The Modeling of Unusual Rate Sensitivities Inside and Outside the Dynamic Strain Aging Regime

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Kwangsoo Ho1

Erhard Krempl

Mechanics of Materials Laboratory,Rensselaer Polytechnic Institute,

Troy, NY 12180-3590

The Modeling of Unusual RateSensitivities Inside and Outsidethe Dynamic Strain Aging RegimeThe viscoplasticity theory based on overstress (VBO) is altered by introducing anmentation function into the dynamic recovery term of the growth law of the equilibrstress, which is a tensor valued state variable of the model. It is a measure of thestructure of metals and alloys. The flow law is unaltered but affected by the equilibstress whose growth is modified by the augmentation function. The augmentation fudoes not affect the initial, quasi-elastic region but exhibits a strong influence onlong-time, asymptotic solution that applies approximately when the flow stress is reain a laboratory experiment. On the basis of the long-time asymptotic solution pos(stress increases with strain rate), zero (no influence of strain rate) and negativesensitivity (stress decreases with an increase in strain rate) can be easily modeled.simplest case the augmentation function is a constant and the rate sensitivity reequal throughout the deformation. The modeling of a change from one kind ofsensitivity, say negative, to another, say positive can be accomplished by makinaugmentation function dependent on the accumulated, effective inelastic straintheory is applied to model the varying rate sensitivity of a modified 9Cr-1Mo steel, wchanges from negative, to zero and to positive with temperature. The unusual ratetivities are modeled well together with relaxation and strain rate change tests.@DOI: 10.1115/1.1286233#

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Introduction

When the classical material modes of plasticity and creep wapplied to loading, unloading and reloading histories major dciencies of these models became apparent, Pugh et al.@1#. Experi-ments showed that prior plastic deformation had an effectcreep and prior creep had an effect on subsequent plastic dmation. However, creep and plasticity theories were formulaindependently and therefore these interaction effects could nomodeled.

This fact gave rise to the development of ‘‘unified’’ theoriewhich do not have separate repositories for creep and plastand therefore have no principal problems in modeling the intetion phenomena. The book edited by Krausz and Krausz@2# givesan account of the recent status of the ‘‘unified’’ state variamodels.

The state variable theories recognize that the inelastic defortions is influenced by the ever-changing internal structure andthe current structure and the current loading conditions determthe current response. The state variables represent some feof the changing microstructure and their growth mirrors tchanging internal structure. Forward integration of the coupnonlinear set of first-order differential equations yields thesponse of a material element to a given input.

These state variable theories have been applied to modelininelastic deformation behavior of metals and alloys at ambientat elevated temperatures. Application of the state variable theowas mostly in the regions where the deformation behavior is nmal, i.e., there is an increase in stress when the loading ratecreases. In addition, normal creep and relaxation behavior isserved, the creep rate magnitude may decrease or increasestress magnitude usually decreases in a relaxation test. T

1Now at Yeungnam University, Korea.Contributed by the Materials Division for publication in the JOURNAL OF ENGI-

NEERING MATERIALS AND TECHNOLOGY. Manuscript received by the MaterialDivision July 8, 1999; revised manuscript received December 26, 2000. AssoTechnical Editor: D. Marquis.

28 Õ Vol. 123, JANUARY 2001 Copyright ©

rom: http://materialstechnology.asmedigitalcollection.asme.org/ on 08/03

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properties are enhanced with an increase in temperature. Inciple, the state variable theories predict the normal behavior w

As the temperature increases normal behavior may ceasexist for a certain range of temperature which is identified asdynamic strain-aging region. It is characterized by serrated yieing, rate insensitivity, or even negative rate sensitivity andincrease in strength with a concurring decrease in ductility wtemperature, see Mulford and Kocks@3# and Kishore et al.@4#.Negative rate sensitivity is considered to be a prerequisiteserrated flow in the context of dynamic strain aging and foroccurrence of the Portevin le Chatelier effect, see Miller aSherby @5#, Mulford and Kocks @3#. Another example of thepathological behavior is given by Ruggles for type 304 stainlsteel, see Ruggles and Krempl@6#. The ratcheting in zero-to-tension loading was significant and rate-dependent at ambtemperature. At 600°C no ratcheting was found under identtest conditions. The excessive cyclic hardening of Haynes-superalloy is another manifestation of dynamic strain aging,Rao et al.@7#.

Regions of dynamic strain aging are also found in many otengineering alloys, especially ferritic and carbon steels. To pform stress analyses in these regions the ‘‘pathological’’ deformtion behavior must be represented in a constitutive equation.

