Load vs time isalso monitored continuously on chart. At the start of each test, both the Bascom Turnerand chart recorder are zeroed on a sensitive (20 pound) load scale, while the specimen isfixed in the top grip but not yet attached to the lower grip.For low temperature testing, very accurate temperature control is necessary, as theloads developed in the rigid system are particularly sensitive to temperature variation. Inthese tests, cooling is provided by chilled alcohol with magnetic stirring. The containeris a foam insulated, 4000 L pyrex beaker. Fine temperature adjustment is made byadmitting small amounts of liquid nitrogen, from a pressurized liquid nitrogen dewer.through the inlet of a small diameter copper tube immersed about 5 cm below the surfaceChapter 2. Experimental Methods^ 22of the alcohol. Temperature control of this apparatus is within ±0.2 °C. Load cyclingwas used to avoid overstressing specimens during initial cooling.Specimen extension is determined from the difference between the total crossheaddisplacement and all elastic deflections of the "machine" (these include all other deflec-tions, besides the elastic and inelastic elongations of specimen gauge length, such as thecrosshead, grips, linkages, load cell, and specimen shoulders). The total extension of thespecimen gauge length at a particular time t isAlt = — It (2.21)where .k is the crosshead speed, P the specimen load and K the machine stiffness, whichis the composite spring constant of the elastic elements of the machine. K is determinedto be 7.5 MN/m by an elastic loading method using a strain hardened specimen (seeAppendix A.2). K is assumed to be constant throughout the tests.The use of Eq. [2.21], by definition, confines all inelastic deflections to the specimengauge length including deformation that may occur outside the reduced (parallel) sec-tion, particularly in the fillets of a tensile specimen. This necessitates that the initialgauge length 10 be treated as an operational length, over which the shoulder—to—shoulderextension is distributed to give a strain equal to the strain measured within the parallelsection [44]. Accordingly, 10 = 2.86 cm is determined from a separate calibration (seeAppendix A.1).2.4 Model SimulationsThe time integration of the equations of the model (Eq's [1.11-1.18]) permits the calcu-lation of all the mechanical and microstructural quantities as they develop during thedeformation testing of IF iron. Among the principal mechanical quantities calculatedare stress, strain, strain rate, slope of the strain vs stress curve and strain hardeningChapter 2. Experimental Methods^ 23Table 2.2: Material constants used in the theoretical calculationsTemperature (K) n 70 (Pa)^ro (m)^a^G (Pa)^E (Pa)^b (m)298^2.0 3.6x109 4.5x10 ^0.33 7.18x10'° 2.11x10 ^2.48x10°273 to be determined^It^7.27x 1010^II^//253^to be determined II^7.32x 1010^// IF198 to be determined^fl^7.36x 1010^//^IIcoefficient. Also calculated are the microstructural quantities including the generationrate of mobile dislocations, rate of trapping in the network, mobile density, network den-sity, dislocation velocity and mean lifetime. The integration is done numerically by aniterative procedure using a 486-33 MHz microcomputer. The time interval of integrationAt depends on the stress rate (or crosshead speed). For convenience, the interval chosenis the time to achieve a stress increase of about 0.01 MPa in one iteration. This intervalinsures that the maximum loss of the mobile dislocation content to the network in anyone iteration is less than 1.0 % of the mobile density, i.e. AO* < 0.01.The initial values of some of the principal variables are (1) strain and stress = 0, (2)network dislocation density pn = 109 m- 2 and mobile dislocation density pm = 10 I11-2and (3) velocity v = 0 up to the elastic limit stress, then by definition, v = 10-9 m/s.All these quantities increase with further increases of stress. The choice of the initialdislocation network density reflects a typical low residual content of an annealed metal.Since it is likely that most of these residual dislocations will be immobile (either theyare severely pinned or do not lie on a glide plane), pm will be practically zero. However,assigning to pm a particular low value is not a critical decision since pm will increase veryrapidly once dislocation generation begins. However, setting pm = 0 causes a division-by-zero-error for some of the calculations.The material constants that have been employed in the calculations are listed inTable 2.2. Note that n, 70 and r, at the lower temperatures are yet to be determined fromChapter 2. Experimental Methods^ 24experiment and model calculations. The model material is a single crystal in comparisonwith the polycrystalline experimental material, and calculations are of shear stress andshear strain; the factor 2.75 is used for conversion to tensile stress and 1/2.75 for tensilestrain. The Burgers vector b = 2.48 x 10'° m. For each temperature, the value of theshear modulus G is calculated in the direction of slip and on the slip plane, according tothe analyses given for cubic metals [42];GH==C44 - -1 H32C44 + C12 - C115(2.22)where the Cij values are the appropriate temperature dependent elastic constants for iron[43]. The Young's modulus listed is the polycrystalline modulus at ambient temperature(its precise value is not important).Simulations have been done for four distinctive kinds of deformation tests. At constantstress rate the model generates (1) soft machine tensile curves of strain vs stress. If atsome point the stress is reduced by a small amount, then the material continues to deformbut at a slower rate under a lower effective stress. For different stress rates and stressdecreases, the model calculates (2) the strain rate ratio, and (3) the recovery time ofthe strain rate to its prior maximum value. At constant crosshead speed, the modelgenerates (4) conventional hard machine tensile curves of stress vs strain. (The results ofcrosshead speed change tests or load relaxation tests can also be calculated but have notbeen done in the present work.) The model accounts for the microstructural differencesfor tests done at different crosshead speeds so that stress differences between curves canbe compared, at the same strain.In the soft machine simulations, the stress rises steadily at a specified constant rate.In a stress drop experiment, a stress decrease is imposed by a conditional statement ata specified strain of 1.0 %. The strain rate ratio is calculated using the strain ratesChapter 2. Experimental Methods^ 25developed just prior to, and just after the stress decrease. After the stress decrease, thegeneration rate of mobile dislocations is set to zero, i.e. en = 0, and remains zero untilsuch time as the magnitude of the stress is restored to its prior maximum value; then,generation resumes according to Eq. [1.12].The hard machine simulation is identical except that the stress rate is derived fromthe crosshead speed, as well as from the elastic constants of the machine and specimenand the inelastic deformation rate. Consequently, the stress rate is not constant buttends to be high at the start of the stress vs strain test, and subsequently decreases withfurther strain. The analysis of Holbrook et al [44] is used to calculate the stress rate;. , cr \1= [— — (1 —/og M ) (2.23)where X is the crosshead speed, la is the initial operational gauge length, C the combinedspecimen—machine modulus, M the effective machine modulus, g = 1 + et a variablestretch ratio, and the inelastic strain rate.Chapter 3Results and Discussion3.1 Strain versus StressStrain vs stress curves were obtained in the soft tensile machine for four temperatures,25, -20, -50 and -75 °C, and a stress rate of 1.0 MPa/s. These curves (Figure 3.2(a))show strain, which is the dependent variable in a soft tensile machine test, plotted onthe ordinate versus stress, the independent variable, on the abscissa. The stress hasbeen normalized with respect to the shear modulus at 25 °C so that the temperaturedependence of the deformation resistance is more usefully shown; i.e. & . a x G298/GT.Inelastic yielding is identified by a rapid initial rise in strain rate' (Figure 3.2(b)) tovalues ranging from about 2.0x10-4 s-1 at 25 °C to 4.5x10-4 s-1 at -75 °C. This strainrate increase appears to be the soft machine equivalent of the yield point, seen in a hardmachine. In this pure iron, it is quite weak. (At this stress rate, the calculated elasticstrain rate is 4.74x10-6 s-1 and is barely detectable.) The 0.2 % offset yield stress(normalized) is very sensitive to the deformation temperature, increasing from about81 MPa at 20 °C to 214 MPa at -75 °C.Strain hardening is shown by the inverse slope of these curves, 1/S = da/de. Itappears to be generally weaker than that described by a parabolic law.2 That is, S is'Figure 3.2(b) shows the slope S = de/do of the strain vs stress curve; the equation for the strainrate is i = Ser.2The usual relation describing strain hardening of body-centred cubic materials is a parabolic curveof the form:T = kei.