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A state variable description of mechanical propertiesD. Stone, Che-Yu Li

To cite this version:D. Stone, Che-Yu Li. A state variable description of mechanical properties. Revue de PhysiqueAppliquee, 1988, 23 (4), pp.639-647. �10.1051/rphysap:01988002304063900�. �jpa-00245812�

https://hal.archives-ouvertes.fr/jpa-00245812https://hal.archives-ouvertes.fr

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A state variable description of mechanical properties

D. Stone and Che-Yu Li

Department of Materials Science and Engineering, Cornell University, Bard Hall, Ithaca,NY 14853-1501, U.S.A.

(Reçu le 15 juin 1987, accepté le 9 novembre 1987)

Résumé.2014Une approche utilisant des variables d’état pour décrire la déformation inélastique d’unsolide cristallin est traitée. Cette approche est basée sur la formulation developpée par Hart[1]. Nous démontrons qu’elle décrit des processus de déformation communément observés tels que lefluage. Par contre, la description d’autres processus tels que: déformation inhomogène ou influ-ence de l’interaction entre dislocations et atomes en solution sur le processus de déformation va

au-delà du but recherché par cette théorie.

Abstract.--A state variable approach based on Hart’s [1] formulation for describing nonelasticdeformation of crystalline solids is reviewed. It is shown to describe commonly observed deforma-tion phenomena, such as creep. Presently, deformation phenomena such as inhomogeneous flow andthose involving strong dislocation-solute interactions are not within the scope of this approach.

Revue Phys. Appl. 23 (1988) 639-647 AVRIL 1988,Classification

Physics Abstracts81.40L - 62.20F - 62.20H - 62.205 - 61.70L

The nonelastic properties of a crystallinesolid depend upon prior deformation and thermalhistory. To fully account for the history effectscan be a cumbersome task in stress analysis andmechanical testing. If the deformation propertiescan be demonstrated to be uniquely characterizedby state variables, and if the manner in which theproperties evolve along an arbitrary thermal ormechanical path can be shown to depend upon thestate variables as well, time dependent stressanalysis and materials testing can be greatly sim-plified.

The appropriate state variables are not easilyidentified based solely on theoretical arguments.The experimental effort required to establish themcan also be difficult. In part this is because a

variety of mechanisms contribute to nonelasticdeformation, and the delineation of these mechan-isms requires extensive experimental effort. Forexample, the separation of the grain boundarysliding contribution from grain matrix deformationduring creep at elevated temperatures is an im-portant but difficult task.

This paper reviews a phenomenological approach,originally proposed by E.W. Hart designed to es-tablish state variable flow relationships based ondirect experimental measurements without first re-sorting to a microscopic t"heory of deformation[1]. Many of the state variables involved in thisapproach have been given physical significance,but no general theory based on microscopic pro-cesses has been developed.

REVUE DE PHYSIQUE APPLIQUÉE. - T. 23, N° 4, AVRIL 1988

The existence of state variables characterizingthe structural state of a material is a fundamen-tal basis of Hart’s formulation. In this approachthe load relaxation test has been used extensivelyto determine flow relations at a constant (ornearly constant) structure. It has been foundthat the logarithmic plots of stress versus strainrate, obtained from load relaxation tests of spec-imens with different degrees of workhardening, canbe translated along a linear path to overlap eachother. This is the so-called scaling relation-ship, not hitherto predicted by dislocation basedtheories [1]. Mathematically it means that theplots belong to a one-parameter family of curves,a concept first proposed by E.W. Hart. Thisparameter is called the hardness, a*, and is astate variable. It is uniquely defined by thecurrent value of stress, strain rate, and tempera-ture, which are other state variables. In prac-tice, it is not only important to know the exis-tence of a*, but also to be able to experimentallymeasure this parameter.