The present paper introduces a method of modeling negarate sensitivity and zero rate sensitivity in the context of a stvariable model that normally represents positive rate sensitivKrempl and his students have developed the viscoplasticity thebased on overstress~VBO!. It is shown that the theory as modifieby Ho @8# can represent the pathological behavior in principThe proposed modifications complement similar work by Namura @9# and Yaguchi and Takahashi@10#.

Experiments and Models With Unusual Rate EffectsFigure 1 from Yaguchi and Takahashi@10# illustrates the nor-

mal and the pathological behavior that the paper addresseshows the stress-strain and relaxation behaviors of the mod9Cr-1Mo steel between 200 and 600°C. The peak stress denthe stress measured at 1.5 percent strain in tensile tests~filled

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2001 by ASME Transactions of the ASME

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symbols! run at one of four different strain rates. After reachinthis stress a 24-hr relaxation test follows and the stresses aend of the relaxation tests are indicated by the open symbols

The strain rate sensitivity at 200°C is slightly negative andalmost zero at 400°C. At 500°C and above positive rate sensitiis observed and it increases with temperature. The relaxationin 24 hr is highest at 600°C and it decreases with decreatemperature. Surprisingly, significant relaxation is still observed400 and 200°C. In this temperature region rate sensitivity is zor negative. Another characteristic of the high temperature reation behavior continues from the highest to the lowest tempture. The stress at the end of the 24-hr relaxation tests is, withexception of the 500°C results, lowest for the fastest prior strrate. This observation seems to say that relaxation is not affeby the mechanisms that cause rate dependence to disappear

Figure 1 shows that the strain-aging region is between 200500°C. Normal behavior is found above this temperature.though normal behavior is expected again at ambient, the prestudy does not have any data in this region.

There have been few attempts to model such a behavior. Ychi and Takahashi@10# introduce an aging stress that has negatrate sensitivity together with the positive rate dependence ofoverstress. When the positive and negative rate sensitivitiescel, rate independence will be modeled. The theory is showreproduce the data, see Yaguchi and Takahashi@10#.

Nakamura@9# has successfully applied a modified versionVBO to the modeling of the ‘‘pathological,’’ cyclic behavior o316 FR steel at 923 °K. This steel shows rate insensitivity atinitial cycles that changes to normal rate sensitivity aftercycles.

Estrin @11# discusses dynamic strain aging and recognizes ia contribution to the solute effect on the mechanical threshEquation ~71! yields a decrease of the contribution to the mchanical threshold for dynamic strain aging as the inelastic strate increases. The stress may decrease or increase when thdence is inserted into the kinetic equation of Eq.~6!.

Peaks of the flow-stress with temperature together with serryielding are said to be indicators of dynamic strain aging. ‘‘Allthese phenomena are modeled in MATMOD-BSSOL by usinginteractive solute-strengthening variable . . . ’’ see Henshall et al.

Fig. 1 Strain rate sensitivity and relaxation behavior of modi-fied 9Cr-1Mo steel as a function of temperature, from Yaguchiand Takahashi †10‡

Journal of Engineering Materials and Technology


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@12# p. 194. Earlier, Miller and Sherby@5# modeled solutestrengthening in the MATMOD framework and demonstratedinverse strain rate sensitivity.

This paper aims at modeling the type of behavior depictedFig. 1 with one model making allowance for the influencetemperature on the stress-strain and the relaxation behaviors.is made possible by a recent ‘‘innovation’’ of Ho, see Ho@8#, Hoand Krempl@13#. Ho found a consistent way to model positivand negative rate sensitivity by introducing an augmentation fution or constant in the dynamic recovery term of the growth lfor the equilibrium stress. The authors have not seen a comparmodification with other models. After an explanation of thmethod in principle, the one-dimensional VBO with augmentatfunction is introduced, material constants will be determined anumerical simulations will be performed so as to demonstratecapabilities of modeling regular and pathological behavior oftype shown in Fig. 1. A small strain, isotropic version of thmodel is given in Ho@8#, Ho and Krempl@13#, and Ho andKrempl @14#.