^ (3.24)This gives rise to a slope of a strain vs stress curve which is not independent of stress, as is characteristic26Chapter 3. Results and Discussion^ 27108.02.00^100^200^300^400STRESS (MPa)(a)1210co 8.06.0UJCI.0 4.0CO2.000^100^200^300^400STRESS (MPa)(b)Figure 3.2: Soft tensile machine (a) strain vs stress, and (b) the slope (i/&) vs stresscurves at four temperatures, and a constant stress rate of 1.0 MPa/s. The curves havebeen normalized with respect to the 25 °C shear modulus so that the temperature de-pendence of the deformation resistance is more usefully shown; i.e. er = a x G298/GT and= S x GT/G298.Chapter 3. Results and Discussion^ 285.0e, 4.0'zr 3.0CCL1J 2.0El)CC•zZ1.0CCCl)0200^250^300TEMPERATURE (K)Figure 3.3: Temperature dependence of the strain hardening rate. The curves have beennormalized with respect to the 25 °C shear modulus; i.e. 1/,' 1/S x G298/GT.Chapter 3. Results and Discussion^ 29not linear in the stress but rises ever more rapidly as the stress and strain increase. Forexample, at -75 °C and a stress of 250 MPa, the change of slope with stress (dS/da) is1.2x10-6 MPa-2. At 300 MPa, this change of slope is 3.7x106 MPa-2 and at 350 MPa,7.0x106 MPa-2 (Figure 3.2).As well, the strain hardening appears to be sensitive to the deformation temperature,particularly for strains up to about 8.0 % (Figure 3.3). Below this level of strain, the strainhardening rate (1/S = do-/dc) decreases with decreasing temperature. Above about8.0 % strain, the strain hardening rate is roughly constant. A similar observation of thetemperature dependence of the strain hardening rate for iron was reported by Keh [39],who additionally reported that the strain hardening rate approached a minimum at about-75 °C and a maximum value at about 25 °C. (Curiously, Keh observed that the strainhardening rate rises upon further decrease of temperature, below -75 °C.) As a result ofthe observed temperature dependence of strain hardening rate, the separation betweenstrain vs stress curves, at constant strain (Figure 3.2(a)), decreases with increasing strain.3.2 Stress Sensitivity MeasurementsThe principal data, from which n and To are determined, are measurements of the sen-sitivity of the strain rate to a small, abrupt decrease in the applied stress. A typicalresult is shown in Figure 3.4, which contains the segment of the original experimentaldigital recording of the extensometer output voltage pertaining to the stress drop; thedigital recorder presents pseudo-analog plots of the extensometer output voltage and itstime derivative. The smooth line shows extension and the noisy line, the time derivativeof linear hardening, but proportional to the stress;M25 = —2 T.k2(3.25)Chapter 3. Results and Discussion^ 30calculated by the digital recorder, shows extension rate. In the instant of the stress drop,the extension-time curve falls abrubtly by an amount comparable to the expected elasticcontraction of the specimen, and the slope also falls abruptly to an approximately con-stant value. At later times, and not shown, the slope (extension rate) recovers to its priormaximum value and above; in most of these tests, this happens only after the specimenhas suffered appreciable additional strain.The noise of the time derivative curves is considerable, and seems to increase withthe density and flow rate of particulate matter, i.e. with the stress rate. In addition, inprior experience with creep testing using the identical equipment, we have observed thatthe loading curves are somewhat noisier than creep curves. These effects suggest thatat least part of the noise maybe attributed to small variations in the flow of particulatematter, this will likely result in minor differences in inertial loading, and will cause themachine to ring. As well, ringing may result from the initial impact of the flow of shot(in a high loading rate test) onto the bottom of the loading container, and may not bedamped-out in the duration of a short test. Another contributing factor to the noiseis the high resolution of the extensometer which is not less than about 10-5 strain or2.5x10-4 mm. The extrapolated lines in Figure 3.4 show attempts to smooth these databy linear regression, a task which is easily performed by using the analytical functionsprovided by the digital recorder.The response of the strain rate to the abrupt stress decrease is measured by the ratioR of the strain rate just before the stress decrease to the strain rate 2 just after; thusR = t /i2 will always be greater than unity. The logarithmic stress sensitivity of thestrain rate is given bydln^ln Rn =  (3.26)dln^Acylcr•In this test, the strain rate ratio and the extension rate ratio are identical because theChapter 3. Results and Discussion^ 31--,- ,^_ _,___ ^ , !-+I-IMMI1111 .1-1-,---.-"--_ la- 1 --L---:'- t  - -1---7- itI-__.1- f^i-^'^-1 ^,MEMt-,--,-^I-^- ,.---^4- - — -',_ _-I- -•,_-.-,i. M-_-:-a-•-, _ _....._,_ ,, - .^.- 4 -1^..^,-'AIL-Albal&--- PPI...s.wite- '---4- A__1.-^• -1--- :::^:_ ,1 IV4Niimin• zi.41 ^'-4 i-,_ . _4_44_i_4^t^r-..^40,^.,^f^,^L__, _-,_ -L.' ---^,. LTIMEFigure 3.4: The original experimental extension and extension rate vs time curves froma typical stress decrease test. The temperature is -50 °C, the stress 305 MPa, the stressrate 0.022 MPais, and the stress drop 7.5 MPa. The estimated strain rate ratio is 2.54.ztu_14,0Chapter 3. Results and Discussion^ 32specimen length is unchanged in the instant of the stress decrease, thus R = 102. Thethe extension rates, 11 and 12, just before and just after the stress decrease are measuredby short linear extrapolations of the derivative curves to the time at which the stress isreduced. This point is taken to be the time of the final recorded extension just prior tothe stress drop, which is indicated by the peak in the extension vs time curve. In thisparticular test (Figure 3.4), the measured extension rate just before the stress is loweredis 3.625x107 m/s and just after, 1.375x107 m/s. The strain rate ratio is 2.54 and thelogarithmic stress sensitivity is 39.5.3.2.1 Effect of Stress and StrainFor a given stress drop, the strain rate ratio is found to be independent of stress andstrain at a particular stress rate and temperature. This result seems to be general for lowtemperature stress sensitivity measurements performed in the soft tensile machine [3,45].Thus, an average of several measurements on a single specimen at increasing stress andstrain can be expected to give a good estimate of the strain rate ratio. A typical resultis shown in Figure 3.5(a). Here, the temperature is -75 °C, the stress rate 0.021 MPa/s,and the stress drop 11.5 MPa. The values of the strain rate ratio are scattered aboutthe mean value of 2.56. The estimated standard deviation is 0.064. From Eq. [3.27],it is evident that a constant strain rate 'ratio arises from an effective stress and velocityexponent which are independent of stress and strain. The constancy of the effective stressis an important result with which to test the theory.The constancy of the strain rate ratio contrasts with the variability of the stress sensi-tivity n*, which rises proportionately with increasing stress (Figure 3.5(b)). According tothe analysis of the measured stress sensitivity (Eq. [1.20]), n* will be linear in the stress ifnlue is constant, independent of stress and strain. The slope of the line is 0.089 MPa-1.3.0-75°C2.56 ± 0.064 = 1.0 MPa/sAog = -11.5 MPa2.0?^150 200^250^300^350 •STRESS (MPa)(a)3025II 20SLOPE = 0.089 MPa-115d. 1.0 MPa/s= -11.5 MPa150^200^250^300^350STRESS (MPa)(b)Chapter 3. Results and Discussion^ 33Figure 3.5: Stress dependence of (a) the strain rate ratio, and (b) the stress sensitivity.The temperature is -75 °C, the stress rate 0.021 MPa/s, the stress drop 11.5 MPa, andthe estimated strain rate ratio 2.56 ±0.064. The constancy of the strain rate ratio withstress suggests that the effective stress is constant in this experiment.Chapter 3. Results and Discussion^ 34Extrapolation of the stress sensitivity line back to the value at zero strain has been sug-gested by some [32,35] to give the velocity exponent n. However, it has been shown [3]that the extrapolated value of n* and the actual value of n are in gross disagreement forIF iron at 25 °C. This discrepancy is attributed to a flow stress which is larger than theeffective stress, even at yield.3.2.2 Effect of the Magnitude of the Stress DropFor a given deformation temperature and stress rate (which give a particular value of theeffective stress), the strain rate ratio should be approximately exponential in the stressdrop; namely,R exp (nAn , (3.27)as seen by equating Eq's. [1.20] and [3.26]. Experimentally, this relation has been con-firmed previously for IF iron at 25 °C [3]. Model calculations will show that Eq. [3.27]is a good approximation if Ao- is small relative to o-e, which is usually the case when thevalue of R is small, i.e. R < 3.0. Otherwise, R is given by the following exact equation3:R=Acr —nCre(3.28)Although in this present study, the number of variable stress drop data points obtainedfor a particular stress rate and temperature are few, the data do appear to be consistentwith Eq. [3.27] (Figure 3.6). In this figure, lines have been fitted through experimentalpoints representing three values of stress drop, but for a fixed temperature and stressrate. A forth point is gained for a zero stress drop for which the strain rate ratio is equalto unity, by definition. A single filled point indicates the average of six to seven tests,3This expression can be derived by considering the appropriate strain rate equations: (1)^=Prn b(creiror just before the stress decrease, and (2) 2 =^b[(0e — Acr)/rojn just after the stressdecrease; R = For the case in which Acr- is small relative to 0e, the term 1— Aol exp(— Acr /a, )and Eq. [3.27] holds. This approximation becomes evident when exp(Acr/ae) is expressed as a seriesexpansion.4.03.5 -20 °C3.23.00 2.52.214.1 2.0•ccccs1.5cr)1.2 •1.0 o^2.0^4.0^6.0^2.0^4.0^6.0^8.0 0^2.0^4.0^6.0^8.0^10^12STRESS DECREASE (MPa) STRESS DECREASE (MPa) STRESS DECREASE (MPa)(a) (I)) (c)Chapter 3. Results and Discussion^ 35Figure 3.6: Strain rate ratio vs stress decrease at several stress rates and three temper-atures; (a) -20 °C, (b) -50 °C, and (c) -75 °C. In this and subsequent figures, the filledpoints are averages of at least six tests performed on a single specimen, and unfilledpoints are individual supplementary tests.Chapter 3. Results and Discussion^ 36done on a single specimen but at increasing stress and strain. These data will be referredto as the principal data. Unfilled points lying one above another on several lines indicatea group of single, supplementary tests performed at different