In contrast, often-used flow laws relating min-imum creep rate to applied stress are obtained ex-perimentally without the assurance that the stressdependence of the creep rate corresponds to thatof a constant structural state. In fact, such acorrespondence is highly unlikely. To interpretthe minimum creep rate data additional informationon the structural state of the material and itsvariation during creep are required.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01988002304063900

http://www.edpsciences.orghttp://dx.doi.org/10.1051/rphysap:01988002304063900

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Flow stress-strain rate data obtained in loadrelaxation tests covering a wide range of strainrates and temperatures are illustrated in log-logform in Figures 1 and 2 [2,3]. One striking fea-ture of these data is that the strain rate sensi-

tivity, as defined by dlog a/dlog 03B5, varies withtemperature and stress. Superficially, the loadrelaxation data differ significantly from creepdata presented in familiar power law form. A pur-pose of this paper is to demonstrate that theconstant structure load relaxation data in Figures1 and 2 are consistent with deformation data ob-tained in other experiments. A good example isthe creep data reported in the form of creep maps,as shown in Figure 3 [4].

It is important to note that, in addition toconstant structure load relaxation data that spec-ifies o* and the flow relations, one also requiresa workhardening law to establish the evolution ofa* along an arbitrary mechanical path [5]. Aworkhardening law does not a priori involve thestate variables c* and o. If the workhardeningrelationship is not a sole function of state vari-ables, it will not be easy to integrate the defor-mation equations along an arbitrary path in astress analysis. In this case, experiments wouldbe required to determine the workhardening be-havior for each deformation path.

A State Variable Approach

The most basic phenomenon to have been charac-terized by state variables is time-dependent,

Figure 1. Constant structure load relaxation dataof 316 Stainless Steel at several temperatures.The strain rate sensitivity is seen to be a func-tion of strain rate and temperature. After ref.

[2].

plastic deformation in polycrystalline metals in-volving grain matrix processes only. The experi-mental basis for establishing the state variablespertaining to plastic deformation will be reviewedinitially.

Figure 2. Load relaxation data of Al at constantstructure, after ref. [3].

Hart [6] first realized that the load relaxa-tion experiment can be used to obtain flow stress-strain rate data at a constant Structure. Suchdata usually cover a five aecade strain rate win-dow with minimal scatter, revealing the character-istic shapes of flow curves [7]. To begin a loadrelaxation run, the specimen is loaded at a con-stant displacement rate, preferably in a screwdriven machine, to a predetermined level of load.The crosshead is then fixed and the load allowedto relax as the specimen continues to deform. Asthe load relaxes the strain rate decreases. Therate of load relaxation is related to the plasticstrain rate of the specimen through the machineconstant, or stiffness. For a machine of highstiffness, the total elongation required to sig-

Figure 3. Constant shear strain rate contours ofNi for steady state-like (SSL) power law creep.See ref. [4].

nificantly relax the load is small, on the orderof 0.1% to 0.3%, such that little workhardeningoccurs. Thus, the specimen is able to maintain anearly constant structural state provided time-dependent processes, such as thermal recovery andstrain aging, are not important.

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In the case that structural changes or changesin the rate parameters are suspected to play arole, one can perform a reloading experiment todetect any time-dependent change in the structuralstate [6]. The reloading is made to a level ofstress below the initial stress of the previousrun. If, after a transient period, the reloadingdata are found to converge onto the previous loadrelaxation curve, one may conclude that the struc-tural state or the rate parameters have remainedunchanged during the initial relaxation run. Al-though the load relaxation test does not guaranteeconstant structure flow data, the reloading relax-ation run can be used to determine whether time

dependent effects are present. Figures 4 and 5show initial and repeated load relaxation data ofFerrovac-E Iron and an Fe-0.10% Ti alloy, respec-tively, at T = 30°C [8]. In the former case thereloading data do not approach the initial loadrelaxation data at low strain rates. This is be-lieved to be caused by the effects of strain aging

Figure 4. Loading and reloading relaxation dataof Ferrovac-E Iron demonstrating the case wherereloading data do not overlap initial relaxationdata due to strain aging. After ref. [8].

Figure 5. Loading and reloading relaxation dataof Fe-0.10% Ti alloy. These data demonstrate thecase where the structure of the material remain

unchanged throughout relaxation. After ref. [8].

during load relaxation. In the case of the Fe-0.10% Ti alloy, the reloading data fall well ontop of the initial relaxation data suggesting theabsence of time-dependent changes.

The availability of constant structure load re-laxation curves with their clearly defined shapeshave allowed the discovery of the scaling rela-tionship. Figures 6 and 7 are typical load relax-ation data obtained at a high and low homologoustemperature, respectively [9,10], exhibiting scal-ing in that the individual curves can be rigidlytranslated along a straight line to overlap oneanother.