One-Dimensional VBO With Augmentation FunctionLet s ande be the true stress and strain, respectively, and leE

and Et be the elastic and the tangent modulus at the maximstrain of interest, respectively. The overstress isX5s2G and itrepresents the stress that is available for generating inelastic srate. The equilibrium stressG is thought of as a measure of thobstacle strength of the material. The purpose of the kinemstressf is to set the slope at the maximum strain of interest andmodel the Bauschinger effect. The slopeEt is based on the inelastic strain rate and is related to the tangent modulus based onstrain rateEt by Et5Et /(12Et /E). The positive shape functionc@G# controls the transition from the initial quasi-elastic regionthe fully developed inelastic flow and is bounded byEt,c,E.In its effects the isotropic stressA is similar to the isotropic hard-ening of rate-independent plasticity.

The constitutive equation consists of the coupled nonlinearferential equations and can be written as

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The constantsm and r in Eq. ~2! control the transients and thmagnitude of the static recovery term, respectively. In Eq.~2! theterm (2rG) on the right-hand side represents the softening tcan occur at elevated temperature and this term gives the grolaw of the equilibrium stress the so-called Bailey-Orowan formsee the description in Ho and Krempl@14# and Tachibana andKrempl @15#.

The constantm is multiplied by the overstress rateX which iszero when the asymptotic solution is reached, see Eq.~6!. There-fore, the dimensionless factorm influences the transient behavioupon a jump in strain or stress rate. This can happen in the cas

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strain rate cycling, at the start of relaxation, and upon unloadAn analysis of the VBO behavior upon a sudden change in ragiven by Krempl and Kallianpur@16#, Krempl and Nakamura@17#. The major change in VBO is the addition of the augmention constant or functionb by Ho @8# to model positive, negativeor zero rate sensitivity. Ho@8#, Ho and Krempl@12# discuss theeffects ofb on the modeling capabilities of VBO. While there iswide range forb the sumA1lG is required to be positive at altimes, see Nakamura@9#.

Asymptotic Behavior. The constitutive equations admlong-time asymptotic solutions that are mathematically validinfinite time. The formulas that are applicable for the asymptolimits describe the behavior when inelastic flow is fully estalished with sufficient accuracy. The asymptotic solutions canobtained by converting the set of differential equations into ingral representation and formally calculating the limits, see Cnocky and Krempl@18#, Krempl @19#, see Fig. 1 and Ho@8#. Forthe uniaxial case the following relations apply for a constant strrate and when no static recovery term, i.e.,r 50 in Eq. ~2!, ispresent.

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$G2 f %5$A1bG%H s2G

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where$ % denotes the asymptotic, long-term value of a quantEquation~5! shows that the stress, equilibrium stress and kimatic stress grow at the same rate in fully developed inelaflow. The slope is given by Eq.~3!. It is seen from Eq.~6! that theoverstress is nonlinearly related to the total strain rate by thecosity functionk. The asymptotic limit for the stress minus thkinematic stress contains the rate-independent~plastic! and therate-dependent contributions~viscous! given by

$s2 f %plastic1$s2 f %viscous5$A%$~s2G!/G%

1$~11b!G%$~s2G!/G%. (9)

Equation~6! reveals that the overstress is not directly affectedthe newly introduced quantityb. It has, however, an influence othe equilibrium stressG. Equation~7! has the difference$s2 f %on the left-hand side sinces has no asymptotic solution unlessEtis zero, buts2 f has one for any value ofEt . The growth law forthe kinematic stressf is rate independent, see Eq.~3!.

It should be noted that the overstress in the asymptotic sdepends nonlinearly on the strain rate due to the viscosity funcand is not influenced by the presence of the augmentation funcb. That means that the creep and relaxation behaviors aredirectly affected even if the augmentation function is mademodel negative rate sensitivity. This property corresponds toqualitative behavior of the 9Cr-1Mo steel as shown in Fig.Numerical experiments will show that VBO is capable of moding the relaxation behavior found in Fig. 1.

To see what influence the augmentation functionb can have weconsider Eq.~7! and Table 1. Forb50 the normal behavior ofVBO is reproduced. In this case the stress minus the kinemstress is the sum of the rate-independent isotropic stress$A% andthe viscous contribution due to the overstress. For other valueb negative rate sensitivity~b,21!, zero rate sensitivity,~b521!,can be modeled as seen in Table 1.

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Numerical SimulationsThe numerical experiments are intended to elucidate the p

erties of the model in general. VBO will then be applied to tdata shown in Fig. 1 and other results from Yaguchi and Tahashi @20#; Yaguchi and Takahashi@10#, pertaining to the 9Cr-1Mo steel. This alloy shows regular rate sensitivity, but patholocal behavior in the dynamic strain-aging regime, i.e., negativezero rate sensitivity and increasing strength with an increastemperature.