stress rates on a singlespecimen. The experimental points have been fitted by least squares lines which havebeen forced through the origin. Their slopes are equal to ln R/Acr n/o-e4, which is ameasure of the semi-logarithmic stress sensitivity of both the strain rate and dislocationvelocity, i.e. din qdcr = dlnv/da. (Recall that in this test, because the mobile densityis taken to be unchanged during the stress decrease, the stress sensitivities of the strainrate and dislocation velocity are identical.) Additionally, this fitting procedure providesa useful method of smoothing measured R values, which requires fewer measurementsthan the technique (used in this study) of averaging several measurements for a singlevalued stress decrease.3.2.3 Effects of Stress Rate and TemperatureThe variation of slope in Figure 3.6 characterizes the temperature and stress rate de-pendences of the semi-logarithmic stress sensitivity, as defined by the ratio n/ a,. Tobetter show these variations, a cross plot (Figure 3.7) is constructed using the fittedslopes vs stress rate. At -20 °C, the stress sensitivity is both large and sensitive to stressrate, increasing from 0.10 MPa-1, at 1.0 MPais, to 0.37 MPa-1, at 6.6 x 10-3 MPa/s.With decreasing temperature, both the stress sensitivity and the its variation with stressrate decline. For example at -75 °C, the stress sensitivity varies from 0.060 MPa-1, at1.0 MPa/s, to 0.096 MPa-1, at 6.5 x 10-3 MPa/s.41n the thermal activation literature, Ore =V*IkT where V* is the activation volume.4.0 I^I^II^I^II^I^I^I__--_----_----_-----_--------Chapter 3. Results and Discussion^ 370^I^1^I^I^I^I^I^I^I^I 10-3 2^5 10-2 2^5 10-1 2^5 1.0 2.0 5.0STRESS RATE (MPa/s)Figure 3.7: Stress sensitivity of the dislocation velocity, as defined by din v/do- = nlo-e, vsstress rate for three temperatures. Recall that in a stress decrease experiment, because themobile density is taken to be unchanged during the stress decrease, the stress sensitivitiesof the strain rate and the dislocation velocity are identical.Chapter 3. Results and Discussion^ 383.3 Calculation of Dislocation Velocity ConstantsFor each temperature, the velocity equation constants, n and ro, are determined froma theoretical fit of a strain rate ratio vs stress rate profile for a single valued stressdecrease, as shown in Figure 3.8. The fits have been made to the principal strain rateratio data obtained from several tests performed on a single specimen. (Alternatively,fits could have been made to smoothed R vs & data obtained from the slope of linesin Figure 3.6, for a single valued stress decrease.) The fitted curves are included inFigure 3.8. Although these theoretical curves appear to be continuous, they have beenobtained by joining discrete strain rate ratios calculated using a particular stress dropand several stress rates. Then, using the fitted values for n and r0, additional theoreticalcurves are calculated for supplementary stress drops; their good agreement attests to thepredictive capability of the model.Fitting is done by generating a great number of strain rate ratios for each experimentalstress rate using a computer program with two nested control loops, which incrementseparately the values of n and r0. Initially, a wide range of n and To values is tried usingcoarse increments. Then, as the region of possibility in n and To parameter space (whichcontains the best fit values of n and To) is reduced, finer increments are chosen. Smartchoices for n and To are made according to the observation that n and To separatelycontrol two aspects of the strain rate ratio: (1) the smaller the value of n, the moresensitive the strain rate ratio will be to stress rate, i.e. the slope of the R vs à- curve issteeper and R increases more rapidly at lower stress rates; and (2) the larger the valueof r0, the smaller is the strain rate ratio, i.e. the entire R VS Gr curve shifts downwards tosmaller strain rate ratios, while its slope remains essentially unchanged.In addition, the best fit values of n and To were determined by minimizing the Chisquare (x2), which is a weighted least-squares calculation. That is, for each set of n andChapter 3. Results and Discussion^ 394.0rz.'1.0 0 -20 ° CSTRESS DROP^FIT5 10-2 2^5 10-, 2^5 1.0 2.0 5.0 10STRESS RATE (MPa/s)(a)1.05 10-2 2^5 10-, 2^5 1.0 2.0 5.0 10STRESS RATE (MPa/s)(b)3.0z 2.01.05 10-2 2^5 10-, 2^5 1.0 2.0 5.0 10STRESS RATE (MPa/s)(c)aFigure 3.8: Strain rate ratio vs applied stress rate for various stress drops at (a) -20 °C.(b) -50 °C and (c) -75 °C. Experimental points, theoretical curves. Theoretical fits havebeen made to the filled points which are mean values of R for several tests. A scatterbar represents a single standard deviation on either side of the mean R value. Unfilledpoints represent supplementary single tests of the predictive capabilities of the theory.Chapter 3. Results and Discussion^ 4070 values and a series of calculated R vs O- values, X2 is given by the sum of the squaresof the weighted residuals between the measured and calculated strain rate ratios; theweights used are the estimated standard deviations of the measured R values at eachstress rate:The best fit of an experimental R vs 6- curve produces a minimum x2. (A rule of thumbfor a moderately good fit is X2 v or the number of degrees of freedom [46]; in these fitsv = N — 2, where N = 5 is the number of R vs er values being fitted.)Confidence intervals were also determined by calculating the region bounded by per-turbations equal to Ax2 = 1 away from the best fit values of n and To (Figures 3.9(a-c).)The boundary was calculated by separately incrementing n and To and discarding thosevalues which gave Ax2 > 1. The projection of the Ax2 = 1 boundary onto each axis givesthe 68.3% confidence interval (two standard deviations wide) for n and 70, respectively[46].The results of this curve fitting exercise, as represented by Figures 3.8 and 3.9 andsummarized in Figure 3.10, establish the dynamic constants for IF iron. With decreasingtemperature, To declines from 1.7 x 109 Pa at -20 °C to 5.1 x 108 Pa at -75 °C, whilen increases from 3.2 to 6.8. The values at ambient temperature are n = 2.0 and 70 =3.6 x 109 Pa, which were determined previously [3].It is implicit in the calculation of the strain rate ratio, as it is influenced by themagnitude of the stress decrease and stress rate, that the value of the effective stressis determined simultaneously when the values of n and 70 have been determined. (Thisdetermination can be done because the value of the effective stress is established bythe values of n and 70 and the stress rate. In comparison, the influence of the strainhardening rate, which depends upon the value of the mean free path 7.0 is negligible.x2^1\1"' ^R(e'i; n To))St(3.29)3.63.4111.0 3.2g"-Q>- 3.02.81.0^1.2^1.4^1.6^1.8^2.0^2.2^2.4^2.6VELOCITY EQUATION CONSTANT, To (109 Pa)FIT^-20 •Cn = 3.2To= 1.7x109 Pa-),4= 1- 0.520^ 4. 0.675029-0225Uj005.25 04.64.44.24.00.60 0.70 0.80 0.90 1.0 1.1 1.2 13›.< 4.8FIT^-50 "Cfl = 4.6To= 0.91 x 109 PaAx2,--^Ii-FIT^-75 *C ^ I _n = 6.8To = 0.51 x 109 Pa x2= I0 105 5130Chapter 3. Results and Discussion^ 41(a)VELOCITY EQUATION CONSTANT, To (109 Pa)(b)8.0Lu 7.5><CLLU 7.06.56.0a.Lu 5.50.35^0.40^0.45^0.50^0.55^060^0.65^0.70VELOCITY EQUATION CONSTANT, To (109 Pa)(C)Figure 3.9: The optimized Chi-squared fit of the dislocation velocity constants and theircalculated confidence intervals for (a) -20 °C, (b) -50 °C and (c) -75 °C. The confidenceregion boundary shown corresponds to ,Ax2 = 1 larger than the minimum Chi-squared.The projection of the boundary onto each axis gives the 68.3% confidence interval (twostandard deviations wide) for n and To, respectively.NMI•- •^ --•■I^IN--•4^NM- A _A • -1200^250^300TEMPERATURE (K)0Chapter 3. Results and Discussion^ 42ZFigure 3.10: Temperature variation of the dislocation velocity constants.Chapter 3. Results and Discussion^ 43In these simulations, ro was taken to be the value determined at 25 °C (ro = 4.5 pm)[3]; doubling this value, for example, increased cre by less than 5 %, at 1.0 % strain.)Calculations show that the effective stress will be both small, in comparison to theapplied stress, and stress rate sensitive if n is small. In particular, Alden [3] has shownthat this is nature of IF iron at 25 °C for which the calculated tensile value of the effectivestress is very small (e.g. the theoretical tensile value is only 7.7 MPa at a stress rate of1.0 MPa/s). This is also the nature of IF iron at -20 °C. Not only is the effective stresssmall in comparison to the applied stress (albeit about 4.4 times larger than the 25 °Cvalue, at 1.0 MPa/s), but its value is still very much stress rate sensitive, as implied bythe strong stress rate dependence of both the strain rate ratio (Figure 3.8(a)) and stresssensitivity of the dislocation velocity (Figure 3.7).A separate decrease of To will also lower the value of the effective stress. However,despite the declining value of To at low temperature, this effect is overshadowed by thestrong temperature dependence of n. For example, the calculated tensile value of theeffective stress at -75 °C and a stress rate of 1.0 MPa/s is 116 MPa. In addition, thelarge n value is responsible for the lowered stress rate depedence of both the strain rateratio (Figure 3.8(c)) and stress sensitivity of the dislocation velocity (Figure 3.7). (Thereis a probably a practical upper limit to the value of n which can be determined byapplication of this technique. This limit will be reached before the stress rate sensitivityof the strain rate ratio becomes vanishingly small.)3.4 Determination of Static Strength ConstantsSo far, the theory has been used to calculate the stress sensitivity of the dislocation veloc-ity of IF iron, for several temperatures, as it is influenced by stress rate and the magnitudeChapter 3. Results and Discussion^ 44of the stress decrease. The success of these