constitutive modeling of the viscoplastic deformation in high temperature forging of titanium alloy...

Materials Science and Engineering A245 (1998) 29–38

Constitutive modeling of the viscoplastic deformation in hightemperature forging of titanium alloy IMI834

Min Zhou *Department of Mechanical Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK

Received 23 June 1997; received in revised form 20 August 1997

Abstract

The experimental observation reveals that the deformation mechanisms in high temperature forging of titanium alloy IMI834are primarily attributed to deformation-induced recrystallization, which results in a decrease in flow stress after its peak value hasbeen obtained. In this paper, evolution of mean grain size is formulated and internal variables are used to characterize theresulting flow softening. An automated procedure for the determination of the material constants is also developed. The modelhas been successfully applied to predict the true stress–strain behavior of IMI834 at the isothermal forging conditions. © 1998Elsevier Science S.A. All rights reserved.

Keywords: Constitutive modeling; Dynamic recrystallization; Forging; Titanium alloy

1. Introduction

In order to achieve the required creep and fatigueresistance in the critical components of jet engines,compressor discs, for example, a careful control of themicrostructural development in titanium alloy IMI834through dynamic recrystallization during high tempera-ture forging is necessary [1,2]. The deformation behav-ior of IMI834 during forging is characterized by astrong temperature dependence of the flow stress andthe substantial flow softening, which is mainly owing todynamic recrystallization [1]. Whilst the isothermalforging process is adopted to control the microstruc-ture, the flow softening due to dynamic recrystallizationbecomes the predominant characteristic of the deforma-tion in IMI834 under the processing conditionsinvestigated.

The deformation processes and microstructuralchanges in titanium alloys have been the subjects ofmany authors [1–7]. The deformation-induced recrys-tallization in a–b titanium alloys under isothermalforging conditions was modeled by Immarigeon andKoul [3]. It was suggested that realistic modeling re-

quires that the contribution of each mechanism todeformation, such as intragranular flow and grainboundary sliding, be considered as a dynamic variable.On microstructural development in high temperatureprocessing, including isothermal forging of titaniumalloys, Flower [2] discussed the relationships betweenmicrostructures and mechanical properties and tried tomodel high temperature deformation in forging. It wasindicated that, though the empirical curve fitting ap-proach is generally adopted in the forging industry, theforging of near-a alloys like IMI834 must also considerthe microstructural development during forging so as tobe able to predict forging parameters. The kinetics ofmetallurgical change during forging and the heat treat-ment were investigated by Evans et al. [1] by laboratorytesting of IMI834 samples. Though the relationshipunder the b transus temperature between the mean a

grain size and the b recrystallized grain size has beenformulated and the rate of recrystallization defined intheir work, the modeling of the effects of dynamicrecrystallization on the stress–strain behavior and onthe microstructural development is still a matter offurther investigation.

In a more general review on mechanical and mi-crostructural aspects of dynamic recrystallization bySakai and Jonas [8], the transition from cyclic peak tosingle peak flow was clarified with indications that a

* Present address: Modelling Department, British Steel, SwindenTechnology Center, Moorgate, Rotherham, South Yorkshire, S603AR, UK.

0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S0921-5093(97)00707-7

M. Zhou / Materials Science and Engineering A245 (1998) 29–38M. Zhou / Materials Science and Engineering A245 (1998) 29–3830

single peak in flow stress is associated with grain refine-ment whereas multiple peaks with grain coarsening. Itwas shown that, at 940°C, on a 0.16% C steel tested intension at the relatively high strain rate (1.48×10−1

s−1), grain refinement took place, and once steady-stateflow was established an equilibrium grain size wasproduced. Whatever grain refinement or grain coarsen-ing occurs during recrystallization, flow softening in thestress–strain behavior is normally produced.

