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CONTINUUM MECHANICAL UNIT CELL MODELSFOR STUDYING THE THERMOMECHANICALBEHAVIOR OF HIGH SPEED TOOL STEELS

H.J. Bohm, T. Drabek and A. EckschlagerChristian Doppler Laboratory

for Functionally Oriented Materials Design,

Institute of Lightweight Structures and Aerospace Engineering,

Vienna University of Technology, Vienna, Austria

Abstract The thermomechanical behavior of high speed tool steels is studied by acontinuum micromechanical approach. The simulations are based on three-dimensional unit cell models that contain a number of randomly positionedspherical carbides embedded in a thermoelastoplastic steel matrix. Models ofthis type allow effects of the geometrical arrangement of the primary carbidesto be investigated for a wide range of loading conditions.In the present study emphasis is put on modeling the thermal residual stressescaused by the thermal expansion mismatch between matrix andcarbides. Thepredictions indicate that even though such thermal residual stresses can reachconsiderable magnitudes, their influence on the overall stiffness of tool steelstypically is small and their effect on the strength also is minor.

Keywords: continuum micromechanics, unit cell models, thermal residual stress

495

496 6TH INTERNATIONAL TOOLING CONFERENCE

LIST OF SYMBOLS

α coefficient of thermal expansionE Young’s modulusεeqv,p accumulated equivalent plastic strainν Poisson numberh hardening coefficient (Ludwik law)m Weibull modulusn hardening exponent (Ludwik law)Pi fracture probability ofi-th particleσ1 maximum principal stressσeqv von Mises equivalent stressσf characteristic strengthσm mean stressσy flow stressσy,0 initial yield stressV0 reference volumeξ carbide volume fraction

INTRODUCTION

Due to their combination of high stiffness and strength withsubstantialfracture toughness and wear resistance, high speed tool steels (HSSs) are agroup of materials of major technological importance. Manyaspects of theirthermomechanical behavior are not yet fully understood, however, which hasstimulated a continuing interest in experimental and theoretical research.HSSs derive their application relevant properties from their heterogeneousstructure. At length scales of the order of a few micrometersthis takes theform of primary carbides embedded in a martensitic–austenitic steel matrix,which, in turn, contains much smaller secondary carbides.

The present study concentrates on modeling at the length scale of theprimary carbides, where HSSs may be viewed as a special classof particlereinforced ductile matrix composites and can be studied by the methods ofcontinuum micromechanics. The basic idea underlying such approachesis to describe the thermomechanical behavior of inhomogeneous materialson the basis of the material properties and the geometrical arrangements of

Unit Cell Models for Thermomechanical Behavior of Tool Steels 497

their constituents. A typical aim of micromechanical analyses is to studyproblems that are difficult or impossible to answer by experiments, such asobtaining information on the influence of the geometrical arrangement ofthe primary carbides on the stiffness and strength properties of tool steels.

The most important approaches in continuum micromechanicsare, onthe one hand, mean field and variational schemes for analytically or semi-analytically estimating or bounding the thermomechanicalresponses of in-homogeneous materials and, on the other hand, numerically-based methodsthat aim at studying specific phase arrangements (“microgeometries”) at ahigh level of detail. The most important representatives ofthe latter groupof methods are unit cell analyses that use periodic “model composites” fordescribing materials that are free of damage or show distributed damage, andembedded cell approaches that focus on localized regions ofspecial interest,such as crack tips.

A considerable body of literature on micromechanical studies of ductilematrix composites is available, most of which have been directed at thethermomechanical behavior of aluminum- or titanium-basedmetal matrixcomposites (MMCs). A number of publications, however, havebeen specif-ically directed at modelling the mechanical behavior of tool steels and relatedmaterials: Two-dimensional unit cell models were reportedwhich focus onexploring the initiation of local damage in HSSs by ductile or creep failureof the matrix, by brittle fracture of the particles and/or byinterfacial decohe-sion, see e.g. [1, 2, 3]. Also, a few studies using three-dimensional unit cellshave been published [4, 5]. Macrocracks in tool steels were described byplanar embedded cell models that, on the one hand, used microgeometriesobtained from metallographical sections of HSSs [6, 7] and,on the otherhand, compared specific generic arrangements of carbides [8]. In addition,hierarchical schemes [5, 9] were used to investigate the thermomechanicalbehavior of conventionally produced HSSs, in which the carbides tend to beconcentrated in clusters or layers.