The high homologous temperature data, Figure 6,show a characteristic concave downward shapewhereas the low temperature data are concave up-ward in shape. Composite curves resulting fromtranslations according to the scaling relation-ships are depicted in Figures 8 and 9. Because ofscaling, the curves in Figures 8 and 9 can be con-sidered to belong one parameter families ofcurves. This parameter is the hardness parameter,a*, which is uniquely specified by a constantstructure load relaxation curve. Mathematically,each relaxation curve is represented by a statevariable relationship a* - f(a,c) such that ;, theplastic strain rate and a, the applied stress, areestablished to be state variables. For conven-

ience, a* is measured as the high strain ratelimiting stress of the high homologous temperaturecurves or the low strain rate limiting stress ofthe low homologous temperature curves [2]. For amaterial at a constant structure, a* has the samevalue for both the high homologous temperaturedata and the low homologous temperature data. Itis a scalar and depends weakly on temperature suchthat a*/G is independent of temperature, where Gis the shear modulus. Physically, a* reflects thedislocation cell structure, and has been linked tothe dislocation cell size through a Hall-Petchtype relationship [9].

At intermediate temperatures, such as for the300°C curve in Figure 1, simple scaling is notpossible because both high and low temperaturemodes of deformation contribute and the scalingdirection is different for each mode [2].

For a wide range of crystalline solids, includ-ing ceramics, constant structure load relaxationdata are generally found to display two basic de-formation characteristics, depending, as illus-trated in Figures 8 and 9, on the homologous tem-

Figure 6 . Load relaxation data of Al at two levelsof Ce, demonstrating the scaling directmon at highhomologous temperatures. The load relaxation dataexhibit a concave downward shape. After ref. [9].

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Figure 7. Load relaxation data of 304 Stainlesssteel at two levels of Q*, demonstrating thescaling behavior for low homologous temperatures.The curves are concave upward. After ref. [10].

perature. Hart has associated the high homologoustemperature behavior with dislocation climb andthe low homologous temperature behavior with dis-location glide and has proposed a deformationmodel [1,11]. In this model the driving force fordislocation climb is cr which is a state variable

a

and is close to the applied stress at high homol-ogous temperatures. For dislocation glide, o- isthe driving force, which is the difference betweenthe applied stress and cra. The flow equationshave the forms:

for high and low homologous temperatures, respec-tively, where 03B5* and a* are rate parameters, and Àand M are pure numbers. At low homologous temper-atures or high strain rates, o s a*. The scaling

Figure 8. Composite curve of Aluminum constructedbased on the translation of the load relaxationdata in Figure 6 along the scaling direction. Thesolid curve has the form of Equation 2, with a* =

Figure 9. Composite curve of 304 SS constructedbased on the translation of the load relaxationdata in Figure 7 along the scaling direction. Thesolid curve has the form of Equation 1, with a* =

199 MPa, a*. = 9 x 10 38 /sec, and M = 14.relationship for high homologous temperatures isfound experimentally to be of the form:

Here, e* is a frequency factor, m is a constant,which i s found experimentally to be about 5 forseveral metals [1,12]. Q is experimentally ob-served to correspond to the activation energy fordiffusion. Depending on temperature, Q may varyreflecting either volume diffusion or pipe diffu-sion as the dominant mechanism. The temperaturedependence of a* at low homologous temperatures isfound to be weak compared to that of E* [2].

The slope of the scaling path, the line alongwhich the load relaxation curves can be translatedto overlap one another, is 1/m for Equation 1 and1/M for Equation 2. The high homologous tempera-ture scaling can be verified by noting thatdlog 03B5*/dlog a* = m. A similar considerationshows that 1/M is the slope of the scaling path atlow homologous temperatures. The shape factor X,which, mathematically, determines how sharply con-cave downward the high homologous temperaturecurve turns at low strain rates, appears to be aconstant with a value of approximately 0.15. Thephysical significance of this number remains to beexamined. M, which defines the shape of the lowhomologous temperature data, is generally found tovary between 7 and 20 [12].

At high homologous temperatures, the strainrate becomes singular as oa approaches a*. This

singularity can pose difficulties during numericalintegration. The functional form of Equation 1appears, however, to have fundamental signifi-cance, so that caution should be exercised when

attempting to modify the form of Equation 1 fornumerical convenience.