predictions require that the dynamic resis-tance to dislocation glide, which is equal to the effective stress, be calculated from thetheory. The theory permits this calculation once the dislocation velocity constants havebeen determined. The calculation of these dynamic properties depends strongly on nei-ther the nature of strain hardening (e.g. linear vs parabolic) nor the value of the strainhardening coefficient. In order to calculate other mechanical properties of IF iron, whichdepend additionally on the evolution of the network dislocation microstructure, strainhardening must be considered. These properties include the strain vs stress curve, thestrain rate and its recovery after an abrupt stress decrease, and the strain dependence ofthe stress sensitivity n*. (The emphasis of this thesis, however, is not the measurementand prediction of properties which depend primarily on the evolution of the networkdislocation microstructure.)As indicated in Section 3.1, strain hardening in IF iron is apparently more complexthan can be described by a simple linear or parabolic hardening law. However, moderatelygood fits to the strain vs stress curve up to about 8.0 % strain (Figure 3.11(a)) can bemade using the parabolic hardening law (Eq. 3.24), if the hardening rate is adjustedslightly to account for effects of temperature. The adjustable constant in this fit is themean free path ro, which may be calculated directly from the curvature (dS/do-) of thetensile strain vs stress curve (Figure 3.11(b)).To perform this calculation, the strain hardening coefficient5 is equated to the inverseslope of the strain vs stress curve,1 ^MS^ (3.30)9^2 •Substituting into Eq. {3.30} an equation for the strain hardening coefficient, whichis derived from the theory (Eq. [1.18j), and the empirical equation for the slope of a5The precise statement of the strain hardening coefficient is provided by the derivative of the staticstrength with respect to strain, 9 = drs/de.Chapter 3. Results and Discussion^ 45parabolic strain vs stress curve (Eq. [3.25]), and then rearranging for r, gives:ct2 G2 brok2(3.31)Here, k is the constant in the parabolic hardening law equation (Eq. [3.24]); its shearvalue can be evaluated from the curvature of a tensile strain vs stress curve,1^lt13 dS1k22 [du •(3.32)Fitting the curvature of the strain vs stress curves, shown by the slope of the light linesin Figure 3.11(b), is not as straight forward as the above discussion implies. Inasmuchas the strain vs stress curves do not exhibit well-defined, constant, changes of slope withstress (curvature), there is some uncertainty as to which value of the curvature to choose.(It may be a fact that the increasing curvature of the strain vs stress curves indicatesthat trapping becomes less efficient at higher stresses, and as a result the mean free pathincreases.) Consequently, in order to make reasonably good fits to the experimental stressvs strain curves up to about 8.0 % strain, we take an average of the change of slope withstress over this range of strain, at each temperature. These average curvatures are shownby the solid lines in Figure 3.11(b). The calculated values of 7.0, which are summarized inTable 3.3, increase from 4.5 pm to 5.6 pm, as the temperature decreases from -20 °C to-75 °C; at 25 °C, the curvature of the strain vs stress curve is consistent with 7.0 = 4.5 pmas determined in the previous room temperature study [3]. These adjustments in thevalues of 7-0 do not affect the fits of n and To.In addition to fixing the values of 7.0 to account for the increase of static strengthdue to strain hardening, it is found that in order to fit the experimental strain vs stresscurves, the calculated strain vs stress curves must all be shifted to higher stresses bya constant amount of about 30 MPa (with respect to the shear modulus at 25 °C).This stress which has a shear value of 1.1 x 107 Pa, presumably arises from sources ofChapter 3. Results and Discussion^ 46108.0— EXPERIMENTALCURVES— CALCULATEDCURVES2.0■■1025 *C-20 " ^500 ^-754003000^100^200STRESS (MPa)(a)1210ca 8.01. 6.0L.LJ4.0(1)2.00— EXPERIMENTALCURVES— CALCULATEDCURVES0^100^200^300^400STRESS (MPa)(b)Figure 3.11: Curve fitting to (a) the strain vs stress curves and (b) the slope (0) vsstress curves establishes the initial static strength (r°„) and the mean free path (ro) forthis material. The curves have been normalized with respect to the 25 °C shear modulus.Chapter 3. Results and Discussion^ 47static strength in the annealed material, such as grain boundaries, coarse precipitates orclustering of excess titanium or other solute atoms. (Recall that the static strength dueto residual dislocations in the recrystallized material has already been roughly accountedfor by assuming an initial network density of ion = 109 In-2, which contributes about0.5 MPa to the initial tensile strength.) The result of this final curve fitting exercise isshown in Figure 3.11(a).One final discrepancy between the calculated and experimental strain vs stress curves(and unresolved in this study) is the nature of the elastic-to-inelastic transition. Whilethe experimental curves show a rapid elastic-to--inelastic transition, which is indicated byan abrupt rise in strain rate (Figure 3.11(b)), the theoretical curves show a more gradualtransition. Moreover, this difference in yield behaviour results in a gap between the largerexperimental and the smaller predicted 0.2 % offset yield stress (Figure 3.12). This gapwidens as the temperature decreases, but eventually disappears at larger stresses andstrains.In Figure 3.12, the calculated 0.2 % offset yield strength is given by the sum ofthe dynamic and static components, where the static strength incorporates both theinitial static strength and strain hardening (which contributes about 36 MPa at thislevel of strain). The calculated dynamic strength, which is equal to the effective stress,increases from 7.7 MPa at 25 °C to 116 MPa at -75 °C and evidently accounts for themost of the temperature dependence of the experimental 0.2 % offset yield stress. Theremaining difference, which is largely temperature independent, is mostly accounted forby the calculated static strength. In comparison, the gap between the experimental andtheoretical yield stresses is small. At this time we are not certain as to the origin ofthis discrepancy. Perhaps, if dislocation sources in the annealed material are slightlypinned, then an extra unpinning stress, which may be temperature dependent, will berequired. Then, yielding will be delayed until the unpinning stress is reached; afterwards2000_INIMr■S.-. ■.-■..',aro •■■ ..'0tmilIIMII—WO--• 0.2% OFFSET STRESSCALCULATEDA YIELD STRENGTHO DYNAMIC STRENGTH -O STATIC STRENGTHChapter 3. Results and Discussion^ 48200^250^300TEMPERATURE (K)Figure 3.12: Comparison of the measured and calculated 0.2 % yield stresses. Thecalculated yield strength is given as the sum of both the dynamic and static components.All stress values have been normalized with respect to the 25 °C shear modulus.Chapter 3. Results and Discussion^ 49Table 3.3: Summary of the fitted constantsTemperature(K)ii T0(Pa)ro(m) (Pa)298 2.0 3.6x109 4.5x10-6 1.1x107253 3.2 1.7x109 4.5x10-6223 4.6 9.1x108 5.0x10-6198 6.8 5.1x108 5.6x10-6dislocation multiplication may occur, for a short time, at a higher rate than is given byEq. [1.12]. We note that the resolution of this problem is not the central issue of thisstudy.The results of these two curve-fitting exercises, as represented by Figures 3.8 and3.11, and summarized in Table 3.3, establish the deformation constants for IF iron. It isnow possible to calculate all of the mechanical and microstructural variables for IF iron.As an example of such calculations, Figures 3.13(a) and (b) show the microstructuralparameters which determine the inelastic strain rate and the various components of thedeformation resistance (static and dynamic strengths) of IF iron, as they vary with theincrease of stress and strain, for a strain vs stress test performed at -50 °C and at astress rate of 1.0 MPa/s. (Recall that in a soft machine test, stress is the independentvariable, while strain is the dependent variable. In Figures 3.13(a) and (b), however,strain is plotted on the abscissa so that, in a later discussion, hard and soft machinesimulations can be readily compared.) Notice particularly that the dislocation velocityand the dynamic resistance (effective stress) are nearly constant with stress and strain,after a rapid initial increase at small strain. (The values of both the dislocation velocityand the effective stress increase with stress rate, but only the effective stress increaseswith the values of the dislocation velocity constants (at low temperature), the dislocationvelocity declines.) Despite the constancy of the dislocation velocity, the strain rate rises8686 74,482104 —10'30^2.0^4.0^6.0^8.0^10TENSILE STRAIN (%)(a)52 2-50 'C I= 1.0 MPals807060STRAIN HARDENING(STATIC)—INITIAL STATIC STRENGTHDYNAMIC STRENGTHChapter 3. Results and Discussion^ 50250502001501002.0^4.0^6.0^8.0^10STRAIN (%)(b)•Lc 501—ci)cncc 40z30Cc2010Figure 3.13: Theoretical predictions of (a) microstructural parameters of the inelasticstrain rate, and (b) static and dynamic components of the deformation resistance, for astress vs strain test at -50 °C and 1.0 MPa/s rate of stress increase.Chapter 3. Results and Discussion^ 51as the mobile dislocation density increases (Figure 3.13(a)).Strain hardening is the only component of the deformation resistance (Figure 3.13(b))which increases with stress and strain. The initial static resistance is constant, by def-inition. (Both components of the static strength, as defined by theory, are also rateinsensitive.) The constancy of the effective stress with increasing stress and strain is animportant result and underlies the constancy of the strain rate ratio.3.5 Additional Measured and Simulated Tests3.5.1 