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Materials Science and Engineering A302 (2001) 197205

Temperature, strain rate, stress state and the failure ofHY-100 steel

V. Jablokov a,1, D.M. Goto b, D.A. Koss a,*, J.B. McKirgan c

a Department of Materials Science and Engineering, The Pennsyl6ania State Uni6ersity, Uni6ersity Park, PA 16802-5006, USAb Na6al Surface Warfare Center, Indian Head, MD 20903, USA

c Na6al Surface Warfare Center, Carderock, MD, USA

Received 27 June 2000; received in revised form 10 October 2000

Abstract

The influence of temperature and strain rate on the deformation and failure behavior of HY-100 steel has been examined as afunction of stress state using notched and un-notched axisymmetric tensile specimens. Behavior over the range of temperatures:strain rates from 85C and 1 s1 to 27C and 103 s1 shows an equivalence of decreasing test temperature or increasingstrain rate on deformation behavior in a manner that can be predicted by the thermally activated flow theory. Over the entirerange of temperatures:strain rates, the influence of stress state on failure is such that two void coalescence mechanisms controlfailure; at low stress triaxialities, relatively equiaxed voids grow to impingement, while at high triaxialities, a void-sheet processintervenes linking elongated MnS-initiated voids by a shear instability. The failure strains decrease rapidly with increasing stresstriaxiality ratio in a similar manner for all temperatures and strain rates except for an intermediate stress triaxiality conditionwhere the void-sheet mode of failure extends to lower stress triaxialities under cryogenic test conditions. 2001 Elsevier ScienceB.V. All rights reserved.

Keywords: Fracture; Temperature; Strain rate; Stress state; Failure behavior; HY-100 steel

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1. Introduction

An accurate description of the deformation and fail-ure response of structural materials over a wide rangeof loading conditions is critical to the success of compu-tational models that predict material failure under in-service conditions. To that end, several constitutivemodels have been developed to describe the deformationbehavior of materials as a function of temperature andstrain rate. Prominent examples of such models includethe mechanical threshold model (MTS) [14], the John-sonCook model [5], and the ZerilliArmstrong model[6]. Utilizing a number of constants, these models areable to link the flow stress to temperature, strain rate,and strain by fitting the constants to stressstrain dataover a range of temperatures and strain rates. In addi-

tion to the issue of the deformation response, predictingthe performance of a structural component must alsoaddress the failure conditions, taking into account notonly the imposed strain rate and temperature but alsothe stress state. Thus, for example, the combination ofan appropriate constitutive law combined with a localfracture criterion may be used in a computational anal-ysis to predict the performance of a large structure.

This study examines both the deformation and tensilefailure response of HY-100 steel over a range of tem-peratures, strain rates, and multiaxial stress states. Onegoal is to test the ability of a constitutive model, theMTS model in this case, to predict the equivalence ofdecreasing test temperatures with increasing strainrates. A second objective is to examine the failureresponse of this steel under low temperature:highstrain-rate conditions in which ductile, microvoid frac-ture still controls failure. Specifically, we address theissue of whether decreasing test temperature or increas-ing strain rate has any effect on a failure process thatremains a ductile, microvoid fracture event. As such,

* Corresponding author. Tel.: 1-814-8655447; fax: 1-814-8652917.

E-mail address: [email protected] (D.A. Koss).1 Present address: Siemens Westinghouse Power Corp., Orlando,

FL 32826, USA.

0921-5093:01:$ - see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0921 -5093 (00 )01832 -3

V. Jabloko6 et al. : Materials Science and Engineering A302 (2001) 197205198

Table 1Composition of HY-100 steel base plate used in the study

CElement Mn P S Si Ni Cr Mo Cu

0.26 0.008 0.009Wt.% 0.220.16 2.62 1.32 0.25 0.14

this study may be viewed as an extension of previousresearch examining the failure of HY-100 steel over arange of stress states but confined to quasi-static defor-mation rates at room temperature [7,8]. That studyidentified two ductile failure mechanisms: a void coales-cence process in which relatively equiaxed voids grew toimpingement at low stress triaxialities and a void-sheetmechanism that links large voids initiated at elongatedinclusions at high stress triaxialities. The effect of tem-perature and strain-rate on these failure mechanisms isexplored in this study.