The purpose of this paper is to model the flowbehavior of IMI834 that has undergone dynamic re-crystallization in high temperature isothermal forging.The evolution of the recrystallized grain size and itseffects on the stress–strain behavior in IMI834 areconsidered as an internal variable in a viscoplasticconstitutive model that takes into account a range ofdeformation mechanisms taking place through compet-ing and interacting processes of grain boundary sliding,diffusional flow, dislocation glide and climb processes.Meanwhile, the strain rate dependence of the steadystate flow is described, the temperature-dependence ofthe material constants determined and the flow diagramand techniques to identify the material constants dis-cussed. The constitutive equations were validated by aset of typical IMI834 data at the normalized strain ratesof 0.01, 0.1 and 1.0, and for each strain rate at thetemperatures of 940, 965, 985, and 1005°C, respectively.The comparison between the experimental data and thepredicted stress–strain behavior is made and the resultsare summarized.

2. Consideration of dynamic recrystallization in aconstitutive model

2.1. Constituti6e equations

The characteristic form of the true stress–true strainbehavior of a material deformed at elevated tempera-tures reflects that the stress increases immediately fol-lowing application of load and viscoplastic flowcommences at the strain rate dependent yield stress. Aunified viscoplastic constitutive model [7] has been de-veloped to predict the deformation behavior of a classof titanium alloys undergoing deformation through arange of mechanisms. The hyperbolic sine relationshipbetween the viscoplastic strain rate and stress was usedwhich is typical of materials that show a change inpredominant deformation mechanisms from low tohigh stress levels at elevated temperature. For example,at low stress levels, and correspondingly low viscoplas-tic strain rates that is typical of diffusion processes, thesinh relationship degenerates to a linear relationship.However, at high stress levels, and hence high strainrates, the relationship between the two becomes highlynonlinear, which is characteristic of dislocation con-trolled deformation processes.

The total true strain rate is assumed to be separatedinto an elastic strain rate and a viscoplastic strain rateand any anelastic effects are ignored, which leads to

o; =o; e+o; vp. (1)

The elastic strain is related to the true stress by theHooke’s law. The viscoplastic strain is history depen-dent and its rate at any time is defined as a hyperbolicsine function of the stress, s, and the internal variables,X and R, as follows

o; vp=a sinh[b(�s−X �−R−k(o; ))]X: =Co; vp−gXp;

p=& t

0

�o; vp� dt

, (2)

where the temperature dependence of the flow behavioris described by assuming that the material constants a,b, C and g, are functions of temperature; k is a strainrate dependent initial yield stress; p is the accumulatedviscoplastic strain; X is an internal variable to modeldirectional kinematic hardening [7,9] developed by thecompeting processes of dislocation pile-up and dynamicrecovery through climb or annihilation events; R isanother internal variable to describe the isotropic soft-ening behavior induced by dynamic recrystallization,which will be discussed below.

Fig. 1. Schematic of the evolution of recrystallized grain sizes (a), andthe effects on time history of stress (b)

M. Zhou / Materials Science and Engineering A245 (1998) 29–38M. Zhou / Materials Science and Engineering A245 (1998) 29–38 31

Fig. 2. Flow diagram showing the procedure for the determination ofthe material constants

structure during deformation increases to a very highlevel, which eventually drives nucleation of recrystal-lization. The deformed grains are then continuouslyreplaced by a series of new equiaxed, dislocation-freegrains, leading to the elimination of large numbers ofdislocations by the migration of high angle boundariesand thus flow softening.

This process may be viewed as three primary stageswhich are characterized by the evolution of the recrys-tallized grain size, as schematically shown in Fig. 1. Thegrain size is defined as a mean homogenized grain sizein a material cell that contains enough grains to makea continuum assumption valid. The first stage indicatesthat the microstructure of initial grain size, d0, remainsprimarily unchanged until the threshold strain, orec, orthe time of onset of nucleation, trec, of recrystallization,indicated by the horizontal line d=d0 in Fig. 1(a). Theviscoplastic stress–strain behavior in this stage is strainhardening dominant, varying from yield stress, k, topeak flow stress, sp. The second stage starts from thenucleation at trec, which is followed by the growth ofthe newly recrystallized grains of size drec, that is char-acterized by the reduction of the mean grain size dm andthe resulting flow softening of stress. The third stagedescribes the steady-state of recrystallization in whichthe original grain structure has been completely re-placed by a series of new equiaxed, strain-free grainsand a perfect viscoplastic flow is approached with asteady-state value of the recrystallized grain size, dF,which corresponds to the steady-state flow stress, ss, inFig. 1(b).