The increase in strength with increasing temperature canmodeled by letting the isotropic stress increase. For normal mrial behavior and low homologous temperature, the isotrostress is the repository for modeling of cyclic hardening/softeniwhich takes many cycles and therefore a long inelastic strain plength to evolve. If the growth law forA is properly adjusted forthe cyclic behavior then its evolution in a tensile test can be vsmall andA can be considered a constant. Therefore, the isotrostress is kept constant in the simulations. The situation maydifferent whenA is modeling the softening effects at high temperature as it is done in Eq.~4!, which is not homogeneous in thrates and so time under load plays a role. The balancing ofgrowth of the isotropic stress is not emphasized in this paper. Hthe effects of rate are of prime interest.

Hypothetical Material. The material constants given iTable 2 are used. Each time only two numerical tensile tests wconstant strain rates of 1026 1/s and 1023 l/s are performed so asnot to crowd the graphs. In principle there is no limit on thmagnitude of the strain rate when VBO is used in the numerexercises.

Figure 2 shows two stress-strain diagrams at different strates usingb50.2, which is a positive rate sensitivity but differenfrom the normal VBO sensitivity ofb50. The stress levels of thestress-strain curve and of the equilibrium stress-strain curvecrease nonlinearly with strain rate. For normal VBO t

Table 1 Rate sensitivity indicators

Augmentationfactor

Ratesensitivity

b,21 Negativeb.21 Positiveb521 Zerob50 Positive, normal

VBO behavior

Table 2 Material constants for hypothetical material

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asymptotic equilibrium stress would be rate independent.overstress for the fast straining is, as expected from Eq.~6!, largerthan for the slow straining.

For b521 the model depicts rate insensitivity in thasymptotic state, see Fig. 3. Note that the rate insensitivity is ofor the stress, the rate sensitivity of the equilibrium stressincreased compared to Fig. 2.

Negative rate sensitivity is modeled in Fig. 4~b521.2!. It isnoted that both the stress and the equilibrium stress exhibit ntive rate sensitivity.

In all three cases there are rate-dependent transitions betinitial, quasi-elastic behavior and the fully established inelasflow. Also in each case the evolution of the kinematic stress isindependent and is independent of the value ofb. Surprisingly,

Fig. 2 Positive rate sensitivity, bÄ0.2. The stress togetherwith the equilibrium and the kinematic stresses are plotted. Thedashed and solid curves are for strain rates of 10 À6 1Õs and10À3 1Õs, respectively.

Fig. 3 Zero rate sensitivity, bÄÀ1. The stress together withthe equilibrium and the kinematic stresses are plotted. Thedashed and solid curves are for strain rates of 10 À6 1Õs and10À3 1Õs, respectively.



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the overstress is also independent, which is in accordance withprediction of Eq.~6!. Measuring$s2G% at 5% in each of Figs.2–4 shows thatb has no effect. The stress level of each strestrain curve adjusts itself to allow the same overstresses in ecase, see Figs. 2–4.

In the present formulation, creep~defined as stress rate is zer!and relaxation~defined as strain rate is zero! behavior are onlydependent on the overstress, see Eq.~1!, and Eq.~6!. It followsthat creep and relaxation are not affected by the rate sensitchanges in the tensile tests shown in Figs. 2–4. This is only truthe relaxation or creep tests are started in the flow stress regiois seen from Figs. 2–4 that after the initial quasi-elastic regionoverstress will depend on the augmentation factorb until the flowstress-region is reached.

Fig. 4 Negative rate sensitivity, bÄÀ1.2. The stress togetherwith the equilibrium and the kinematic stresses are plotted. Thedashed and solid curves are for strain rates of 10 À6 1Õs and10À3 1Õs, respectively.

Table 3 Material constants for modified 9Cr-1Mo steel



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The examples given in Figs. 2–4 are for a constant augmetion factor. It is possible to makeb a function. When the augmentation factor is made to depend on accumulated strain, thesensitivity can be made to change from normal to negative orversa, see Ho@8#, Ho and Krempl@13#. It is also possible to ‘‘turnrate dependence on or off.’’ This feature is needed to modelobservations of Nakamura@9#. It is also possible to make thaugmentation factorb depend on inelastic strain rate. In this cathe well-known increase in rate sensitivity at strain rates exceing 104 1/s can be modeled, see Ho@8#. By makingb depend onboth the accumulated inelastic strain and the inelastic strainthe rate-dependent stress-strain behavior of PMMA was modfrom zero to a true strain of one at four different strain raranging from 10 to 1023 l/s, see Ho and Krempl@13#. If rate-insensitivity or negative rate sensitivity are modeled then thispect of dynamic strain aging can be made to appear in cerstrain and strain rate regions as observed by Mulford and Ko@3# and Kalk and Schwink@21#. See Ho@8#, Ho and Krempl@13#,

Fig. 5 Simulation of stress-strain curves showing negativestrain rate sensitivity at 200°C, data from Fig. 1

Fig. 6 Simulation of relaxation behavior at 200°C, data arefrom Fig. 1



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Ho and Krempl@14# for modeling of these effects. In each cathe initial quasi-elastic region remains quasi-elastic.