A number of recent studies have pointed out that considerable errors maybe introduced into predictions for the overall responses and especially forthe microscale stress and strain distributions when two-dimensional modelsare used for describing particle reinforced composites [10, 11, 12]. It isalso of interest that relatively small volume elements (i.e. unit cells contain-ing a limited number of particles) can give excellent results for the elastic[13] and good results for the inelastic [11] behavior of overall isotropic ma-


terials that contain spherical particulate reinforcements. Accordingly, themost promising micromechanical modelling approach for HSSs appear tobe three-dimensional unit cell or embedded cell models in which a numberof randomly positioned carbides approximate the phase arrangements of theactual materials.

The major difficulty encountered in micromechanical analyses of HSSs liein the limited availability of reliable stiffness and strength data for the steelmatrix, the carbides and the interface between them. In addition, studiesbased on highly resolved microgeometries typically incur high computa-tional costs, especially when three-dimensional models are used.

METHODS

The model microgeometries underlying the present study arespace fillingperiodically repeating arrangements of identical spherical carbides embed-ded in a matrix, 15 of which are randomly positioned in three-dimensionalunit cells. Appropriate particle positions were generatedby a Random Se-quential Adsorption algorithm, compare e.g. [15], modifiedto support peri-odic geometries. Typical unit cells of this type can be seen in Fig. 3. Despitethe rather small number of particles involved, such models closely approachoverall isotropic behavior in the elastic range, compare [11]. Even thoughthe use of such microgeometries obviously involves a considerable degreeof idealization, they are thought to be fairly realistic descriptions for thearrangements of carbides in tool steels produced by powder metallurgicalroutes.

The thermomechanical responses of the unit cells were evaluated by theFinite Element method, the preprocessor MSC/PATRAN V.8.5 (MacNeal–Schwendler Corp., Los Angeles, CA, 1998) and the analysis code ABAQUSV5.8/Standard (Hibbit, Karlsson and Sorensen Inc., Pawtucket, RI., 1998)being employed. The models were meshed with 10-node tetrahedral ele-ments, periodic boundary conditions were enforced by appropriate constraintequations, and material as well as geometrical nonlinearities were accountedfor. Element counts were of the order of 50.000 per unit cell.

The constituent material laws used in the analyses correspond to isotropicthermoelastic carbides embedded in an isotropic thermoelastoplastic steelmatrix described by aJ2 continuum plasticity model. The strain hardeningof the matrix was approximated by a modified Ludwik hardeninglaw


σy = σy,0 + hεneqv,p , (1)

whereh andn are the hardening coefficient and the hardening exponent,respectively,σy andσy,0 represent the actual flow stress and the initial yieldstress of the matrix, andεeqv,p stands for the accumulated equivalent plasticstrain. Because no data were available for the temperature dependence of thematerial parameters of matrix and carbides the same material behavior wasemployed for the whole temperature range considered; this approximationtends to overestimate thermal residual stresses and to underestimate plasticstrains in the cooled-down HSS. The interfaces between the constituentswere assumed to be perfectly bonded. Generic material parameters whichhad been used for modeling DIN S6–5–2–5 HSSs in a number of previousstudies, compare e.g. [5, 14], were employed, see Table 1.

Table 1. Material parameters used for the thermoelastoplastic steel matrix and the thermoe-lastic primary carbides in the unit cell models of HSSs

E ν σy,0 h n m σf α

[GPa] [ ] [GPa] [GPa] [ ] [ ] [GPa] [K−1]

Steel matrix 210 0.30 2.75 1.5 0.50 — — 14.0 × 10−6

Primary carbides 450 0.25 — — — 5 3.66 6.0 × 10−6

The results of the Finite Element analyses were evaluated, on the one hand,in terms of homogenized material properties such as overallstrain vs. tem-perature and stress vs. strain curves as well as effective moduli and effectivecoefficients of thermal expansion. On the other hand, the microscale dis-tributions of the stresses and strains within the unit cellswere described interms of phase averages and the corresponding standard deviations. Theevaluation of the latter type of results was based on an option of ABAQUSthat allows the volume corresponding to each integration point to be output,so that the phase average of some functionf can be approximated as


< f >=1

V (j)

∫

V (j)f(r)dV ≈

1

V (j)

N(j)∑

l=1

flVl withN(j)∑

l=1

Vl = V (j) .

(2)Herefl andVl are the function value and the integration weight (in terms

of a volume), respectively, associated with thel-th integration point withina given phase (j), which contains a total ofN (j) integration points.