2. Experimental and computational procedure

The material used in this study was HY-100 steelbase plate supplied by the Carderock Division of theNaval Surface Warfare Center with composition shownin Table 1. The heat treatment of the base plate mate-rial involved an austenitization at 900C for 1 hfollowed by a quench and temper at 630C for 1.5 h,resulting in a mixture of tempered martensite andbainite as shown in Fig. 1. Chemical banding is alsopresent in the microstructure due to segregation ofsubstitutional elements such as Ni and Cr [9]. The smallvolume fraction (0.015%) of MnS inclusions presenttended to concentrate within the Ni- and Cr-rich bands,which are elongated in the rolling direction [9]. TheMnS inclusions were either in the form of 12 mmdiameter equiaxed particles or, as shown in Fig. 1,elongated stringers (30100 mm long and 23 mmthick aligned parallel to the rolling direction).

All of specimens used in this study were machinedsuch that the loading direction corresponded to thelong transverse direction of the plate. To determine theinfluence of stress states on the failure initiation behav-ior, circumferentially notched tensile specimens weretested using four different notch acuities, each having adifferent radius of curvature, r, but the same initialminimum notch diameter constant (2R7.62 mm).The outer diameter of the notched specimens was equalto 15.24 mm while the smooth bar specimens had adiameter of 7.62 mm and a gage length of 25 mm. Thenotch geometry, as characterized by an R:r ratio, hadthe following range of values: R:r2.0 (D-notchspecimen), 1.0 A-notch specimen), 0.5 (B notch spec-imen), and 0.25 (E notch specimen). The specimenstested at dynamic strain-rates had the same geometriesbut were scaled down so that the minimum notchdiameter (2R) was equal to 4.47 mm.

Tensile testing at cryogenic temperatures was per-formed at 85 and 40C and at strain rates ofeither 103 or 1 s1. To calculate the cross-head speedrequired to produce the desired initial strain-rate, theeffective gauge length of the notched specimen wasassumed to be roughly equal to the radius of curvature,r. This assumption is based on finite element analysisthat showed that the axial strain in the notch decreasesrapidly at distances of approximately 90.5r from theminimum diameter section for all specimen geometries[10].

Material failure in this study is defined as that condi-tion in which the material damage is sufficiently severesuch that the stress-carrying capacity is measurablydegraded. Thus, using the procedure described else-where [7,8], failure initiation was determined experi-mentally as that point at which the load-diametriccontraction curve showed an abrupt drop during thetensile test. This test procedure had the sensitivity todetect roughly a 2% load loss due to damage accumula-tion, which is consistent with a 3% area fraction ofmicrovoids present on the fracture surfaces of speci-mens that were strained to failure and subsequentlyfractured by cleavage at liquid nitrogen temperatures[8]. In addition to the determination of failure strains,fracture strains, of, were also calculated using initial, do,and final diameter, df, measurements taken from thebroken tensile specimens and using the expression: of2 ln(do:df).

Fig. 1. Elongated MnS inclusion present within the tempered marten-site microstructure of the HY-100 steel.

V. Jabloko6 et al. : Materials Science and Engineering A302 (2001) 197205 199

Compression testing was also performed to obtainthe stressstrain behavior using cylindrical specimenswith a height and diameter equal to 6.35 mm. Prior toplacing the specimens on the platens, molybdenumdisulfide lubricant was applied on the specimen end-faces to decrease frictional effects. In order to obtain atotal strain value of 0.5, the test was performed intwo increments to decrease frictional effects. Followingthe first strain increment to a strain value of 0.25, thetest was interrupted, the samples were re-lubricated,and the second increment started until a total strainvalue of 0.5 was achieved on the specimens.

Having measured the specimen diameter contractionat failure initiation and knowing the stressstrain re-sponse of the material, we subsequently used finiteelement analysis to determine the local stressstrainconditions at the center of the minimum cross-sectionalarea of the notch where failure initiated. Axisymmetricfinite element analyses were performed for each speci-men geometry to determine the stress and strain statesat failure initiation at the center of the minimum cross-sectional area of the notch. The finite element analysiscode ABAQUS was used to determine the mean stress,sm, equivalent stress, seq, and equivalent plastic strain,oeq, to identify the failure initiation condition in termsof the stress triaxiality ratio, sm:seq and the equivalentplastic failure strain, of,eq at the failure initiation site.Fractography was performed after tensile testing usingscanning electron microscopy.