The changes of grain size discussed in Fig. 1 can beseen to be limited to the case of grain refinement andsingle peak flow [8]. The model has been built based onthe investigation by Evans et al. [1], which revealed thepresence of a dynamic recrystallization process by ob-servations of the b grain size in the forged structureswhere the b grain size had been reduced in all IMI834specimens. It was also demonstrated that, at some largestrain, e.g. o=0.4, a constant stress had been achieved.

2.2. E6olution of recrystallization and the effects onstress–strain beha6ior

The microstructural processes in IMI834 during forg-ing are very complex because this near-a titanium alloyis increasingly being processed within the a+b field toproduce a+b structures containing a low volume frac-tion of primary a phase [2]. The influence of the a

particles is intended to constrain rapid grain growthduring recrystallization and hence to control greatly,the resulting b grain size [1,10]. This two phase mi-crostructure offers a good compromise between theconflicting requirements for creep and fatigue resis-tance. Though the processes are not totally understood,it may be accepted that dislocation density in the grain

Fig. 3. Stress–strain behavior at a given temperature and variousstrain rates with dashed lines (R excluded) representing dynamicrecovery and solid lines (R included) representing dynamic recrystal-lization.


Fig. 4. The predicted normalized stress–strain curves (lines), compared with experiments (symbols), at the temperatures 965°C (a) and 1005°C (b)with various normalized strain rates.

It was reported [3,8,11] that a threshold strain orec

exists below which no recrystallization occurs. In acomputer simulation of dynamic recrystallization madeby Luton and Sellars [12], it was indicated that thenucleation of new grains, and therefore the initiation offlow softening, involves a critical strain that increaseswith increasing strain rate and decreasing temperature,and it was assumed that the nucleation of all new grainsoccurs at orec:op where op is the strain associated withthe peak stress. In another investigation on characteri-zation and modeling for forging deformation of a near-a titanium alloy Ti-6242 with the a–b startingmicrostructure made by Dadras and Thomas [13], aconstant strain of o=0.04, was taken as the onset atwhich the flow softening took place. A similar assump-tion is followed in the present work, that recrystalliza-

tion initiates when the strain at peak stress has beenobtained.

For the grain refinement process shown in Fig. 1,nucleation and growth rate of the mean grain sizevaries from zero to negative values and back to zeroagain, indicating that the grain size remains unchangedbefore recrystallization and a steady-state grain size isobtained after undergoing recrystallization, which maybe represented by

d: = −a(d−dF)�o−orec�n o; , (3)

where the sign of �x� denotes that �x�=x for x\0and �x�=0 for x50. The steady-state grain size, dF,and the two material constants, a and n, may betemperature dependent. The variation of the grain sizewith respect to strain can be obtained by the integration


Fig. 5. Normalized saturated stress vs. normalized strain rate in logarithmic coordinates with the prediction indicated by lines and the experimentaldata indicated by symbols.

of Eq. (3) which evolves from the initial grain size, d0,and saturates to the final recrystallized grain size, dF, atlarge plastic strain.

Eq. (3) describes the dependence of nucleation andgrowth rate of recrystallization on the present values ofinstantaneous grain size d, deformation o, deformationrate o; , threshold strain orec, and steady-state grain size,dF. As strain rate increases, dislocation densities in-creases, which drives nucleation and grain growth. Asthe deformation is accumulated, dislocation energy isstored which initiates recrystallization, indicating a de-formation-induced recrystallization. Also, as the steady-state grain size is approached, recrystallization tends toslow down.