Recovery of Strain RateAfter the abrupt stress decrease, in a soft machine stress drop test, the applied stresscontinues to rise at a constant rate and the strain rate gradually recovers to its priormaximum value. Both the times and shape of this recovery are of particular interest,and provide a basis on which to test the model's central hypothesis concerning the evo-lution of the mobile dislocation density [3]. Although several stress drop tests have beenperformed in procuring values for n and To, separate strain rate recovery tests have beendone, primarily because recovery times are long in this material. This tendency causesspecimens to suffer appreciable additional strain prior to the recovery of the strain rate.Recovery tests have not been performed at every deformation temperature for whichthe material constants have been determined. Instead, supplementing tests already per-formed at 25 °C [3], a few representative tests have been done at -50 °C using a stressrate of 1.0 MPa/s and four stress drops ranging from 2.5 MPa to 10 MPa, or about 3.3 %to 13 % of the calculated effective stress. The stress drops were applied after a prestrainof about 3.0 %, at 225 MPa, at which the model first matches the experimental strainhardening rate (see Figure 3.11(a)).The results of these experiments show that recovery times of the strain rate are alwaysChapter 3. Results and Discussion^ 52much longer than the times required to restore the applied stress to its prior maximumvalue. For example after a stress decrease of 2.5 MPa, the recovery time of the strain rateis 12.3 s. After a 7.5 MPa stress decrease, the recovery time is 23.0 s (Figures 3.14(a) and(b)). As reported in the prior study [3], these times increase, but not proportionately,with the magnitude of the stress decrease (Figure 3.14(b)). The usual shape of therecovery consists of (1) an initial region of low slope, which ends approximately withthe time required to restore the applied stress to its prior maximum value, followed by(2) a region of higher, but falling slope (Figure 3.14(a)). The model makes quantitativepredictions of both the shape and times of this recovery.These quantitative predictions, particularly of the shape of the recovery, provideconfirmation of the equations governing the evolution of the mobile dislocation density[3]. Recall that in the theory, the mobile dislocation density is related to the rate ofstress increase and not directly the level of stress (or strain) itself. (Perhaps the mostpersuasive evidence for this hypothesis is from the study of the low temperature creepof 304 stainless steel [5] in which the loading stress rate, used to attain the creep stress,is found to affect strongly the amount of subsequent creep strain.) In the present study,the theory predicts that the mobile density will be unchanged in the instant of the stressdecrease. Then, in strict adherence to Eq. [1.11], the total density will not increase againuntil the applied stress is restored to its prior maximum value. There is a delay time forthe generation of new (mobile) dislocations, but the attrition of mobile dislocations tothe network (which is the microstructural origin of strain hardening) continues. Becauseof discontinued generation and continued trapping of mobile dislocations, following thestress decrease, the mobile density declines. Then, at time equal to Ao-/O-, dislocationgeneration recommences (Figure 3.15(a)).The dislocation velocity, on the other hand, falls immediately in response to thestress decrease, but, afterwards, gradually recovers as the applied stress continues to_-— EXPERIMENTALCURVESCALCULATED -CURVES_10^20TIME (s )(a)30^40010 120^2.0^4.0^6.0^8.0STRESS DECREASE (MPa)(b)3530255.00Chapter 3. Results and Discussion^ 536.0—...-7(,),rcz. 5.0,_.I.L.iI--•‹C 4.0CCZ• 3.0CCI—(I)2.0Figure 3.14: Recovery of strain rate after the abrupt stress decrease. The theory predictsboth (a) the initial slow recovery rate and (b) the recovery time. The temperature is-50 °C, stress rate = 1.0 MPais and stress = 225 MPa just prior to stress drop. In (a)the magnitude of the stress decrease is 7.5 MPa and the recovery time is 23.0 s.2.0 0 3020TIME (s)(c)10 406.0Chapter 3. Results and Discussion^ 54• 6.5(2)- >- 5.5QW- 5.0- WCC)• 4.510z :9t 9.0• oZ.1 8.0• 70QCO 06.014.15.020TIME (s)(a)20TIME (s)(b)0^100^1030^4030^40Figure 3.15: Theoretical prediction of the microstructural parameters governing the re-covery of the strain rate, following a stress decrease; (a) the mobile dislocation density.(b) dislocation velocity, and (c) inelastic strain rate. The temperature is -50°C, stressrate 1.0 MPa/s, stress 225 NIPa and stress drop 7.5 MPa. (The stress continues to riseat constant rate, following the stress decrease.)Chapter 3. Results and Discussion^ 55rise (Figure 3.15(b)). Despite the continued recovery of the dislocation velocity, theinitial recovery of the strain rate will be slow due to the declining mobile density. Oncedislocation sources are reactivated, after the delay time, the strain rate rises rapidly toits prior maximum value (Figure 3.15(c)).3.5.2 Constant Crosshead Speed TestsPrevious tests and simulations were done with the soft tensile machine in which the stressrate is controlled and the extension is measured. In the present section, stress vs straincurves are obtained using a conventional hard tensile machine in which the speed of acrosshead is controlled and the load is measured. The control of the crosshead speedmakes hard machine tests distinctive in at least two respects:(1) The stress rate is variable; it depends on the crosshead speed, the elastic deflectionsof the machine and specimen, and the inelastic deformation of the specimen. For example,in a stress vs strain test, the stress rate is high during elastic straining and subsequentlydeclines as inelastic strain increases.(2) The mobile dislocation density and the dislocation velocity are inextricably linked:an increase in one of these variables requires the decrease in the other. For example, rapiddislocation multiplication at small strain may cause a yield drop in a stress vs strain test,or a lessening of the increase of flow stress in an instantaneous upward crosshead speedchange test, thereby lowering the measured rate sensitivity.The primary reason for conducting stress vs strain tests at several crosshead speedsand temperature is to obtain experimental curves which can then be compared to theo-retical predictions. In particular we are interested in predicitions of the rate sensitivityof IF iron, as indicated by the relative differences between stress levels Of of curvesobtained for different crosshead speeds, but at constant strain. (A similar, study of therate sensitivity of the flow stress could have been performed in the soft tensile machine.Chapter 3. Results and Discussion^ 56but at different stress rates, since the nature of the rate sensitivity of the flow stressdoes not depend on which testing machine is used.) However, we must note that themeasurement of AU f cannot be used to establish the rate sensitivity of the flow stress,as defined by m = do-/A, primarily because of uncertainty in the constancy of structure.(Recall that in the theory, the total dislocation density depends on the level of stress,and the mobile fraction is determined by a competition between stress rate dependentgeneration and velocity dependent trapping.) An alternate method would be to performinstantaneous crosshead speed change tests, but even in these tests the constancy of thestructure has been questioned [34,36].Another reason for performing hard machine tests and comparing the results withtheory is to demonstrate the versatility of the model in which the stress rate is the centralvariable controlling the change of structure (Eq. [1.14]). These tests also approximateindustrial processes which often impose shape changes at constant rate. However, wenote that industrial deformation rates are typically 2 to 5 orders of magnitude greaterthan the maximum rate which can be attained in the conventional hard testing machine.Measured Stress versus StrainFigure 3.16 shows several stress vs strain curves obtained at four temperatures (25, —20,—50 and —75 °C) and several crosshead speeds spanning four orders of magnitude (8.5 x10-4 mm/s to 8.5 x 10-i mm/s). Observe in this figure that the scales are the same, sothat the temperature dependence and rate sensitivity of the flow stresses can be readilycompared. The following features are observed:(1) At small strain, yielding is identified by a stress drop, the magnitude of whichincreases with crosshead speed and at lower temperature. For example, at ambienttemperature and low crosshead speed (8.5 x 10-4 mm/s), the stress drop is vanishinglysmall. At -75 °C and high crosshead speed (8.5 x mm/s), it is about 60 MPa.4001004.0^6.0STRAIN (%)(C)2.0 8.0 10A 3.0K /0-2s4O 3.0 x 10-3 0O 3.0 x 10-4 0o 3.0 x 10-5 •-50 b _•_Chapter 3. Results and Discussion^ 57Figure 3.16: Measured hard tensile machine stress vs strain curves for four crossheadspeeds (X), ranging from 8.5 x 10-4 to 8.5 x 10' mm/s, and at four temperatures:(a) 25 °C, (b) -20 °C, (c) -50 °C and (d) -75 °C. The initial effective gauge length/0 = 2.86cm.Chapter 3. Results and Discussion^ 58As well, the sensitivity of both upper and lower yield stresses to the crosshead speedincreases at low temperature.(2) At strains above about 1.0 %, strain hardening appears to be approximatelyparabolic. However, the curves cannot be superimposed onto each other by paralleltransfer because the strain hardening rate falls with high crosshead speed and low tem-perature. As a result, the separation between flow stresses of neighbouring curves (Aaf ),at constant strain, decreases with increasing strain.