3. Results and discussion

3.1. Stressstrain response

The low temperature deformation behavior of a ma-terial may be analyzed in the context of thermallyactivated motion of dislocations such that materialstrain rate, stress, and temperature are coupled. Givensuch an analysis, the resulting deformation kinetics canbe predicted by an appropriate constitutive relationshipsuch as the Mechanical Threshold Stress, MTS, model[14]. In particular, the MTS model may be used topredict the equivalence in the stressstrain responsebetween a quasi-static test at cryogenic temperaturesand a high strain-rate, dynamic test at elevated temper-atures. Such an analysis suggests that deformation re-sponse at cryogenic temperatures and low strain-rates isequivalent to that under dynamic strain-rate test condi-tions at a higher temperature. Thus, one purpose of thisstudy is to test that hypothesis. Appendix A describesthe application of the MTS model to HY-100 steelutilizing only room temperature test data in order toobtain the appropriate constants. The discussion belowaddresses the ability of the MTS model formulated inthis manner to predict specific cryogenic temperatures

Fig. 2. Stressstrain behavior of HY-100 steel under five test condi-tions. Data at 25C and 4700 s1 is from Ref. [4].

and strain-rates in which the stressstrain responses ofHY-100 steel should be equivalent.

A comparison of stressstrain behavior of the HY-100 steel as determined experimentally at several tem-perature and strain-rate conditions is shown in Fig. 2.Addressing the issue raised above and as described inAppendix A, the MTS model predicts that the stressstrain behavior at 40C and 1 s1 and 85C and103 s1 should be equivalent. Fig. 2 supports thisprediction by showing that the stressstrain behavior at85C and 103 s1 coincides extremely well to thestressstrain behavior determined at 40C and 1s1. The MTS model also predicts that the deformationresponse at 85C and 1 s1 should be equivalent tothat at 25C at 7290 s1 (see Appendix A). Therefore,the stressstrain behavior at 85C and 1 s1 shouldbe almost equivalent to that at 25C and 4700 s1, acondition for which experimental data are available [4].As shown in Fig. 2, the experimental stressstrainresponse of HY-100 at 25C and 4700 s1 [4] agreesreasonably well with that at 85C and 1 s1. Asignificant difference between these two deformationconditions is an increased rate of strain hardening ofthe material when tested at 25C and 4700 s1. Never-theless, taken as a whole, the data in Fig. 2 support theMTS model in its ability to predict an equivalencebetween temperature and strain rate in controlling thestressstrain response of HY-100 steel.

3.2. Failure response

The effect of temperature and strain-rate on failure atdifferent stress triaxialities was determined through ten-sile testing of the circumferentially notched specimens.The computed local values of the failure strains atfailure initiation (at the center of the notch as deter-mined by finite element analysis) are shown as a func-tion of stress triaxiality ratio and test temperature inFig. 3a. The data show no effect of temperature onfailure strain except for the intermediate stress triaxial-ity B-notched specimens (R:r0.5 and sm:seq$0.850.90) in which case there appears to be a small decrease


in failure strains with decreasing temperature. In addi-tion, at a given temperature, the results in Fig. 3a showno effect of strain rate on failure; the failure strains at103 s1 strain rate are the same as those at 1 s1 fora given specimen geometry.

The fracture strains, as determined from the initialand final diameters of the specimens, are shown in Fig.3b for quasi-static strain rates over a range of tempera-tures as well as for dynamic strain-rate tests of notchedspecimens at 25C. As in the case of the failure strains,the fracture data in Fig. 3b reveal that at intermediate(A-notch) to high (D-notch) stress triaxialities the frac-ture strains are not sensitive to temperature or strain-rate, even at high strain rates of :103 s1. At lowstress triaxialities (B-notch), however, there is a slightdecrease of the fracture strains with decreasing temper-ature, as was observed in the case of the failure straindata.