The form of Equation 1 is not accountable basedon current single dislocation based deformationtheories. Although the detailed mechanisms remainto be developed, it is apparent that this form isinsensitive to detailed atomic and dislocationstructures. This points to deformation processesinvolving collective motion of dislocations withwell defined statistics [13].

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The hardness parameter, o* will increase as aresult of deformation. To describe this evolution

along an arbitrary mechanical path requires a work-hardening relationship that can not be assumed,without experimental verification, to be a func-tion solely of state variables 6* and oa. Inde-

pendent experiments have been performed, however,to demonstrate the adequacy of these state vari-ables to describe workhardening [5,14]. The formof the workhardening law is established based ondata obtained from a combination of load relax-ation and tensile tests.

The absolute workhardening parameter, definedas:

has been demonstrated to be of the form [5,11]:

where A and a0 are constants and 6 is a pure num-ber. At low homologous temperatures, when defor-mation is controlled by glide-friction, Qa/Q* = 1.It can be shown that, for 03C3o ~ 0, at these temper-atures 1/6 is the conventional strain hardeningexponent. Based on the Considère condition, thevalue of a* is A when a specimen begins to neck ina tensile test.

At high homologous temperatures r becomes sen-sitive to strain rate, which is viewed as a mani-

festation of dynamic recovery. An explicit ratedependence can be obtained by substituting theexpression for aa/a* in terms of e*/E based onEquation 1. If we define r* = (A/(03C3*-03C3o))03B4 and a*= 03B5*(0393*)1/03B403BB , Equation 5 becomes [5]:

Like the hardness parameter, a*, r* is an athermalterm and a*/s gives the rate dependence of r. Athigh strain rates, r = r*, which governs workhard-ening under conditions where glide friction con-trols. The form of the workhardening function(Equation 6) is consistent with current conceptson workhardening that the rate independent partreflects the mechanical effects alone, whereas therate dependent part results from the effects ofdynamic recovery [5].

Applicability of the State Variable Approach

The yield stress of a polycrystalline solid isoften reported as a function of temperature.These types of data correspond to those repre-sented by equations 1 and 2 with the appropriateactivation energies. This is a straightforwardcomparison [12]. Sometimes the ultimate tensilestress is reported. In this case, at least forlow homologous temperatures, the Considère cri-terion can be used based on the workhardeningcorrelation. For creep data, a somewhat moreinvolved discussion will have to be made.

In principle, the flow relationships may beintegrated to describe plastic flow along an arbi-trary deformation path. In fact this has beendone for a variety of situations [12,15]. Thesedetailed analyses are outside the scope of thepresent paper. Instead, a part uf a deformationmap created by Frost and Ashby [4] will be ex-amined without resorting to a detailed analysis toillustrate the applicability of the present statevariable approach. In particular, we are inte-rested in the power law creep regime of the defor-mation map for nickel, Figure 3. Each curve ofthe map in this regime is the locus of temperatureand flow stress required to produce a particularsteady state-like (SSL) creep rate. Above about

T/Tm = 0.6, where T m is the absolute melting tem-perature, the data shown correspond to a SSL creeprate having a fifth power dependence on stress andan activation energy of volume self-diffusion.Below about 0.5 T/T m a higher stress exponent isobserved along with an activation energy corres-ponding to pipe diffusion. The stress dependenceat low temperatures reflects, in part, that of

pipe diffusion (a Q2 dependence [4]). By exclud-ing this effect, therefore, in both temperatureranges the SSL creep data can be considered tohave the same stress dependence.

The fifth power stress dependence of the SSLcreep rate coincides with the stress exponent ofthe scaling relationship, Equation 3 at highhomologous temperatures. This correspondence canbe taken as a convenient starting point for thediscussion.