2.3. Modeling of recrystallization by internal 6ariables

The relationship between the steady-state grain sizeduring dynamic recrystallization and the deformationstress was investigated by Derby [14]. It was indicatedthat consideration of a dynamic balance between thenucleation rate and grain boundary migration rate canprovide a physical basis for the empirical relation ofssdl

F=constant, where ss and dF are the stress andgrain size associated with steady-state, as shown in Fig.1. For transient effects of the microstructural changesduring recrystallization on the stress–strain behavior,internal variables may be introduced to characterize thephenomenon of flow softening.

The material in forging exhibits multiaxial viscoplas-tic behavior which can generally be described with thehelp of the yield surface, characterizing the elastic andinelastic deformation. It may be reasonable to assumethat the variation of the yield surface due to dynamicrecrystallization, is a uniform expansion or contractionof the initial yield surface which can be represented by

an isotropic internal variable, R. The expansion of yieldsurface leads to further strain hardening whilst thecontraction of yield surface leads to strain softening.Additional to expansion or contraction, the translationof the yield surface may occur which may be repre-sented by a directional kinematic hardening variable, X,in Eq. (2), which represents internal stress generated byhardening through dislocation pile up andentanglements.

The yield surface is defined as (k+R) where k is theinitial yield stress. The internal variable R representsinternal stress generated by softening due to dynamicrecrystallization. The flow stress drops after the peakstress has been reached, which is associated with theevolution of the mean grain size described in Eq. (3).The internal variable R may be able to characterize thisflow softening with its rate equation defined as

R: = −b(R+Q)�o−orec�m p; , (4)

where b and m are temperature-dependent constants, pis the accumulated viscoplastic strain defined in Eq. (2).

The integration of Eq. (4) shows that R approachesits saturation value (−Q) at large plastic strain at arate determined by the constants b and m, which indi-cates a contraction of the yield surface. As the satura-tion stress ss increases with increases in strain rate andwith decreases in temperature, the dependence of thesaturation back stress Q on strain rate may be approx-imated by a power law function as follows,

Q(o; )=qo; r, (5)

where q and r are temperature-dependent constants.The rate equation of R in Eq. (4) is subjected to the

constraint o\orec, suggesting that no recrystallizationinitiates until after the threshold strain orec, which isconsistent with the investigation [1] indicating that no


Fig. 6. The effects of the isotropic variable R on the normalized stress–strain behavior at the temperatures 965°C (a) and 1005°C (b) with variousnormalized strain rates.

recrystallization takes place before the peak in thestress–strain curve and, thereafter, decreases in stressresult from a continuous replacement of original grainsby the recrystallized grains.

2.4. Temperature dependence of material constants

The strain rate dependence of the deformation be-havior due to dynamic recrystallization has been dis-cussed in the preceding sections. The temperaturedependence of the deformation may be described by thetemperature-compensated material constants defined as

(Ci)j=ai exp�bi

Tj

�i=1, 2,…, I ; j=1, 2,…, J (6)

where Ci refer to the material constants a, b, C, g, b, m,q and r, respectively; I is the total number of constantsto be determined, J is the number of temperatures, Tj isthe jth temperature, ai and bi are temperature-indepen-dent constants corresponding to Ci.

3. Determination of the material constants

The above phenomenological constitutive equationsuse two internal variables to characterize the flow be-havior due to dynamic recrystallization. As no recrys-tallization occurs in the first stage in Fig. 1, the strainhardening is therefore assumed to be the result of thedirectional kinematic hardening only and the effects of


Fig. 7. Experimental (symbols) and predicted (lines) normalized stress–strain curves at temperatures of 940°C (a) and 985°C (b) with variousnormalized strain rates.

the isotropic variable R are disregarded. The flow soft-ening in the second stage starts with the nucleation ofdynamic recrystallization at the threshold time trec whenthe dislocation density in the b matrix becomes suffi-ciently high to permit the nucleation of recrystallizationduring deformation. This regime is assumed due toboth kinematic hardening and isotropic hardening, andrecrystallization is primarily isotropic which is de-scribed by contraction of the initial or subsequent yieldsurface. The steady-state flow in the third stage showsthat the growth of recrystallized grains reaches asteady-state, which is characterized by the saturatedstate of the isotropic and kinematic hardening vari-ables, R and X.