In summary, these ductility data show that, exceptfor the intermediate stress triaxiality (B-Notch) speci-mens, both the failure and the fracture strains show nosensitivity to temperature or strain rate for a givennotch geometry over the range of conditions from 103

s1 and 25C to 1 s1 and 85C. In the case of the

Fig. 4. A failure limit diagram showing the equivalent plastic strain-to-failure and average stress triaxiality ratio combinations at failureinitiation for HY-100 steel.

B-notch specimens, there appears to be a small increasein ductility with increasing temperature.

An alternate method of viewing the failure strainresults is in the form of a failure limit diagram, whichdepicts the equivalent plastic strain at failure initiationand average stress triaxiality ratio for a given specimengeometry. As shown in Fig. 4, the failure limit diagramfor the HY-100 steel tested in the long transversedirection may be interpreted in terms of the presence oftwo distinct failure regions [7,8]. At low stress triaxiali-ties (sm:seqB0.85) corresponding to the Region I fail-ure locus, the failure strain has a strong sensitivity tothe stress triaxiality. Previous research indicates thatfailure in this region is dictated by the coalescence ofequiaxed voids [7,8]. At higher stress triaxialities (sm:seq\0.85) corresponding to the Region II failure locus,a weak sensitivity of failure strain on stress triaxiality isobserved. At least at quasi-static strain rates, previousfractographic results link this region to void coalescenceby a void-sheet mechanism. In this region, elongatedinclusion-initiated voids induce a deformation localiza-tion between them, resulting in rapid void linking and azig-zag fracture surface.

As depicted in Fig. 4, the cryogenic and high strain-rate tests indicate that while the failure strains are verydependent on the stress triaxiality ratio, they are rela-tively independent of the strain-rate and temperature.The principal difference between the failure behavior atambient and cryogenic conditions is at the intermediatestress triaxiality condition (B-Notch: sm:seq:0.850.90) where the Region II mode of failure appears to beextended to the lower stress triaxiality condition.

3.3. Fractography

Based on failure at room temperature under quasi-static strain rates [7,8], previous research has identifiedvoid coalescence by the void-sheet mechanism as theprocess controlling the Region II failure in Fig. 4. Inthe present study, fractography performed for a widerange of temperatures:strain rate conditions (Fig. 5),

Fig. 3. The dependence of (a) the fracture strain and (b) the failurestrain on test temperature and strain-rate. See text for definitions offracture and failure strains.


indicates that the fracture process for Region II contin-ues to involve the void-sheet linking of elongated voidsnucleated at large MnS (1575 mm long) stringers.For example, fractographs taken in the failure initiationregions of the high stress triaxiality (D-Notch) speci-mens tested over a wide range of conditions, such as atthe 85C and 1 s1 and 25C and 103 s1 casesshown in Fig. 6(a,b), indicate failure due to void linkingby a void-sheet mechanism. Fig. 6 as well as profileviews of the fracture surface depict shear planes con-necting the large, elongated MnS-stringer-nucleatedvoids and forming a ridgetrough profile [7,8].

As analyzed by Bandstra et al., void-sheet failure is aresult of a localized shear instability that developsbetween the two neighboring, elongated primary voids[12,13]. The void-sheet deformation localization processalso results in the eventual nucleation of secondarymicrovoids within the deformation localization bandjoining the primary voids. Thus, Fig. 5 also shows thepresence of (a) secondary microvoids of a few micronsin size nucleated at equiaxed MnS particles near theprimary voids and (b) even smaller microvoids (1 mmin diameter) nucleated at smaller, equiaxed (pre-sumably) carbide particles.

Compared to Region II failure, the Region I failurelocus in Fig. 4 displays a stronger sensitivity of failurestrain on the stress triaxiality for the lower stress triax-iality (sm:seqB0.85) specimens. In Region I, failureinitiation at room temperature and quasi-static strainrates occurs by damage accumulation consisting of thenucleation of relatively equiaxed voids, their growth,and coalescence [7,8]. After specimen fracture, large,deep equiaxed primary voids, indicating extensive voidgrowth, are present in the failure initiation region of thefracture surface. Fig. 6(c) shows a typical example ofthis mode of fracture, specifically in this case for thelow stress triaxiality B-notch condition at roomtemperature.