(3) The separation between adjacent curves (Acrf ) is sensitive to crosshead speed andtemperature. For example, at 25 °C, Acrf is small and increases successively by a factorof about 2 with each factor 10 increase of crosshead speed. At lower temperature, themagnitude of Ao-f increases, but the factor which multiplies successive stress differencesdiminishes. At -75 °C, the Acri's are nearly identical between adjacent curves.Hard Machine SimulationsSimulations of the hard machine tensile tests are similar to those done for the soft ma-chine except, in the hard machine simulation, the stress rate is not explicitly stated butis derived from Eq. [2.23]. Consequently, the model predicts some differences in themicrostructural parameters of the inelastic strain rate and levels of effective stress. Atypical simulation is done for -50 °C at a crosshead speed of 8.5 x 10-4 mm/s The mobiledensity increases with stress and strain. The dislocation velocity, which has a maximumvalue at small strain, falls in order to maintain a nearly constant but slowly decliningstrain rate (Figure 3.17(a)). Consequently, the effective stress, which reaches a maxi-mum value at small strain, slowly declines at large strain (Figure 3.17(b)). The strengthcontributed by strain hardening is assumed to be entirely static, i.e. temperature andrate independent.The calculated stress vs strain curves are presented in Figure 3.18, along with the—DYNAMIC STRENGTHSTRAIN HARDENING(STATIC)-50 t50.= 3. Ox10-4s. 1-411- INITIAL STATIC STRENGTHChapter 3.^Results and Discussion8621059210" 'EA.I-7 _1--50 tkt= 3.0 x104s44 8 8i-c[sz6 6-cc^2 4 4 Tc8LU104 2 2LUco^  10 -8 •••••2.0^4.0^6.0^8.0^10TENSILE STRAIN (x,)(a)5010^ 00 o 2.0STRAIN (%)(b)Figure 3.17: Theoretical predictions of (a) microstructural parameters of the inelasticstrain rate, and (b) static and dynamic components of the deformation resistance, for astress vs strain test at -50 °C and a nominal strain rate of 3.0 x 10-4 s-1.  .80 25070• 601.1.1.A:c• 50c, 40177-vr• 30LL. 2020010015010Chapter 3. Results and Discussion^ 60experimental curves for comparison. The differences between the level of the calculatedflow stresses, at constant strain, represent only differences in the magnitude of the effec-tive stresses; the calculated static strength, which is indicated by the dashed line, doesnot vary with crosshead speed (i.e. ro is assumed to be constant). The theory predictsboth the magnitude of the flow stresses and, consequently, the change in effective stressas it varies with strain rate and temperature, particularly at small to moderate strains(Figures 3.18 and 3.19). For example, at 25 °C and 2.0 % strain, the measured stressdifference between curves obtained at 3.0 x 10-5 s-1 and 3.0 x 10-4 s-1 is 3.3 MPa com-pared to the theoretical value of 3.7 MPa. At -50 °C, the measured stress difference at3.0 x 10-3 s-1 and 3.0 x 10-2 s-1 is 41.6 MPa and the theoretical value is 44.8 MPa.(The theory predicts that these values decrease slightly at larger strains due to a fallingeffective stress (Figure 3.17(b)). However, the decline of the measured stress differences isgreater than the theoretical values because of the additional decrease of strain hardeningrate with crosshead speed, which the theory does not account for.) At -75 °C, the the-ory does not predict a constant stress difference for each factor 10 increase of crossheadspeed. However, the factor of increase for successive theoretical stress differences is small(approximately 1.3).Yield DropsDespite the considerable success of the theory in predicting stress differences, there aretwo notable differences between the experimental and theoretical stress vs strain curves.One discrepancy is the failure of the model, as already noted, to predict yield drops.Instead, the calculated curves rise continuously with stress and strain because the theoryassumes that there is always a sufficient number of operable sources for the dislocationdensity to increase according to Eq. [1.14 However, in order for there to be a stressdrop, there must be a deficiency of dislocation sources in the early stages of strain; for8.0 8.0 104.0^6.0STRAIN (%)2.0^4.0^6.0STRAIN (%)10 2.0(a) (b)(c) (d)- 75 •C2.0^4.0^6.0^8.0^10STRAIN (%)400300CLCI)(I)200100I^II^ EXPERIMENTAL CURVES= CALCULATED CURVESChapter 3. Results and Discussion^ 61Figure 3.18: Comparison of measured and calculated hard tensile machine stress vs straincurves for four crosshead speeds ranging from 8.5 x 10-4 to 8.5 x 10-4 mm/s, and fourtemperatures: (a) 25 °C, (b) -20 °C, (c) -50 °C and (d) -75 °C. The light dashed linesshow measured curves, while the heavy solid lines show the results of model calculations.The calculated static strength is rate insensitive, by definition, and is indicated by theheavy dashed line.EXPERIMENTAL POINTSTHEORETICAL LINESE = 4.0 %/0-4^/ 0 -3^10-2STRAIN RATE, ).ClIc, (s4)400350a.u) 300Cf)CC(1) 250014.20010-1Chapter 3. Results and Discussion^ 62Figure 3.19: Temperature and strain rate dependences of the measured and theoreticalflow stresses, at 4.0 % strain.Chapter 3. Results and Discussion^ 63instance, sources may be pinned by impurity atoms. Note that in the present theory,the mobile density rises very rapidly at small strain (Figure 3.17(a)). Alden [37] hassuggested that the theory may be modified by relaxing Eq. [1.11], in the early stages ofstrain, so that the total dislocation density falls behind its "natural" value (Eq. [1.11]).Once sources are activated, then rapid multiplication may follow, making up the shortfallin total density and causing a temporary drop in stress. Conceptually, the yield dropeffect may be included, but the details of the necessary modifications have yet to beworked out.Strain HardeningThe second difference is between the measured and calculated slopes of the stress vs straincurves. Although the strength contributed by strain hardening may be entirely staticin nature, the strain hardening rate (do/d€) falls with increasing crosshead speed anddecreasing temperature, i.e. with increasing flow stress. This discrepancy is particularlynoticeable for the high crosshead speed tests. However, better agreement is found betweenexperimental and theoretical slopes of the low crosshead speed tests. In particular, fortests performed at crosshead speed X = 8.5 x10-4 mm/s, the strain rate and flow stressesare comparable to those developed in soft machine tests at 1.0 MPa/s. This agreement,however, is expected since it was the slope of the 1.0 MPa/s strain vs stress curves thatwas used to quantify the strain hardening rate (i.e. select ro).To explain the temperature and strain rate dependence of the strain hardening rate,we suggest that trapping by stable attractive junction formation may become less efficientat high flow stresses. Consequently, in a material such as iron, in which the frictionstress (effective stress) is strongly temperature and rate sensitive, strain hardening willbe weaker at low temperature and high strain rate (or stress rate) than at ambienttemperature or slow strain rate. (This is contrary to the usual observation, that the netChapter 3. Results and Discussion^ 64hardening rate is weaker at the high temperature and slow rate because of the onset ofa competitive thermal softening process, e.g. dynamic recovery.) In order to match theweaker strain hardening at high flow stresses, the mean free path (r0) must increase.For example, to make a good fit to the slope of the experimental curve at -75 °C anda crosshead speed of 8.5 x 10-1 mm/s, the value of 7.0 must be increased from 5.6 pm(Table 3.3) to about 10 pm.Unless adjustments in the values of 7'0 are made to better characterize strain hardeningat high crosshead speed, a comparison of the measured and calculated flow stresses willbe complicated by the observed rate dependence of the strain hardening rate. However,the difference between flow stresses of adjacent stress vs strain curves, at constant strain,may be compared in order to show variations in effective stress. The reasons for thisconclusion are twofold: (1) the microstructural origins of the dynamic and static strengthsare different, and therefore the magnitude of the dynamic strength is not affected by themagnitude of the static strength (Eq. [1.1]); and (2) it appears that differences in thestatic strength between adjacent stress vs strain curves are small, at small to moderatestrains.Effective StressAnother comparison of flow stress differences is sometimes made between the so calledathermal plateau stress, measured at and above ambient temperature, and the yield stressat lower temperature. In such a comparison the stress difference is used to estimate theeffective stress at the lower temperature; strain hardening is taken to be similar between0 and 0.2 % strain. However, in this material the value of the effective stress at ambienttemperature is small but not zero. In particular, at 25 °C, 3.0 x 10-5 s-1 and 0.2 % offsetstrain, the theoretical value of the effective stress is 4.72 MPa (at 25 °C. If this valueis close to the actual value of the effective stress, then it represents only 5.4 % of theChapter 3. Results and Discussion^ 65flow stress; the residual stress is presumably static. Consequently, if one assumes thstthe entire yield stress for this test (87.2 MPa) is static, a small, if detectable, error isintroduced into the estimation of the effective stress at lower temperature. The estimatedeffective stresses are shown in Figure 3.20 along with values calculated from the theory.Close agreement is found between theoretical and estimated values of the effective stress,at 25 °C and -20 °C. However, increasing discrepancies as large as about 50 MPa are foundat lower temperature and higher strain rate tests. The magnitude of these discrepanciesincreases with the size of the yield stress drop. Notice in particular, that the larger theyield drop, the greater is the strain at which the lower yield stress is determined. Forinstance, at -75 °C and 3.0 x 10 s-1, the lower yield stress is determined at about 0.6 %inelastic strain; at 3.0 x 10' s-1, the strain is about 0.3 %. Consequently, part of thediscrepancy between the estimated and theoretical values of the effective stress may beassociated with differences in levels of strain hardening.The Nature of the Rate Sensitivity BehaviourBy definition, the semi-logarithmic strain rate sensitivity of the flow stress at constantstructure is given bydo-^cre772 = ^ =^;n(3.33)which is the inverse of the semi-logarithmic stress sensitivity of the dislocation velocity.Consequently, the flow stress is rate sensitive if the ratio, o-e/n, is large (e.g. small n andlarge ro). Conversely, the flow stress is rate insensitive if the ratio, o-e/n, is small (e.glarge n or small ro). However, if only n were known, then the rate sensitivity behaviourof IF iron would appear not to follow these rules. For example, despite a small valueof n at ambient temperature, the sensitivity of the flow stress is small, as indicated bythe small difference between flow stresses in Figure 3.18(a). Then, at lower temperature,25°C _Chapter 3. Results and Discussion 66300MOLOWER YIELDSTRESS DIFFERENCE250 -1- THEORETICALDYNAMIC STRENGTH-•••...