The failure relevant loads on the carbides were assessed by amodifiedWeibull concept [16], in which for a given loading conditioneach particleis assigned a fracture probability,Pi, of the form

Pi = 1 − exp

[

−1

V0

∫

Vi:σ1(r)>0

(

σ1(r)

σf

)m

dV

]

. (3)

In this expressionσ1(r) stands for the spatial distribution of the maximumprincipal stress within thei-th carbide as obtained by the micromechanicalanalyses,Vi : σ1(r) > 0 denotes the region of the particle for which thismaximum principal stress is tensile,V0 is a reference volume that was setequal to the carbides’ volume for the present study, whilem andσf are theWeibull modulus and the characteristic strength of the particles. The valuesused for the latter two parameters are listed in Table 1. Equation (3) wasevaluated by the approximate quadrature scheme given in eqn. (2).

It is worth noting that for the model described above the damage-freethermoelastoplastic material behavior does not depend on the absolute sizeof the carbides; such size effects must be explicitly introduced by an appro-priate choice of the elastoplastic material parameters. The particle fractureprobabilities, in contrast, show such a dependence becauseeqn. (3) intrin-sically gives rise to higher values ofPi whenVi is increased. For a broaderdiscussion of modeling issues in the use of three-dimensional multi-particleunit cell models see e.g. [11, 12]. Additional information on Weibull modelsfor brittle particles embedded in a ductile matrix is given in [17].

DISCUSSION OF RESULTS

The thermal residual stresses in the model tool steel were evaluated forcooling down from a stress free temperature of 600◦C to room temperature.All predictions in this section pertain to a carbide volume fraction ofξ=0.15.


Most of the results presented in the following were obtainedby ensembleaveraging over a number of runs.

RESIDUAL STRESS STATE

Figure 1 depicts the predicted microscale von Mises equivalent residualstresses in the steel matrix after the cooling down process.The stress levelsare grey coded on three parallel section planes within the unit cell, with darkshades of grey corresponding to high stress levels. A highlyinhomogeneousdistribution of the residual stresses is clearly evident, the lower coefficientof thermal expansion of the carbides giving rise to approximately concentricregions of elevated tensile hoop stresses and compressive radial stressesaround each particle. Marked local stress maxima typicallydevelop betweenclosely approaching carbides.

In Table 2 predictions for the microscale von Mises equivalent stress,σeqv, the maximum principal stress,σ1, the mean (hydrostatic) stress,σm,

COOLED DOWN STATEEQUIVALENT STRESS [GPa]

2.5

2.0

1.5

1.0

0.5

Figure 1. Predicted residual von Mises equivalent stress in a tool steel cooled down froma stress free temperature of 600◦C to room temperature.


and the accumulated equivalent plastic strain,εeqv,p, are listed in terms ofphase averages± the corresponding standard deviations for matrix(m) andcarbides(p). The phase averaged equivalent stress in the matrix can be seento reach about 18% of the initial yield stress, but the stronginhomogeneityevidenced by the large standard deviations nevertheless leads to a smallamount of local yielding, which gives rise to a nonzero valueof the phaseaverage ofε(m)

eqv,p. As expected, the phase averages of the mean stress aretensile in the matrix and compressive in the carbides, the absolute value beingabout 5 times higher in the latter on account of the phase volume fractions.Interestingly the standard deviations of the mean stress are relatively smallin both constituents.

Table 2. Predicted residual state in a tool steel (ξ=0.15) cooled down from a stress-freetemperature of 600◦C to room temperature

σ(m)eqv σ

(m)m ε

(m)eqv,p σ

(p)eqv σ

(p)1 σ

(p)m

[GPa] [GPa] [×10−6] [GPa] [GPa] [GPa]

0.50±0.36 0.16±0.02 0.02±3.97 0.31±0.19 -0.78±0.06 -0.92±0.02

For assessing the influence of changes in the material parameters it isworth noting that in general the magnitude of the self equilibrated resid-ual stresses in a particle reinforced material subjected tocooling downgrows with increasing temperature difference, with increasing thermal ex-pansion mismatch between carbides and matrix, and with increasing matrixyield stress. Finally, it should also be mentioned that the effective coeffi-cient of thermal expansion of the HSS was evaluated from the unit cells asα∗=1.26×10−5K−1, which is identical to results obtained from the Mori–Tanaka mean field theory [18].

INFLUENCE OF THE RESIDUAL STRESS STATE ONTHE UNIAXIAL MECHANICAL BEHAVIOR

In a further step, the mechanical responses under uniaxial tensile loadingup to an applied stress 3.0 GPa (which exceeds the initial yield stress of thematrix by about 9%) were simulated for an initially stress free (“virgin”)HSS and for a cooled-down HSS. The latter case corresponds tothe “as


cooled-down” condition, i.e. possible reductions of the residual stress dueto relaxation are not accounted for.