We note that along the scaling direction, oa/Q*and 03B5/03B5* are constants. The form of the scalinglaw is the same as that for power law creep, sug-gesting that SSL creep occurs for a value of ala*,that is (at least to a close approximation) insen-sitive to temperature and strain rate. Thus, athigher applied stresses, higher values of a*operate during SSL creep. If we substitute 03B5*/03B5 =constant into the workhardening relationship,Equation 6, we obtain an expression relating r tor* at SSL creep

The above equation, along with the workhardeningrelationships, Equations [5] and [6], can be usedto examine how SSL creep is established. Figure10 shows typical creep curves for an annealedmaterial at high (a»ci), intermediate (03C3 ~ a*),and low (a«a*) initial stresses respectively (oiis the initial hardness). According to the pres-ent view, the shapes of these curves--that is, theextent of primary creep--is determined mainly bythe workhardening relationship and does not re-quire the presence of thermally induced, staticrecovery. During the primary creep regime, thespecimen workhardens so that the value of o* in-creases. According to Equations (5) and (6), therate of work hardening decreases at high c* lead-ing to a SSL behavior. This process can be veri-fied by numerical integration of Equations 1, 2,and 5 to simulate experimental data [12,15,16].

Consider that, in Figure 10, the initial hard-nesses, ci, of the three specimens are the same,near the level of the intermediate stress. The

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high stress is of the order of A in Equation 5.For a high applied stress, the initial r* is largebecause of the low initial a*. According to Equa-tion 6, the initial value of r is large also be-cause of the high initial strain rate. Thus, ini-tially c* increases rapidly, which reduces Ë. Asthe rate of work hardening decreases, a SSL creepbehavior results. At SSL creep, r/r* is large,but r* is significantly lower than its initialvalue. A large transient stage, or primary creep,is expected before the occurrence of SSL creepbecause of the large increase in a*.

At an intermediate applied stress, the initialvalue of r is smaller because of a lower initialstress or strain rate. Thus the value of a* neverbecomes very large. The value of r will, however,become sufficiently low to cause SSL behavior, buthere the final value of r/r* (according to Equa-tion 7) is small as r* remains high due to low a*.The small change in a* leads to a shorter tran-sient stage as experimental data demonstrate [17].

At the lowest stress (assuming this stress issufficiently low) r is not large enough to produceappréciable workhardening. Because the materialcan not workharden, this behavior falls outsidethe power law creep regime with an exponent of 5.

Mathematically, the concept of steady statecreep requires that the rate of change in thehardness parameter becomes zero. Indeed, SSLcreep behavior may be consistent with the rate of

workhardening, r, being low. During SSL creep at

Figuré 10. Typical high homologous temperaturecreep curves at high, intermediate, and low levelsof stress.

high stresses the workhardening rate is near r*,but r* is low because a* is high. At intermediatestresses r* is high because a* is low, but therate of workhardening is low because r/r* is low.This type of compensating effect, which does notrequire the aid of static recovery, is apparentlyconsistent with the power law creep behaviorexhibited by the nickel data.

In Figure 3, the dashed curves represent theflow stress as a function of temperature, at con-stant strain rate, with a/a* = 0.5 substitutedinto Equation 3. This value of a corresponds to03B5/03B5* = 115. Actual values of a* and 03B5* were es-tablished at 600°C to be 47 MPa and 1.9x10 9/sec,respectively, based on experimental data [18].These values are needed to establish 03B5*o in Equa-

tion 3. The coefficient of volume diffusion for

Ni is given by 1.9 x 10-4 exp(-284/RT) m2sec-1,whereas the pipe diffusion coefficient is 3.1 x

10-23 exp(-170/RT) m4sec-1 (4) for the high andlow temperature branches respectively. The acti-vation energies are in units of kJ/mol. For thelow temperature branch m is expected to increasebecause of the added dependence of the rate ofpipe diffusion on o*. To include this effect, weused an expression relating the stress, o, at SSLcreep to the density of dislocations [4]:

where b is the Burgers’ vector. Since ala* = 0.5at SSL creep, Equation 8 may be used to obtain therelationship between o* and p:

The above information is sufficient to construct

the dashed curves in Figure 3.A low rate of workhardening would result in SSL

creep. Another mechanism was proposed by Hart[11], whereby SSL creep results from scaling andthe nature of Equation 1. Because of the form of

Equation 1 and the scaling relationship (Equation3), curves of different hardness are expected tocross at sufficiently low strain rates as depictedin Figure 11. Accordingly, at a stress above the

Figure 11. Constant structure flow curves, of theform of Equation 1, displaying the crossing-overbehavior created by scaling. The horizontal linerepresents the mechanical path taken by a specimenduring creep under a constant stress. The curves1-4 are in order of increasing hardness. Thecreep rate, which is determined by the intersec-tion between the dashed line and the constanthardness curve, is seen to decrease, then increaseas the specimen workhardens.