•-• _75 0C.■"..••••••_A-_.--• "...-- •A '....,•••...-- ' " 0•. •*150 C.. . °.41■MM1-•• ••..--**---20 °C. JO...411■1150 _01 0 -5^1 0 -4^1O -3^.^10-2^10-1STRAIN RATE, X/I0 (S)Figure 3.20: Comparison of the estimated and calculated values of the effective stressas they vary with temperature and strain rate. The estimated values are given by theincreases of yield stress at low temperature in comparison to the yield stress at 25 °C.Chapter 3. Results and Discussion^ 67the sensitivity increases (Figures 3.18(b-d)), despite an increasing value of n (Table 3.3).The reason for this discrepancy is the strong temperature dependence of the effectivestress, which dominates the rate sensitivity behaviour of IF iron.It is useful for this discussion to rearrange Eq. [3.33] to give the fractional change ofthe effective stress accompanying a change of strain rate,Acr ln R=cre(3.34)At ambient temperature, the fractional change of the effective stress may be large, becauseof a small value of n, but the magnitude of the change of flow stress is small since theeffective stress has a small value (Figure 3.20). At lower temperature, the fractionalchange of the effective stress declines, because of a large value of n, but the magnitudeof the stress change will be large as the effective stress increases strongly with decliningtemperature.Variation of Microstructural Variables with Declining TemperatureThe strong temperature dependence of the effective stress also suggests variations inthe values of the microstructural variables. As the temperature decreases, the frictionaldrag on moving dislocations increases. Then as a result of the theory, the dislocationvelocity decreases and the mobile dislocation density increases (at constant strain rate).These predictions are not intuitively obvious because of an increase of flow stress at lowertemperature. For an example, theoretical values of the effective stress, dislocation velocityand mobile density are obtained at 2.0 % strain, 3.0 x 10-4 s-1 and various temperatures(Table 3.4). At -75 °C, the effective stress is about 17 times larger than at ambienttemperature, the dislocation velocity about 10 times smaller, and the mobile density 10times greater. Unfortunately, it is not possible to directly verify these predictions.Chapter 3. Results and Discussion^ 68Table 3.4: Calculated mechanical and microstructural variables. e = 2.0 %, nominal= 3.0 x 10-4 s-1.Effective Dislocation MobileTemperature Stress stress velocity density(K) (1\4Pa) (1\4Pa) (m/s) (m-2)298 150 7.08 5.11x10-7 6.06x1012253 177 31.4 1.30x10-7 2.38x1013223 210 70.7 7.47x10-8 4.16x1013198 256 121 5.93x10-8 5.26x1013Chapter 4Further Discussion4.1 Summary of ResultsIn this present study we have extended an earlier ambient temperature investigation [3]to measure and predict the dynamic properties of IF iron at three temperatures, -20,-50 and -75 °C. These properties include: (1) the stress sensitivity of strain rate, as isinfluenced by stress, stress rate and the magnitude of the stress decrease; (2) the natureand recovery time of the strain rate following a stress decrease; and (3) the relative levelof hard machine stress vs strain curves as a function of crosshead speed.In order to make predictions of these results, it is necessary to know the dislocationvelocity and mobile density as functions of the stress rate, stress and time. The theorypermits the calculation of these values providing that the material constants are known,which are descriptive of the dynamic resistance opposing the movement of mobile dislo-cations and their mean distance of travel prior to being lost to the dislocation network.In conjunction with measurements of the stress sensitivity of the strain rate, we firstobtain the primary constants, n and 70, which establish the dislocation velocity and themagnitude of the effective stress. Then, from a fit of the strain vs stress curve, thesecondary constants are obtained which establish the strain hardening rate (ro) and theinitial static strength (r:).A principal result of this work may be summarized in a theoretical plot of dislocationvelocity vs effective stress (Figure 4.21). The value of n establish the slope of the lines.6910-91.0^2.0^5.0^10^2^5^102EFFECTIVE (SHEAR) STRESS (MPa)1 0-62251^I I^-00_a =5.0 MPa/s A1.0 MPa/s p0.18 MPa/sA-20 °C /-^0.057 MPa/s p^0^0.024 MPs /p -50 °C /^0—^I^0- / -00/0.0066 MPa/s_-i_Chapter 4. Further Discussion^ 70Figure 4.21: Dislocation velocity vs effective (shear) stress as determined by experimentand theory.Chapter 4. Further Discussion^ 71and ro the relative position. The value of n increases from 2.0, at ambient temperature,to 6.8, at -75 O. The value of To actually declines with falling temperature, from 3.6 x109 Pa to 5.1 x 108 Pa. The value of the frictional resistance on moving dislocations(dynamic strength) increases as the temperature decreases; this result is shown by thelarger effective stresses required to move dislocations and a tendency for the dislocationvelocity to fall. At larger effective stress (low temperature and high deformation rate),the model also predicts that mobile dislocation density tends to increase, relative tosimilar tests done, say, at higher temperature.4.2 Comparison with Prior StudyThe values of the dislocation velocity constants determined in this study are yet to beconfirmed by direct methods. Presently, the only available direct determination of themobility of dislocations in iron is the study done by Turner and Vreeland [17], whomeasured the velocity of individual edge dislocations in high purity iron using an X-raymethod. Their findings are shown in Table 4.5. The values of n are low and less sensitiveto temperature, in comparison to the values obtained in this study (Table 3.3). Thedirectly measured values of 70 are within the same order of magnitude as the valuesdetermined in this study, but Turner and Vreeland's 70 values increase with decreasingtemperature over the same range of temperatures for which the values, obtained in thisstudy, decrease.Using Turner and Vreeland's constants, various mechanical and microstructural vari-ables at 2.0 % strain were calculated for hard machine stress vs strain curves at a nominalstrain rate equal to 3.0 x 10-4 s-1 (Table [4.5]). At ambient temperature, the effectivestress, dislocation velocity and mobile density are comparable to the values calculatedusing the results from the previous study [3] and shown in Table [3.4]. In particular theChapter 4. Further Discussion^ 72Table 4.5: Calculated mechanical and microstructural variables using dislocation velocityconstants determined by direct methods for edge dislocations[17]; e = 2.0 %, nominal= 3.0 x 10-4 sTemperature(K)ii To(Pa)0'(MPa)Cre(MPa)V(m/s)Pm(m-2)373 2.56295 2.8 3.0x108 149 5.45 7.87 x 10-7 3.93 x 1012198 2.97 1.36x 109 156 21.4 2.18 x 10-7 1.43 x 101377 7.35 3.14x 108value of the effective stress using Turner and Vreeland's constants is slightly smaller thanthe value calculated from our constants, by the ratio of 5.45/7.08. At 198 K, however,this ratio is 21.4/121. Consequently, Turner and Vreeland's constants are not charater-istic of the strong temperature dependence of the effective and flow stresses observed inIF iron.It is known, however, that the mobility of edge dislocations is higher than screwdislocations. For instance the velocity in impure LiF [4] was determined to be 50 timeshigher for edge dislocations. Furthermore, from studies of microstrain, the long-rangemotion of edge dislocations has been associated with the microyield stress [47]. Themotion of screws is then required for macroyielding. Consequently, the mobility of slowermoving screw dislocations will determine the strain rate at large strains and, apparently,the dislocation velocity constants determined in this study are for screw dislocations.In Figure 4.22, the weak temperature dependence of the effective stress, calculated at0.2 % offset strain using Turner and Vreeland's edge constants, agrees well with thetemperature dependence of the microyield stress, measured in a prestrained IF iron [47].(The elastic limit stress in this material is less than 30 NIPa.) This behaviour contrastswith the strong temperature dependence of both the the measured macroyield and thecalculated effective stresses, using the dislocation constants determined in this study.- ANELASTIC LIMIT0--.—.—.--......CALCULATEDEFFECTIVESTRESS -(EDGES)—04.\\ LOWER YIELD STRESS\\-^\A.CALCULATED \EFFECTIVE^\- STRESS(SCREWS).N. 0.2% OFFSET'■'.....^STRESS.....',..--Chapter 4. Further Discussion^ 73250200500200^250^300TEMPERATURE (K)Figure 4.22: Comparison of the calculated effective stresses (0.2 % offset) and theyield stresses associated with the long range motion of edge and screw dislocations.::-...' 