The material constants involved in the abovemodified constitutive equations include those in Eq. (2),

(a, b, C, g), and those in Eqs. (4) and (5), (b, q, r). Alsoincluded are the strain rate dependent yield stresses inEq. (2), ki (i=1, 2,…, K) where K is the total numberof strain rates for a given temperature. As indicated inthe preceding sections, the constants b and m describethe rate at which R saturates to its steady-state value(−Q). In the absence of the constant m, Eq. (4)represents a softening which is similar to the isotropichardening rule [9]. The identification of these constantsis shown in the flow diagram of Fig. 2 in which sevensteps are followed.

The first step is designated to identify the constants a

and b in the rate equation of viscoplastic strain, whereX and R are assumed to be excluded and ki are given bythe data base. The second step optimizes C and g withX being involved and R excluded. The constants a and


b in this step are taken over from the first step. Thethird step is followed if the previous two steps fail toproduce satisfactory predictions of the hardeningregime of the stress–strain behavior, in which all con-stants are optimized again with their values identified inthe previous two steps being taken as initial values inthe optimization. These three steps are carried outbased on the data in the strain range [0, orec] wherestrain hardening is assumed mainly due to directionalkinematic hardening developed within the deformedgrains during forging. The detailed optimization proce-dure was discussed in [7].

The internal variable R that characterizes the effectsof dynamic recrystallization is involved in step 4, wherethe constants b, q, r, are optimized using the data in thestrain domain [orec, omax] where omax is the maximumstrain considered.

Though this procedure is quite straightforward, thestrain rate dependence of the initial yield stress has yetto be established. It may be assumed that a power lawapplies between them but its reliability is to a largeextent determined by the accuracy of the yield stressdata which are not distinct particularly for nonferrousmaterials. Steps 5–7 in Fig. 2 are therefore developedthat will make the stress–strain behavior of a materialbe predicted from the constitutive equations for anygiven strain rate and temperature without having toknow the corresponding initial yield stress.

As shown in Fig. 3, the deformation behavior mayreach a saturated state for materials that experiencedynamic recovery, indicated by dashed lines, or mayundergo substantial flow softening for materials thatexperience dynamic recrystallization, indicated by solidlines, which depends on the applied strain rate, temper-ature and the material itself [11]. These phenomenamay be described by the above constitutive equations.The first term on the right hand side of the back stress,X, in Eq. (2), is a hardening term and the second is aterm for recovery. If the internal variable R, that de-scribes dynamic recrystallization, is excluded in Eq. (2),a steady-state flow stress can be achieved immediatelyafter the regime of strain hardening, which can bederived from Eq. (2) at the condition of large plasticstrain,

sis=ki+Xs+1b

sinh−1�o; ia

�i=1, 2,…, N (7)

where Xs is the steady-state value of the kinematicvariable X which saturates to the value C/g with in-creasing plastic strain, ki is the initial yield stress corre-sponding to the total strain rate o; i which is arranged inorder as

o; 1Bo; 2B ······Bo; Nk1Bk2B ······BkN

, (8)

where N is the total number of strain rates.

The strain rate dependent initial yield stress is as-sumed to be

ki=k0+Dki, (9)

where k0=k1 and Dki is the increment corresponding toki which is further assumed to follow the same law asEq. (7)

Fig. 8. Variation of the material constants with temperature (a) a¦, b¦and r ; (b) g, k0 and b ; (c) C and q.


Dki=1b %

sinh−1�o; i−o; 1a %

�. (10)

Eqs. (9) and (10) are substituted into the first term onthe right hand side of Eq. (7) to yield the followingapproximation,

si=�

k0+Cg

�+

1b¦

sinh−1� o; ia¦�

, (11)

where si has replaced sis, indicating that it is a functionof the new constants a¦ and b¦, which may be deter-mined by minimizing the following function

F(a¦, b¦)= %N

i=1

(si−sis)2, (12)

where sis is calculated from Eq. (7) by the constantsgenerated in the step 3 in Fig. 2.