crossing point, a specimen of a higher hardnessstructure will flow at a lower strain rate (asexpected), but at an applied stress below thecrossing point, a higher hardness specimen willflow at a higher strain rate contrary to common

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expectation. Because of these characteristics, itis possible that, in a constant load creep testthe strain rate will first decrease, then in-

crease, as the material workhardens as indicated

in Figure 11. The resulting minimum creep rate

might be identified with steady state creep. Theminimum strain rate thus obtained would correspondto the condition o/Q* = const. (0.264 for m = 5)which also leads to power law creep. Whether this

mechanism operates for the case of nickel is ques-tionable because the stress levels of the highstress portion of the data in Figure 3 are toohigh for the crossing of the relaxation curvesaccording to measured relaxation data. For thelow stress portion of the same data, the rate ofworkhardening is too low for this criterion to

operate.

Additional State Variables

Up to this point the discussion of flow behaviorhas pertained only to the homogenous plastic defor-mation in the grain matrix by dislocation processesat strains in the platic range. Nonelastic defor-mation often involves other phenomena that requirethe establishment of additional state variables(if, indeed, such phenomena can be incorporatedinto a state variable formulation). A number ofphenomena that have been categorized include: dis-location anelasticity, microplasticity and theBauschinger Effect, and grain boundary anelas-ticity and sliding.

A discussion of these phenomena, how they havebeen incorporated into a state variable descrip-tion, and how they are examined experimentallyhave been reviewed in reference [16]. Here, weshall briefly describe the current view of each.

Whenever a stress is applied to a crystallinesolid, the resulting rearrangement of dislocationconfigurations will lead to a small anelasticstrain. Hart [1] proposed that plastic deforma-tion occurs when dislocations at the head of the

pileup leak past the barriers confining them,either by thermally induced climb or glide under asufficiently high applied stress. The drivingforce for dislocation climb is proportional to theanelastic strain, a, which is a state variable de-fined by 3a = Qa, where a is the anelasticmodulus. The value of a is determined by thenature of the dislocation configurations.

Microplasticity [19] refers to a collection ofphenomena that occur before macroplastic yieldingas illustrated in Figure 12. In this figure aschematic tensile curve has been expanded to showthe stress versus nonelastic strain below about

0.5% strain. The flow behavior predicted based ona combination of the relationship 3a = oa andEquation 1 for plastic flow is shown as a solidline, whereas the more realistic material behavioris depicted as a dashed curve. Jackson et al.[19] introduced a concept of microplastic flow,that describes better the dashed curve in Figure12. Physically, the modification requires thattwo types of dislocation pileups contribute to thestored strain. Long range configurations of dis-locations form the first type of pileups againststrong barriers. On a more local scale, a secondtype of pileup is found among weak barriers be-tween the long range barriers as illustrated inFigure 13. Microplasticity results when disloca-tions leak past the weak barriers to form thelong-range pileups responsible for a . The flow

Figure 12. Initial portion of a tensile curve (0.5% strain) illustrating the behavior containinganelasticity only (solid curve) and real materialbehavior (dashed curve). The difference betweenthe two behaviors results from the presence ofmicroplasticity.

Figure 13. Schematic representation of thebarrier structure responsible for microplasticity,as proposed by Jackson [19].

behavior in the microplastic regime can be ex-amined by performing load relaxation tests at theappropriate stress levels. Physically, a distri-bution of barrier strengths exists. The twobarrier model can be considered to be a compromisebetween physical reality and a sufficiently simplemodel for which the required parameters can be de-termined experimentally.

Real materials generally show a Bauschingereffect, in that a specimen which has been plasti-cally deformed, e.g., in compression, exhibitsmore strain at the same stress in subsequent ten-sile loading than in compressive reloading. Thisadditional strain can be manifested at quite lowstress levels. Hart’s model, in which an anelas-tic element operates in parallel with a glidefriction element (Eq. 3) predicts a very weakasymmetry between reloading and reversed loading,that ià much less than the observed Bauschingereffect. The microplastic model accounts for the

Bauschinger effect better because of the largeramount of stored strains.