Modified 9Cr-1Mo Steel. To demonstrate that the modifications of VBO are relevant to engineering alloys the theory isplied to modified 9Cr-1Mo steel reported by Yaguchi and Takhashi@20#, Yaguchi and Takahashi@10# and discussed above, seFig. 1.

At 500°C and above the recovery effects become importantthe static recovery term in the growth law for the equilibriustress needs to be introduced, see Eq.~2!. This will give thegrowth law the well-known Bailey-Orowan format. An indicatioof the need for a recovery term is the following observation.some temperature and with some materials it is observedcreep and creep rupture occurs at stress levels that are inquasi-elastic region of the stress-strain diagram. This is the cfor the 9Cr-1Mo steel above 500°C. We can then apply

Fig. 7 Simulation of stress-strain curves showing zero ratesensitivity at 400°C; compare with Fig. 5 of Yaguchi and Taka-hashi †10‡

Fig. 8 Simulation of relaxation behavior at 400°C; comparewith Fig. 6 of Yaguchi and Takahashi †10‡



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methods proposed previously, see Choi@22#, Majors and Krempl@23#, Maciucescu, Sham et al.@24#, and Tachibana and Kremp@15#.

The constants needed to model the 9Cr-1Mo data can be foin Table 3.

From the available data it is clear that the normal VBO is sficient to model the results above 500°C. The rate-insensitiand the negative rate sensitivity require the augmentation fafor modeling. The simulations of the 200 and 400°C dataplotted in Figs. 5–8. It can be seen that the data are simulwell.

Data at 200°C. It is unfortunate that only the endpoints of thlaboratory experiments are available in the numerical experimof Figs. 5 and 6. The numerical simulations clearly show negarate sensitivity in the fully established inelastic flow regioninterest.

Data at 400°C. The simulated results and the experimental dare plotted in Figs. 7 and 8 where experimental data are availfor the entire curves. The simulated stress-strain diagrams srate sensitivity around the knee of the stress-strain diagram,

Fig. 9 Simulation of stress-strain curves with positive ratesensitivity at 550°C; compare with Fig. 3 of Yaguchi and Taka-hashi †10‡

Fig. 10 Simulation of relaxation behavior at 550°C; comparewith Fig. 4 of Yaguchi and Takahashi †10‡



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they show rate-independent asymptotic behavior, see Fig. 7.ferring to Fig. 8 the experimental curves are well matched wthe relaxation experiment. The relaxation curves in Fig. 8 shthat the relaxation stress associated with the slow strain ratsomewhat higher than the experimental results. A close look aexperimental data reveals that there is an increase in the exmental relaxation rate in the vicinity of the 20,000 s. Sincenumerical stress decrease is not experiencing this episode aference exists at the end. The increase in relaxation rate forprior straining does not show this change and the match is qgood.

Behavior above 500°C. Normal VBO ~b50! is sufficient formodeling. The material constants are given in Table 3 andresults of the simulations are reported in Figs. 9–11. The mogives a good match of the nonlinear rate effects on the strstrain diagrams. The difference between simulation and expmental data are greatest around the knee of the stress-scurves. The stress levels at the maximum strain are well matcand so is the relaxation behavior depicted in Fig. 10. Theredifferences in the creep curves in Fig. 11. The simulations doalways show the same evolution of the slope, see the 200 andMPa simulations and compare with the experimental data.

DiscussionThe augmentation constant or function has enlarged the mo

ing capability of VBO considerably. The initial examples of uof the augmentation function pertain to pathological deformatbehavior of metals and to the modeling of rate sensitivity of sopolymers. Other examples are given by Ho@8#; Ho and Kempl@13#; Ho and Krempl@14#.

These examples show the versatility of the augmentation faand its influence on the modeling capability. When the augmtation factor is constant then the property modeled, say rinsensitivity, is the same during a deformation history. Ifb52exp@2500p#22, b varies between ‘‘zero’’ and ‘‘22,’’ therate sensitivity changes between normal and a negative valuthe inelastic path length increases.