In this way, the strain rate dependence of the initialyield stress may be avoided at the expense of the furthernumerical calculation of Eq. (12). Actually, the effectsof strain rate on the initial yield stress have beencompensated by the constants a¦ and b¦, and k(o; ) inEq. (2) has been replaced by a constant k0, which is theinitial yield stress associated with o; 1.

The above treatment of the strain rate dependence ofthe initial yield stress is accomplished by steps 5–7 inFig. 2. Step 5 calculates the saturated stress sis, asindicated in Fig. 3, using the constants generated fromthe previous steps. Step 6 optimizes the constants a¦and b¦ following Eqs. (11) and (12). The last stepproduces the stress–strain curves using the constantsa¦, b¦, C, g, b, q, and r for the given strain rate andtemperature. The flow diagram in Fig. 2 also suggestsan automated computational procedure to predict thestress–strain behavior for any given strain rate andtemperature.

The temperature-independent constants ai and bi inEq. (6) can be obtained from the linear regressionformat given below

log(Ci)j= log ai+0.4343bi

� 1Tj

�, (13)

where ai and bi are estimated from the plots of Ci inlogarithmic coordinates and temperatures in reciprocalcoordinates.

4. Application to IMI834 and discussion of the results

The constant true strain rate compression tests withthe titanium alloy IMI834 were performed at the nor-malized strain rates of 0.01, 0.1 and 1.0, and for eachstrain rate at the temperatures of 940, 965, 985 and1005°C, respectively. The tests showed that IMI834 didundergo flow softening in the ranges of strain rate andtemperature considered. Substantial softening occurred

at high strain rates and low temperatures. It was ob-served that the flow softening is attributed to dynamicrecrystallization [1].

The threshold strain at which nucleation of recrystal-lization initiates is considered in the present work to bea constant, which is taken from the test data base to bea normalized strain orec=0.1. The data for the truestress–true strain behavior were applied to identify thematerial constants in the constitutive equations of Eqs.(1), (2), (4) and (5) following the flow diagram in Fig. 2,which produces the results shown in Figs. 4–8.

The step 3 in Fig. 2 optimizes the constants (a, b, C,g) for the four temperatures, which is based on the databefore the nucleation of the recrystallized grains ini-tiates and assumed to be distinguished by the thresholdstrain orec (see Fig. 1). Fig. 4 shows the results for thetwo temperatures, 965 and 1005°C, of the predictednormalized stress–strain curves, indicated by solidlines, and the experiments, indicated by symbols, atvarious normalized strain rates. It is shown that a goodprediction of strain hardening has been achieved. It isalso indicated that the ways to identify the materialconstants are quite satisfactory.

Fig. 5 determines the two constants a¦ and b¦ in Eq.(11) based on the already determined constants (a, b,C, g) by minimizing the stress residual function in Eq.(12) in order to achieve the strain rate independence ofthe initial yield stress. The saturated stresses resultingfrom Eq. (11) are indicated by solid lines and comparedwith the experimental data indicated by symbols, sug-gesting that good agreements have been obtained. Thestresses predicted by Eq. (11) are independent of theinitial yield stresses.

The isotropic hardening variable R has been intro-duced to describe dynamic recrystallization. Its effectson the stress–strain behavior in IMI834 are illustratedin Fig. 6 for the temperatures 965°C (a) and 1005°C (b).The solid lines include the effects of R which give asatisfactory prediction of the experimental data indi-cated by symbols. The dashed lines indicate that R isexcluded in the constitutive equation Eq. (2) and thesteady-state flow stress is achieved right after the strainhardening regime. The flow softening in this case isattributed to dynamic recovery, typical of aluminiumalloys at hot working temperatures [11], where anequiaxed subgrain structure develops until a steady-state subgrain structure is reached leading to a steady-state flow stress. It appears that the constitutive modelin Eqs. (2), (4) and (5) may not only describe thedeformation due to dynamic recrystallization but alsothe deformation due to dynamic recovery.