At elevated temperatures (> 0.4 Tm), grainboundary sliding can become an important mode ofdeformation [16,20]. This process operates simul-taneously with grain matrix deformation, but mani-fests most strongly within a window of strain ratethat depends on temperature. In general, thegrain boundary sliding rate is less dependent onstress than the grain matrix flow rate. Thus, atstrain rates above this window the grain matrixdeforms more easily than the grain boundary, andthe flow behavior is as described in Equations 1and 2. Below this window the grain boundaries

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slide easily but are constrained by grain matrixflow at sites such as triple points, so that therate of deformation is again controlled by thematrix. At these low strain rates, the grainboundaries act as shear cracks which concentratethe stress at the grain boundary triple points,thereby lowering the stress required to achieve aparticular strain rate by a factor of approxi-mately 0.8 [21]. Within the intermediate strainrate window both grain matrix and grain boundarydetermine the flow behavior. When plotted in alog-log form, flow stress-strain rate data con-taining the effects of grain boundary sliding willbe of a sigmoidal shape. At sufficiently lowstresses, grain matrix only deforms elasticallyand anelastically to accommodate grain boundarysliding, and grain boundary anelasticity resultsas proposed by Zener [22].

Grain boundary sliding operates within thepower law creep regime of the deformation map inFigure 3. To include this additional mode of de-formation would only slightly change the characterof the curves. However, grain boundary slidingcan lead to intergranular fracture that reducesductility in tensile and creep specimens andtherefore is an important subject for study.

Limitations of the State Variable Approach

Two apparent limitations of state variable

approaches will be mentioned. They are the de-scription of flow properties of materials contain-ing mobile solutes, sometimes including vacancies,and inhomogenous flow. In many cases inhomogenousflow is known to involve the interaction of mobilesolutes with dislocations [23]; in other casesinhomogenous flow involves processes such as me-chanical twinning [24].

During inhomogenous flow, the properties of thespecimen can vary spatially as well as with time;therefore, these properties can not be accountedfor in terms of a simple state variable descrip-tion. A range of structural states are expectedto exist within the material at one time.

The effects of solute-dislocation interactions,as manifested by phenomena like strain aging, pro-dùce.a time-dependence in such parameters as é*and a* during a load relaxation test where o* is

nearly constant [8,25]. Often this type of thetime-dependence is deformation history dependentas well, so that the strength of the solute-dislo-cation interaction cannot be specified by currentstate variables including a, c, and a*. Figure 4depicts such history dependencies. The load re-

laxation data shown are produced by a sequence ofreloading relaxation experiments [8]. Since thereloading is below the initial stress of the firstrun, the hardness parameter o* is not expected tochange significantly in this series of experi-ments. It is seen that the reloading data deviatefrom the initial data in the direction of lower

strain rates suggesting the effect of strainaging. Since the déviation increases as each re-loading run is made, the history of reloading isshown to influence strongly the manifestation ofsolute-dislocation interactions. Whether this

type of history dependency can be described bystate variables is not clear at the present.

Summary

A state variable approach for describing non-elastic deformation in crystalline solids has beenreviewed. An important basis for this approachlies in the détermination of flow properties at aconstant structure, as done in a load relaxation

experiment. When a single mode of deformation(either dislocation climb or glide) is control-ling, the flow stress-strain rate data can befound to translate along a straight line to over-lap one another. This scaling behavior estab-lislies that tlie isostructural data belong ta aone-parameter family of curves. This parameter,a*, uniquely determines the flow behavior at agiven temperature and a constant structural state.

In addition to the constant structure flowdata, a workhardening law representing the evolu-tion of a* with plastic strain is required topredict deformation under arbitrary loadings condi-tions. Fortunately, the evolution of o* can berepresented in terms of a* and aa, so that theflow equations are easily integrable. The inte-gration of tlie flow equations allows the descrip-tion of other déformation properties, for instancecreep.

Additional state variables are required forphenomena, such as anelasticity, microplasticityand the Bauschinger effect, and grain boundarysliding and anelasticity. Inhomogenous flow andsolute-dislocation interactions are as of yet un-treatable by using the present state variableapproach.

Acknowledgment

This work is supported by the U.S. Departmentof Energy through the Materials Science Division.