In contrast to the fracture surface shown in Fig. 6(c)for the B-notch specimen failed at 25C and 103 s1,B-notch specimens tested at 85C and 1 s1 showevidence of extensive void-sheeting as indicated in Fig.6(d); no large equiaxed primary voids are present.Void-sheeting was also the predominant mode of voidlinking for low stress triaxiality (B-Notch) specimenstested at 40C and 1 s1, 85C and 103 s1, and25C and 1000 s1. Thus, as summarized in Table 2,fractography indicates that under cryogenic conditionsand at high strain-rates, the void-sheet mode of failurereplaces the equiaxed void coalescence process at theintermediate stress triaxiality. This evidence supportsthe hypothesis that the loss of ductility evident in thefailure limit diagram (Fig. 4) for B-Notch intermediatestress triaxiality specimens is caused by the extension ofthe void-sheet mode of failure at these test conditions.

We can only speculate as to why decreasing tempera-ture:increasing strain rate promotes the void-sheetmode of void coalescence. It is well known that the flowstress of bcc alloys becomes increasingly sensitive totemperature as test temperature decreases. Thus, it istempting to suggest that small amounts of deformation-induced heating occur on a local scale as flow begins tolocalize; this process promotes further deformation lo-calization, especially at cryogenic temperatures wheresmall increases in temperature result in large decreasesin flow stress. Since void linking by the void-sheetprocess is known to depend on the development ofdeformation localization between the primary voids[1114], a localized adiabatic deformation process,such as is speculated here, could explain an extension ofthe void-sheet process to cryogenic temperatures andhigh strain rates.

Finally, Table 2 also indicates that fractography per-formed on the samples tested at cryogenic temperaturesof 85C reveals the presence of small amounts oftransgranular cleavage in certain areas of the fracturesurface. For the high triaxiality (D-Notch) specimenstested at 85C, 103 s1 and 1 s1, and intermedi-ate (A-Notch) stress triaxiality specimens tested at 85C and 103 s1, transgranular cleavage was

Fig. 5. A fractograph showing the fracture surface associated with thevoid-sheet mode of failure. The large, elongated MnS stringer respon-sible for nucleating the primary void is evident as are secondarymicrovoids associated with small equiaxed MnS particles.


Fig. 6. Fractographs from the failure initiation region revealing extensive void-sheeting in the failure initiation region of a high stress triaxialityspecimen (D-notch) tested at (a) 25C and 103 s1, (b) 85C and 1 s1. In contrast (c) shows coalescence of equiaxed voids in a lowtriaxiality specimen (B-notch) tested at 25C and 103 s1, but (d) extensive void-sheeting between elongated voids in the same B-Notch specimenconfiguration when tested at 85C and 1 s1.

confined to the crack propagation region near the baseof the shear lip on the fractured specimen. Only occa-sionally were cleavage facets found within isolatedgrains (1020 mm diameter) that were surrounded byductile fracture in the failure initiation region.

3.4. The influence of stress state on failure and a6oid-growth relationship

As recognized by previous researchers [14,15] andunder conditions where voids nucleate at small strains,

the strong sensitivity of the failure on stress statesuggests ductility limited by void growth. It is wellknown that the rate of void growth is predicted to bequite sensitive to the degree of stress triaxiality [16,17].In our case, the MnS inclusions nucleate voids at verysmall strains [18], indicating that failure is governed byvoid growth and coalescence. If void growth controlsfailure, a relationship of the following form should fitthe Region I and Region II failure data [14]:

ofK exp(Dsm:seq) (1)

Table 2A summary of the fractography for all of the test conditions

Room temperature 85C40C(1 s1)(103 s1 and 103 s1) (103 s1 and 1 s1)

Same as room temperature inSame as room temperatureHigh (D-Notch), of unaffected Extensive void-sheeting; nofailure initiation region; somecleavagecleavage (4%) in propagationarea

Same as room temperatureMixture of void-sheets and Same as room temperature butMedium

sm

s(A-Notch); of equiaxed primary voids; no some 1020 mm grains cleaved

cleavageunaffected in the failure initiation regionat 1 s1; at 103 s1, cleavageobserved in propagation area

Same as room temperature butLarge equiaxed primary voids, At 1 s1, extensiveLow

sm

s(B-Notch); of decreased at void-sheeting presentextensive void growth, and void-sheeting and no cleavage;

at 103 s1, mixture ofsuppressed void-sheeting at 103low o; and temperatures1; void-sheeting present at 102 void-sheets and equiaxed

primary voids: some cleavages1

(40 mm) in initiation regionalso present


Eq. (1) has the form similar to the void growth modelof Rice and Tracey [16] and has been used previously insimilar forms to correlate the tensile ductile fracturedata for many different materials [5,8,14,19,20]. It willbe recalled that the RiceTracey void growth analysisassumes an isolated spherical void and neglects voidinteractions, and that the analysis predicts a value ofthe exponent D of 1.5 [16].