Fig. 7 shows the predicted normalized stress–straincurves, indicated by solid lines, and the experimentaldata, indicated by symbols, for the other two tempera-tures 940 and 985°C, respectively. Good agreementbetween them demonstrates that the constitutive equa-


tions containing the isotropic variable to characterizedynamic recrystallization are able to predict the flowbehavior in IMI834 over a wide range of strain ratesand temperatures.

The temperature dependence of the material con-stants are shown in Fig. 8, in which the symbolsindicate the optimized values for the constants and thesolid lines are obtained from the linear regression anal-ysis of Eq. (13). It is demonstrated that the linearrelationship applies between the constants in logarith-mic coordinates and the temperatures in reciprocalcoordinates for most constants except for g, suggestingthat the temperature-dependence of material constantscan well be characterized by Eq. (6).

5. Conclusions

The experimental investigation has shown that theflow softening in high temperature forging of titaniumalloy IMI834 is attributed to deformation-induced re-crystallization, which has been modeled in this paper byconsidering the effects of evolution of mean grain sizeon the stress–strain behavior in IMI834 and by intro-ducing internal variables to characterize the phe-nomenon of flow softening. The microstructuralchanges due to dynamic recrystallization are consideredto be isotropic in nature and strain hardening caused bydislocation pile up and entanglements is considereddirectional. The successful prediction of the stress–strain behavior in IMI834 has been made over a nor-malized strain rate range of 1.0, 0.1 and 0.01, at thetemperatures of 940, 965, 985 and 1005°C, whichdemonstrate the success in the constitutive equationsand the material constants identification methods devel-oped. The results also demonstrate that the tempera-

ture-compensated material constants defined are goodrepresentatives of the temperature dependence of thematerial constants in most cases.

The flow softening may be caused by dynamic recov-ery or dynamic recrystallization, depending on tempera-ture, strain rate and the material itself. It appears fromthis investigation that the constitutive equations devel-oped may not only be able to describe the deformationdue to dynamic recrystallization but also the deforma-tion due to dynamic recovery.

Acknowledgements

Discussions with Dr F.P.E. Dunne are gratefullyacknowledged.

References

[1] R.W. Evans, G.S. Clark, S.G.Mckenzie, Proc. MECAMAT 91,Large Plastic Deformation, Balkema, Rotterdam, 1993, pp. 369.

[2] H.M. Flower, Mater. Sci. Technol. 6 (1990) 1082.[3] J.P. Immarigeon, A.K. Koul, Proc. Int. Conf. Computer Model-

ing of Fabrication Processes and Constitutive Behavior ofMetals, Ottowa, Ont., 1986, pp. 309.

[4] S.L. Semiatin, G.D. Lahoti, Met. Trans. A 12 (1981) 1705.[5] H. Mecking, D. Dunst, Mater. Sci. Eng. A175 (1994) 55.[6] A.K. Ghosh, C.H. Hamilton, Met. Trans. A 10 (1979) 699.[7] M. Zhou, F.P.E. Dunne, J. Strain Anal. Eng. Des. 31 (1996) 187.[8] T. Sakai, J.J. Jonas, Acta Metall. 32 (1984) 189.[9] J.L. Chaboche, G. Rousselier, J. Press. Vessel Technol. 105

(1983) 153.[10] F.P.E. Dunne, M.M. Nanneh, M. Zhou, Philos. Mag. A 75

(1997) 587.[11] C.M. Sellars, Mater. Sci. Technol. 6 (1990) 1072.[12] M.J. Luton, C.M. Sellars, Acta Metall. 17 (1969) 1033.[13] P. Dadras, J.F. Thomas Jr., Met. Trans. A 12 (1981) 1867.[14] B. Derby, Acta. Metall. Mater. 39 (1991) 955.

.

constitutive modeling of the viscoplastic deformation in high temperature forging of titanium alloy...

Documents