References

1. E.W. Hart, J. Eng. Mat. and Tech., 98,p. 193, 1976.

2. F.H. Huang, H. Yamada and C.-Y. Li, "Consti-tutive Relations Based on State Variables forNonelastic Deformation in Type 304 and 316Stainless Steels," in Characterization ofMaterials for Service at Elevated Tempera-tures, ed. G.V. Smith, Series MPC-7, ASME,p. 83, 1978.

3. D.W. Henderson, R.C. Kuo, and C.-Y. Li,

Scripta Metall., 18, p. 1021, 1984 and F.E.Hauser, Exp. Mech., 6, p. 395, 1966.

4. H.J. Frost and M.F. Ashby, Deformation-Mechanism Maps, Pergamon Press, Oxford, 1982.

5. M.A. Korhonen, S.-P. Hannula, and C.-Y. Li,Metall. Trans. A, 16A, p. 411, 1985.

6. E.W. Hart and H.D. Solomon, Acta Metall., 21,p. 295, 1973.

7. D. Lee and E.W. Hart, Met. Trans. 2, p. 1245,1971.

8. R.-C. Kuo, "Effect of Solute-Dislocation In-teractions on the Deformation Behavior of Al-

Mg and Fe-C Solid Solutions," Ph.D. Thesis,Cornell University, 1986.

9. P.S. Alexopolous, "An Experimental Investiga-tion of Transient Deformation Based on aState Variable Approach," Ph.D. Thesis, Cor-nell University, 1981.

647

10. H. Yamada and C.-Y. Li, Metall. Trans, 4,p. 2133, 1973.

11. E.W. Hart, C.-Y. Li, H. Yamada and G.L. Wire,"Phenomenological Theory: A Guide to Consti-tutive Relations and Fundamental Deformation

Properties," in Constitutive Equations inPlasticity, ed. A.S. Argon, MIT Press, Cam-bridge, Mass., p. 149, 1975.

12. M.A. Korhonen, S.-P. Hannula, and C.-Y. Li,"State Variable Theories Based on Hart’sFormulation," in Unified Constitutive Equa-

tions for Creep and Plasticity, ed. A. Miller,Elsevier, Amsterdam, 1987.

13. S.-P. Hannula, M.A. Korhonen, and C.-Y. Li,Res. Mech., 15, p. 99, 1985.

14. G.L. Wire, F.V. Ellis and C.-Y. Li, ActaMetallurgica, 24, p. 677, 1976.

15. V. Kumar, S. Mukherjee, F.H. Huang and C.-Y.Li, "Deformation in Type 304 AusteniticStainless Steel," Electric Power ResearchInstitute, Report No. EPRI NP-1276, 1979.

16. Che-Yu Li, "State Variable Theories forNonelastic Deformation," in MetallurgicalTreatises, J.K. Tien and J.F. Elliott, eds.,TMS-AIME, p. 469, 1981.

17. W.D. Jenkins and C.R. Johnson, J. Res. Nat.Bar. Stand., 60, p. 173, 1958.

18. R.L. Keusseyan, "Grain Boundary Sliding andRelated Phenomena," Ph.D. Thesis, CornellUniversity, 1985.

19. M.S. Jackson, C.-W. Cho, P. Alexopoulos, andC.-Y. Li, J. Eng. Mat. and Tech., 103, p. 314,1981.

20. M.A. Korhonen, R.L. Keusseyan, L.-X. Li, andC.-Y. Li, "Load Relaxation Studies of Defor-mation Mechanisms at Elevated Temperatures,in Novel Techniques in Metal DeformationTesting, R.H. Wagoner, ed., TMS-AIME, p. 275,1983.

21. F. Ghahremani, Int. J. Solid Structures, 98,p. 193, 1976.

22. C. Zener, Phys. Rev., 60, p. 906, 1941.23. A.H..Cottrell, Phil. Mag., 44, p. 829, 1953.24. F.H. Huang, G.P. Sabol, S.G. McDonald, and

C.-Y. Li, J. Nuc. Mat., 79, p. 214, 1979.25. S.-P. Hannula, R.C. Kuo, D.W. Henderson, and

C.-Y. Li, "Load Relaxation Studies of SoluteDislocation Interactions During StrainAging," in Solute-Defect Interaction, Theoryand Experiment, S. Saimoto, G.R. Purdy, andG.V. Kidson, eds., Pergamon, p. 366, 1986.