Fig. 7 shows that the stress-state dependence of thefailure of HY-100 steel can indeed be predicted for awide range of temperature:strain rates by a relationshipof the form given by Eq. (1), pro6ided that the data areseparated into Regions I and II. However, as is ob-served for this steel at room temperature [8], the D-val-ues obtained when fitting Eq. (1) to the experimentaldata are significantly higher than that predicted by theRiceTracey analysis (experimentally, D2.7 for Re-gion I and D2.3 for Region II, which compares tothe D1.5 predicted [16]).

There are several possible causes for the difference inD-values between the RiceTracey prediction of 1.5and the observed values of 2.32.7. It is possible that,despite the small volume fraction of voids (512%)just prior to failure [18], significant voidvoid interac-tions occur and accelerate void growth, increasing theD-value. However, given the small inclusion content ofthis steel (volume fraction:0.00015) the inclusion-nu-cleated voids would have to interact at distances inexcess of 10 void diameters. Recent computationalmodeling of void-sheet mode of failure indicates strongvoid interactions can in fact occur between cylindricalvoids 2.5 mm in diameter but spaced 70 mm apart(approximately 28 void diameters) [11,12]. In addition,recent experimental measurements of damage accumu-lation during ductile fracture of HY-100 steel show thepresence of a rapid void growth stage initiating at 51vol.% voids, which also suggests stronger than pre-dicted interactions between widely separated voids [18].Thus, we conclude that the use of a relationship such asEq. (1) is a reasonable approach to predicting thestress-state dependence of failure of steels such at HY-100 even though significant void interaction effectsappear to be present.

4. Summary

Relying on constants developed based on room tem-perature deformation behavior, the MTS model pre-dicts the cryogenic stressstrain response of HY-100steel with reasonable accuracy. Specifically the modelpredicts that the stressstrain behavior at 40C and1 s1 and 85C and 103 s1 should be equivalentas should that at 85C and 1 s1 and 25C and 7290s1; our experimental observation agree fairly well withthese predictions.

The tensile failure behavior (i.e. the mechanical con-ditions at failure at different stress states as well as theunderlying void coalescence mechanisms) of HY-100steel is, for the most part, insensitive to strain-ratesover six orders of magnitude in strain rate and totemperatures ranging from 25 to 85C. The onlysignificant effect on ductility occurs at an intermediatestress triaxiality ratio in which there is a small decreasein failure strains at high strain rates:low temperatures.That ductility decrease is associated with void coales-cence as a result of a void-sheet process in which thehigh strain-rate:cryogenic temperature condition favorsa deformation localization between elongated inclusion-initiated voids.

When the failure data are taken as a whole, thedependence of the failure strains on stress state can beadequately predicted by a relationship of the formpreviously developed by Rice and Tracey for voidgrowth [16]. Such a relationship is consistent with voidnucleation occurring at sulfide particles at small strainsin this steel, and, although the form of the relationshipsuggests the presence of void interaction effects, its useimplies a critical void volume fraction or critical voidgrowth rate as a failure criterion.

Acknowledgements

The authors are very grateful to Carl Cady, RustyGray, Rich Thissel, Shuh-Rong Chen, Mike Lopez, andManny Lovato at the Los Alamos National Laboratoryfor their generous assistance in performing the mechan-ical tests at cryogenic temperatures. We also wish toacknowledge many discussions with Dongchul Chae.This research was supported by the Office of NavalResearch.

Fig. 7. The dependence of failure on stress triaxiality as measuredexperimentally and predicted from Eq. (1) for HY-100 steel.