The overall behavior of the stress-strain diagram is not signcantly affected, no unwanted oscillations and no instabilities hbeen observed.

In all the numerical relaxation experiments for the 9Cr-1Msteel the stress level at the end of the 24-hr relaxation periolowest for the highest prior strain rate. This is true whether r

Fig. 11 Simulation of creep behavior at 550°C; compare withFig. 13 of Yaguchi and Takahashi †19‡; Yaguchi and Takahashi†10‡



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insensitivity, negative or normal rate sensitivity is simulated. Tpredictions of VBO correspond to the experimental findings.

Numerical simulations are performed for various strain ratesdemonstrate that there is no limitation on strain rate. A comcated transition from the initial quasi-elastic behavior to the fudeveloped inelastic flow exists. While the transition varies wthe conditions, the asymptotic solution is well known and canused to check on the computed result.

The results reported so far were all correlation. With the cstants known the predictive capabilities of the model can becertained by running numerical experiments not used in therelation. Yaguchi and Takahashi@14# provide strain rate changdata. Figure 12 shows the prediction of VBO for a reductionstrain rate at different strains at a temperature of 550°C. Thedershoot predicted by VBO is larger than that of the experimeThe final stress levels are, however, identical.

Figure 13 shows the prediction of a strain rate reduction

Fig. 12 Prediction of strain rate change behavior at 550°C;compare with Fig. 12 of Yaguchi and Takahashi †10‡

Fig. 13 Prediction of strain rate change behavior at 400°C;compare with Fig. 13 of Yaguchi and Takahashi †10‡



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400°C. At this temperature rate-insensitivity was found in Fig.Both the experiment and the numerical prediction show an unshoot and then a return to the same stress.

ConclusionsThe extended viscoplasticity theory based on overstress~VBO!,

which has been conceived and developed by the authors, iscompetent to model several unusual forms of rate sensitivitygether with relaxation behavior. Positive, zero, and negativesensitivity of the stress-strain curve can be modeled using anmentation constant or function. Even if unusual rate sensitivityrepresented, the model predicts normal creep and relaxationhaviors. Application of the theory to the behavior of a modifi9Cr-1Mo steel showed excellent qualitative and quantitative meling of the mechanical responses including the predictionstrain rate changes at positive rate sensitivity and rate insensity.

Other applications of the augmentation function includemodeling of changes of the rate dependent cyclic hardeninglowed by softening or vice versa, the significantly increased haening at very high strain rates of interest in dynamic plasticity atertiary creep. These subjects will be discussed in forthcompapers.

AcknowledgmentThis work was performed while the first author was a visiti

research scholar at Rensselaer Polytechnic Institute. The seauthor acknowledges the support of the Department of EneGrant DE-FG02-96ER14603.

References@1# Pugh, C. E., Liu, C. K., et al., 1972,Currently recommended constitutiv

equations for inelastic design analysis of FFTF components, ORNL-TM-3602,Oak Ridge National Laboratory, Sept.

@2# Krausz, A., and Krausz, K., 1996,Unified Constitutive Laws of Plastic Deformation, San Diego, Academic Press.

@3# Mulford, R. A., and Kocks, U. F., 1979, ‘‘New observations on the mechnisms of dynamic strain aging and of jerky flow,’’ Acta Metall.,27, pp. 1125–1134.

@4# Kishore, R., Singh, R. N., et al., 1997, ‘‘Effect of dynamic strain aging on ttensile properties of a modified 9Cr-1Mo steel,’’ J. Mater. Sci.,32, pp. 437–442.

@5# Miller, A. K., and Sherby, O. D., 1978, ‘‘A simplified phenomenogical modfor non-elastic deformation: Predictions of pure Aluminum behavior andcorporation of solute strengthening effects,’’ Acta Metall.,26, pp. 289–304.

@6# Ruggles, M. B., and Krempl, E., 1989, ‘‘The influence of test temperaturethe ratchetting behavior of Type 304 Stainless Steel,’’ ASME J. Eng. MaTechnol.,111, pp. 378–383.

@7# Rao, K. B. S., Castelli, M. G., et al., 1997, ‘‘A critical assessment of tmechanistic aspects in Haynes-188 during low-cycle fatigue in the range oto 1000 C,’’ Metall. Mater. Trans. A,28A, p. 347.

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Documents

The Modeling of Unusual Rate Sensitivities Inside and Outside the Dynamic Strain Aging Regime