3.0 x 10-4 s-1.Chapter 5ConclusionsWith modest effort, in comparison to what has been required previously, we have obtainedconstants of a power-law relationship between the dislocation velocity and effective stressfor IF iron at three temperatures, -20, -50 and -75 °C. These determinations supplementan earlier room temperature study [3] of the identical metal, for which the method ofanalysis was developed. Theoretical predictions of the dynamic properties, based onthese results, show excellent quantitative agreement with experiment. Consequently, thetheory provides some understanding of the inelastic deformation behaviour of IF iron:(1) The strong increase of the effective stress below ambient temperature is associatedwith an increasing value of n; the theoretical value of 70 actually declines.(2) The rate sensitivity of the flow stress is linked to the ratio crseln. At 25 °C, thesensitivity of the flow stress of IF iron is small, despite a small value of n, since theeffective stress is small (e.g. a theoretical tensile value of only 7.1 MPa at 2.0 % strainand a nominal strain rate of 3.0 x 10-4 s-1). At lower temperature, the rate sensitivityincreases despite an increasing value of n because of the strong increase of the effectivestress (e.g. at -75 °C, the theoretical value is 121 MPa).(3) The fractional change of the effective stress Au I a, accompanying a change ofstrain rate is inverse to the value of n. At 25 °C, the fractional change is large becauseof the small value of n. At lower temperature the fractional change declines as the valueof n increases.(4) The weak temperature dependence of the effective stress, calculated for the long74Chapter 5. Conclusions^ 75range motion of edge dislocations, correlates well with the small temperature dependenceof the microyield stress. This behaviour contrasts with the present theoretical predic-tion of the strong temperature dependence of the macroyield stress, suggesting that thedislocation constants determined in this study are for screw dislocations.(5) Strain hardening in this metal is generally weaker than parabolic hardening andtherefore cannot be characterized by a single valued mean free dislocation path, ro. (Thetheoretical value of ro is only approximately constant below about 8.0 % strain; at largerstrains ro increases.) The observed decrease of the hardening rate with temperature andincreasing strain rate (or stress rate) is linked to strong temperature and rate dependenceof the flow stress; at higher stresses, the trapping of mobile dislocations may be lessefficient.One shortfall of the theory, in its present form, is its failure to predict a yield drop.However, this phenomenon may not be conceptually excluded if, for example, the numberof initially operable dislocation sources in the annealed material is reduced to a smallnumber by dislocation pinning.References[1] W. C. Leslie, Metall. Trans. 3, 5 (1972).[2] W. C. Leslie, R. J. Sober, S. G. Babcock, and S. J. Green, Trans. ASM 62, 690(1969).[3] T. H. Alden, Metall. Trans. 20A, 1029 (1989).[4] W. .G. Johnston and J. J. Gilman, J. Appl. Phys. 30, 129 (1959).[5] T. H. Alden, Metall. Trans. 18A, 51 (1987).[6] T. H. Alden, Metall. Trans. 18A, 811 (1987).[7] H. Conrad, J. Metals, 582 (1964).[8] J. C. M. Li, in Dislocation Dynamics, A. R. Rosenfield, et al., (eds.), pp. 87-116,McGraw-Hill, New York (1968).[9] K. Ono, J. Appl. Phys. 39 1803 (1968).[10] J. W. Christian, 2nd Int. Conf. on the Strength of Metals and Alloys 1, 31, ASM,(1970).[11] A. S. Krausz and B. Faucher, Reviews on the Deformation Behaviour of Materials6, 105, (1982).[12] E. Orowan, Proc. Phys. Soc. (London) 52, 8 (1940).[13] W. G. Johnston, J. Appl. Phys. 33, 2716 (1962).76References^ 77[14] H. W. Schadler, Acta metall. 12, 861 (1964).[15] H. L. Prekel and H. Conrad, Acta metall. 15, 955 (1967).[16] H. L. Prekel, A. Lawley and H. Conrad, Acta metall. 16, 337 (1968).[17] A. P. L. Turner and T. Vreeland, Jr., Acta metall. 18, 1225 (1970).[18] D. F. Stein and J. R. Low, Jr., J. Appl. Phys. 31, 362 (1960).[19] H. D. Guberman, Acta metall. 16, 713 (1968).[20] D. P. Pope, T. Vreeland and D. S. Wood, J. Appl. Phys. 38, 4011 (1967).[21] R. W. Rhode and C. H. Pitt, J. Appl. Phys. 38, 876 (1967).[22] W. F. Greenman, T. Vreeland, Jr., and D. S. Wood, J. Appl. Phys. 38, 3595 (1967).[23] K. M. Jassby and T. Vreeland, Phil. Mag. 21, 1147 (1970).[24] T. Suzuki and T. Ishii, Proc. Int. Conf. on Strength of Metals and Alloys, Trans.JIM 9, Suppl., 697 (1968).[25] H. Neuhauser and 0. B. Arlan, Phys. Status Solidi (a) 100, 441 (1987).[26] H. Ney, R. Labusch and P. Haasen, Acta metall. 25, 1257 (1977).[27] H. Flor and H. Neuhauser, Res. Mech. 5, 101 (1982).[28] T. Tanaka and H. Conrad, Acta metall. 19, 1001 (1971).[29] J. A. Gorman, D. S. Wood and T. Vreeland, Jr., J. Appl. Phys. 40, 833 (1969).[30] V. R. Parameswaran and J. Weertman, Metall. Trans. 2, 1233 (1971).[31] R. W. Guard, Acta metall. 9, 163 (1961).References^ 78[32] W. G. Johnston and D. F. Stein, Acta metall. 11, 317 (1963).[33] J. W. Christian, Acta metall. 12, 99 (1964).[34] G. M. Pharr and W. D. Nix, Acta metall. 27, 433 (1979).[35] J. T. Michalak, Acta metall. 13, 213 (1965).[36] T. H. Alden, Acta metall. 37, 1683 (1989).[37] T. H. Alden, Mater. Sci. Engng A111, 107 (1989).[38] T. H. Alden, Metall. Trans. 18 A 811 (1987).[39] A. S. Keh, in Electron Microscopy an Strength of Crystals, G. Thomas and J. Wash-burn, (eds.), pp. 231-300, Interscience, New York-London (1963).[40] J. D. Livingston, Acta metall. 10, 229 (1962).[41] J. E. Bailey and P. B. Hirsch, Phil. Mag. 5, 485 (1960).[42] J. P. Hirth and J. Lothe, Theory of Dislocations, Wiley (1982).[43] G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated AggregateProperties, MIT Press (1971).[44] J. H. Holbrook, J. C. Swearengen and R. W. Rhode, in Mechanical Testing forDeformation Model Development, R. W. Rhode and J. C. Swearengen, (eds.), pp. 80-101, ASTM STP 765, ASTM Philadelphia, Pa (1982).[45] T. H. Alden, research in progress (1992).[46] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, NumericalRecipes, Cambridge Univ. Press, Cambridge (1989).References^ 79[47] C. J. McMahon, Jr. (ed.), Microplasticity, Interscience, New York, pp. 121-144(1968).Appendix AHard Tensile Machine CalibrationsA.1 Calibration of Initial Specimen Gauge LengthIt is implicit in the analysis of the displacement of the crosshead, e.g. Eq. [2.21], thatall inelastic deformation is confined to an appropriate initial specimen gauge length,Lo. Since additional inelastic deformation may occur in the fillets, which are between theparallel gauge section and the shoulders of the grip section, Lo is treated as an operationalgauge length, over which all inelastic strain between specimen shoulders is distributedgiving a strain which is equal to the uniform strain measured within the parallel section;i.e.ALtLo =^ (A.35)etwhere AL t is the shoulder-to-shoulder elongation and et is the uniform strain within theparallel section.Determination of the operation length can be easily done with two extensometers,one used to measure ALt and the other et. However, in order to measure the shoulder-to-shoulder elongation, the required 1.5 inch (3.8 cm) extensometer was not on hand.Instead, inelastic elongation was determined by measuring, with a travelling microscope,the displacement of a pair of lines scribed across the shoulders. Applying Eq. [A.35] wouldunderestimate the value of ALt; however, this value may be corrected by including anestimate of the elastic elongation. Two such calibrations were done at 10 % uniformstrain giving values of 10 = 2.85 and 2.87 cm; thus, 10 is taken to be the average value of80^Appendix A. Hard Tensile Machine Calibrations^ 812.86 cm.A.2 Determination of Machine Stiffness ConstantManipulation of the coupling equation (of which Eq. [2.23] is a modified form) can provideseveral techniques for the determination of the machine stiffness K. The technique usedin this investigation is the elastic loading method which is a simple technique that reliesupon a preyield determination of the specimen loading rate P. With this method, thecoupling equation is solved for the combined specimen-machine modulus C by makingthe appropriate substitutions for elastic loading; i.e. j = 0 and a = 01A0 Thus,^C=gq /0 P^(A.36)Then M and finally K are determined from the following equations,and1_ i^1af+717M = gqMoK =(A.37)(A.38)(A.39)It is apparent that the determination of K requires knowledge of the elastic modulusof the specimen a priori, and an independent determination of the stretch ratios g lt110and q = (these definitions will be made apparent in Figure ).To the application of this technique, a specimen with initial area A0 = 2.20 mm2 andeffective gauge length 10 = 2.86 cm was initially prestrained to 3.128 cm total extension.about 8.8 % strain, and then unloaded as shown in Figure A.23(a). An extensometerwith gauge length of 2.54 cm and calibrated to 15 % extension, was used to measure thespecimen extension. This was done, so that upon reloading, a more characteristic valueAir_RELOAD3.0Appendix A. Hard Tensile Machine Calibrations^ 82600 _PRES TRAIN500400-_CI 300'r0--.I2001000 0^1.0^2.0SPECIMEN EXTENSION (mm)(a)_0.02^0.04^0.06SPECIMEN EXTENSION (mm)(b)Figure A.23: Determination of the hard machine stiffness: (a) the total and inelasticelongations are determined from the load vs elongation curve for a specimen prestrainedto 8.8 % uniform strain; and (b) the elastic loading rate is determined from the timederivative of the reloading curve vs specimen extension.Appendix A. Hard Tensile Machine Calibrations^ 83for P would be obtained for a larger load range. An average loading rate of 40.8 N/sfor k = 8.47 x 10-6 m/s (0.02 in/min) is indicated by the reloading curve shown inFigure A.23(b). K was determined to be 7.5 MN/m.

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0086493/manifest

Comment

Related Items