Table 3HY-100 steel MTS model parameters used for determining the appro-priate cryogenic temperature to emulate a dynamic test at 100 s1

UnitsValueParameter

sa 50 MPamo 7.14610

4 MPaMPa K10.9047k:b3

1109o; oi s10.698423goi3:2qi

pi 1:2MPa1338si

1107o; oo s11.6goo1qo

po 2:3MPa20 000uo

6a1.6goos

MPa1000sosoo; oos s11107

given strain rate or temperature and the flow stress at 0K) are expressed below in general form as:

Sj(o; , T)sj:msj:mo

!

1 kT

gojmb3lno; ojo;m1:qj"1:pj

(A.3)

For HY-100 steel, two effective barrier types areconsidered: short-range barriers due to solute and inter-stitial atoms (Si and si), and long-range barriers consid-ering other dislocations and carbide particles (So andso). A minor contribution due to athermal strengthen-ing, e.g. grain boundary, is given by sa.

Plastic strain is implicitly represented in the MTSmodel framework. The MTS framework considersstress as essentially characterizing the material mi-crostructure (stress is an internal state variable) andstrain as describing how the microstructure evolves withincreasing deformation. As such, plastic strain o is givenin terms of stress and can be expressed as:

otanh(a)

uo(tanh2(a)1)so tanh(a)

sos

aln

tanh(a) coshasosos

sinh

asosos

m)soi

sof

(A.4)

where

sos soso exp kTmb3goos

ln o;o; oos

(A.5)

The stress interval, soiB soB sof, is assumed to corre-spond to a constant deformation path, i.e. constantstrain-rate, temperature, stress-state, etc.

Determination of the stressstrain behavior requiresthat the appropriate model parameters are inserted intoEqs. (A.1), (A.2) and (A.3) to calculate the initial flowstress at zero-plastic strain. In the present study, theseparameters as listed in Table 3 have been determinedbased on the room temperature response of HY-100steel. Plastic strain is then determined by applying themodel parameters, given in Table 3 to Eqs. (A.3), (A.4)and (A.5), and incrementally increasing so.

Predicting corresponding temperature:strain-rateconditions using the MTS model relies primarily onevaluating the thermally-activated component associ-ated with short-range obstacles, Si and si. The rate-sen-sitivity of long-range obstacles, in the current HY-100MTS model description, is secondary in comparison toshort-range obstacles. In other words, yielding behavioris markedly more sensitive to temperature and strain-rate than is strain-hardening. In short, an Arrhenius-like diagram is constructed to illustrate the dependenceofsysa

m

Pion

Appendix A

The Mechanical Threshold Stress (MTS) model hasbeen previously employed to determine flow stress be-haviors at different temperaturestrain rate conditions[1,21]. This strength model is based on thermally-acti-vated dislocation motion and references a mechanicalthreshold stress, or the stress necessary to move adislocation in the absence of thermal energy, i.e. at 0 K.Thermal effects on dislocation motion are incorporatedthrough an Arrhenius-based expression which relatesflow stress to strain-rate and temperature, thus givingrise to rate-sensitive constitutive behavior.

The current study has applied the MTS strengthmodel in order to determine the appropriate cryogenictemperature at which the flow stress at room tempera-ture and a dynamic strain rate (103 s1) can beemulated via a quasi-static (103 s1) mechanical test.The form of the strength model and model parameterdescriptions used for HY-100 is presented in detailelsewhere [22]. The MTS model for HY-100 steel issummarized below (see also Table 3) and is given as:

s

m

sa

mSi(o; , T)

simo

So(o; , T)

somo

(A.1)

where sa is the athermal stress component, and thetemperature-dependent shear modulus, m, is given by:

m [MPa]mo2910 MPa

exp204 K

T

1(A.2)

where mo is the shear modulus at 0 K. The correctionfactors that account for the thermal effects, Si and So(essentially the ratios between the component of flowstress attributable to a particular obstacle type at any


kTmb3

lno; ojo;m1:qj

This plot provides a relatively straightforward methodof determining temperature and strain-rate combina-tions which yield identical values ofsysa

m

Piand was used in the current study to identify that thetemperature:strain-rate combinations of 40C and 1s1 produced essentially the identical flow stress behav-ior as 85C and 103 s1. In addition, the deforma-tion response at 85C and 1 s1 should beequivalent to that at 25C and 7290 s1. The textdescribes our experiments to verify the equivalence ofthese temperature:strain-rate conditions.

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