In Fig. 2 the predicted tangent moduli (obtained by differentiating thestress vs. strain curves) for the two cases are shown as functions of the ap-plied stress. With the exception of some very minor differences at appliedstresses between 2.5 GPa and 2.8 GPa, where progressive yielding of the steelmatrix occurs, the curves are essentially identical and theeffective Young’smodulus has a value of approximately 235 GPa in both cases. Evidentlythe simulations indicate that the thermal residual stresses have no significantinfluence on the uniaxial tensile response of HSSs. This result is in contrastto predictions from two-dimensional unit cell analyses, which cannot ade-quately account for the out-of-plane constraint and tend toshow a noticeableinfluence of the residual stress state on the stress vs. strain response [19].

Table 3 lists predictions for some phase averaged microfields in virginand cooled-down HSSs subjected to a tensile uniaxial stressof 3.0 GPa.

Only minor differences are present in the results for the matrix, the equiv-alent and mean stresses and the equivalent plastic strain being a few percenthigher in the cooled-down case. The maximum principal stress and the meanstress in the particles, however, are predicted to be approximately 10% and70% higher, respectively, in the virgin HSS. The former difference alsotranslates into a small but noticeable reduction of the Weibull fracture prob-abilities of the particles in the cooled-down HSS. This effect can be seen inFig. 3, which displays the predicted fracture probabilities of the individualcarbides under uniaxial tensile loading in thex-direction, higher values ofPi being indicated by darker shades of grey. It should be noted that theseplots are strictly for simplified comparisons between virgin and cooled-downHSSs only — when particle fracture is accounted for in micromechanicalmodels on the basis of Weibull statistics, the majority of particles do notreach fracture probabilities much beyond 0.5 before failing, see e.g. [3, 20],and the failure of individual particles leads to stress redistribution within theunit cell.

From the above results on the microfields some tendency can bediscernedfor particle fracture as well as interfacial decohesion to be somewhat morelikely for the virgin material, whereas cooled-down HSSs are slightly moresusceptible to damage initiation in the matrix. The predicted differencesappear to be, however, too small to be of practical relevance.


Figure 2. Predicted effective tangent moduli as functions of the applied uniaxial tensilestress of a virgin (solid line) and a cooled down (dotted line) tool steel.

(a) (b)

Figure 3. Weibull fracture probabilities of the carbides under a uniaxial tensile stress of 3GPa acting inx-direction predicted for a virgin (top) and a cooled down (bottom) tool steel.


It is worth noting that much stronger effects of thermal residual stressesare present in aluminum based MMCs of similar particle volume fractions.Due to the much lower yield strength of the aluminum matrix and to the moremarked elastic and thermal expansion contrasts between matrix and particles,cooling down by 380◦C suffices to cause essentially total yielding of thematrix. If stress relaxation effects are again neglected, there is no strictlyelastic regime during subsequent uniaxial tensile loadingof the cooled-downMMC, which shows a considerably reduced stiffness under uniaxial tensileloading, compare [21]. In analogy, tool steels with a lower matrix yieldstrength than the one given in Table 1 can be expected to show astronger,but still rather limited sensitivity to thermal residual stress effects.

CONCLUSIONS

A three-dimensional Finite Element based multi-particle unit cell modelwas used to study the effects of thermal residual stresses from cooling downprocesses on the overall mechanical behavior of high speed tool steels. Theresults indicate that the stiffness and strength of HSSs arerather insensitiveto such stresses, mainly due to the high initial yield stressof the steel matrix.

The main practical difficulties in using the above multi-particle modelingstrategy lie in obtaining appropriate material parametersand in the highrequirements in computational resources. The approach, however, allows fora considerable degree of flexibility in that different particle volume fractionsand reinforcement shapes can be handled, see e.g. [22], a wide range of loadcases and constituent material properties can be considered, and descriptionsfor progressive local damage, compare e.g. [3, 20], can be introduced.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support bythe Chris-tian Doppler Research Society, Vienna. Special thanks are due to BohlerEdelstahl GmbH & Co KG, whose interest led us to work in this field ofresearch.

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Table 3. Comparison of predicted microfields in virgin and cooled down tool steels (ξ=0.15)subjected to a uniaxial tensile stress of 3 GPa

σ(m)eqv σ

(m)m ε

(m)eqv,p σ

(p)eqv σ

(p)1 σ

(p)m

[GPa] [GPa] [×10−2] [GPa] [GPa] [GPa]

virgin 2.87±0.05 1.03±0.27 0.77±0.38 4.31±0.23 3.80±0.22 0.94±0.13cooled down 2.88±0.04 1.09±0.25 0.81±0.38 4.40±0.22 3.48±0.23 0.55±0.15