Modelling of solidification and heat treatment for the prediction of yield stress of cast alloys

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Modelling of solidification and heat treatment for the prediction of yield stress of cast alloys

Charles-André Gandin, Yves Brechet, Michel Rappaz, Gilles Canova, Michael Ashby, Hugh Shercliff

To cite this version:

Charles-André Gandin, Yves Brechet, Michel Rappaz, Gilles Canova, Michael Ashby, et al.. Modelling of solidification and heat treatment for the prediction of yield stress of cast alloys. La Metallurgia Italiana, Milano: Associazione Italiana di Metallurgia, 2002, pp.37-38. �hal-01570149�

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Ch.-A. Gondín, ^Y. Bréchet, M. Rappaz, G. Canova, M. Ashby, H. Sherclíff

A ^{modet is} developedfor the prediction of the microstructure evolution of cast alloys during its industrial process route. The materials under consideration are binary aluminium-copper alloys. The process are

casting, homogenising and ageing. A suite of models is built, which successively calculates microsegregation taking place duríng solidification and homogenisation, and precipitation occurring during ageing of the alloy. As a final output, the model permits the final ^{yield stress of the material to be}

calculated as afunction of the process parameters (i.e., casting cooling rate, inoculation of the alloy, duration and temperature of the homogenising and ageing heat treatments). This is made possible by introducing a sîructural hardening model between the diffusion driven phase transformation models and

a final mechanical model for heterogeneous materials.

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p${Agf Tr{AhlSf0R$rlA3X*$ i4ffir}frffi temperature). The inputs of the model are thus the average grain size after solidification (equivalent to the final grain Modelling of solidification is based on the resolution of the ^{density), and} the cooling rate during solidification imposed solute conservation equation in spherical coordinates, cou- through a constant enthalpy change over time (Figure 1b).

pled with a heat balance. The size of the spherical domain The outputs of the calculations are the evolution with time represents the average grain size. Since the dendritic nature ^of the phase fractions as well as the evolution with time of of the grains is not accounted for, only the formation of the solute profile in the grain envelope at room temperature.

"globulitic" grains is modelled, whose approximation by Homogenisation is modelled using the same one-dimensio- spheres is well justified. _Such grain structure is often obser- ^{nal solute diffusion model developed} for solidification. It is ved in aluminium casting alloys, a common industri.al prac- ^done by heating the system (through a positive enthalpy tice being to add into the melt very efficient inoculation par- ^{change over time) up to} the desired homogenisation tempe- ticles. The resolution of the conservation equation accounts ^{rature, and by} keeping the system at that temperature for ^a for diffusion in maximum three phases. For instance, the ^{given time (Figure 1b). As a} result, the O-AlrCu phase is primary aluminium solid phase, cr-Al, the liquid, l, and the progressively dissolved, eventually disappears, and the con- secondary 0-AlrCu phase, are simultaneously present during centration profile within the grain is flatten.

the eutectic transformation according to Figure la. During ^Modelling of precipitation is carried out by using the model cooling of the system from an initially liquid state, each new ^{first proposed by Langer and Schwartz} [1] ^{and modified by} solid phase starts to form without nucleation undercooling Kampmann and Wagner [2]. ^The model considers nuclea- once the uniform temperature domain reaches a new equili- tion, growth and coarsening of the precipitates as concomi- brium temperature (i.e., either the liquidus or the eutectic ^tant processes. Rate equations are thus written and solved

Fígure 1 - (a) Aluminium-rich region of the binary Al-Cu ^phase diagram and (b) schematícs of a typical temperature history for ^{a cast} alloy with structural hardening. The probess parameters describing the heat treatments of a cast alloy are defined in (b).

Figura ^{1 -} (a) Regione del diagramma di fase binario ^di Al-Cu ricca di allumínio e (b) schema di temperature tipiche per una lega per getti con indurimento strutturale. I parametri di processo che descrívono i trattamenti termici di una lega per getîi sono definiti ín (b).

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solution strengthening. The model is applied at different lo_

cation within the grain to deduce the ìpace dependence of

the flow stress, o.,(r).

Finally, the globàl yield stress of the material is deduced from a mechanical model developed by Hervé and Zaoui [4j for a heterogeneous material. Thè moíet is based on u ,"jf_

consistent Eshelby-type approach and is extended to a mate_

rial made.9t qn!éricat embedded layers. Each tayer is cta_

racteized by individual mechanical properties cornputed by the structural hardening model.

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,2 - Variation as a function of ^the residual eutectic fraction,

8, ^{of the} normalized conve,ntional yield stress of cast allóys, o o,

with respect to a perfectly

,homogenised and segregation jiee _al"iiy (no eutectíc phase left and no solute gradient),"dJo.r. _The standaid alloy is defined by the alloy compositíon, At_SwtToUu, the average grain síry of tle a-Al phase, Rr=50pp, rhe solidificatio, pror"i,

parameters, Tu=-1!ft t,. T

"n=-lKr-t, ^the homogeiisation process parameters, T

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Tp=400K. rp=l0days. See Figure"lbfor _rhe iefinition of rhe process parameters.

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-Variazione ⁱⁿ funzione ^della frazione eutettíca residua, gs dello sfurzo di snervamento convenzionale normalizzato di

leghe fus.e, dr, rispetto ad una lega perfettamente omogeneizzata e priva di segregafioni (nessuna rimanente fase eutettica e nessun gradiente di soluto), d" _o.r. h ^lega standard è definita da compo^sizione della lega, Al-Swt%oCu, dimensioni media del grano dellalase a-Al, Rr=50p11n, parametri _del processo di

solidificazione. _T'=-.1 yt r. _{T,n=-l Ks-r, parametri} del processo di omoSenliyylione, T *=1.1ft ti Ton-->--. Tn=800K. trllh,

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Tr=500K. tr=l g.iorno. Le variazioni individfiali ,onril!.ro,t, p'r,

quesri parcimetri comprendono R"= lee11a. i _^=-JK, ^,, i ₌ ^j11r,.

T^=.z1oy..rn=l!h. ^To=400K. r,=î0 _siomi. Si'veda tafigì'ra tb per la definizione dei ptirametri df processo.

using a Runge-Kutta algorithm. As a result of the calcula_

tion, the size and volume fraction of the precipitates _{are de_}

termined at each radial position within thè sphàrical grain as a function of the local composition and for à given artificial ageing temperature (Figure _1b).

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The main results of the model are summarised in Figure 2.

For a given alloy composition, with different combinations of the process parameters (see figure caption for the list of

the. parame.ters. investigated, _as well aj the magnitude of their variations), it is found that the variation oí the yield stress of the alloy is a direct function of the residual eut-ectic fraction. This simple final result deduced from several cal_

culations is consistent with the usual industrial practices. In order to optimise-.the yield stress of cast alloys, the process parameters are adjusted so as to maximise the solute òontent of the o,-Al solid solution. In turn, this practice corresponds to a minimisation of the residual _eutectic fraction.

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del which describes the microstructure óhange of alumi- nium-copper alloys during its process route, anà iB effect on the final yield stress. The model is now completed _{and has} been used to calculate the combined effect óf inoculation, cooling rate, temperature and time of the homogenising _and ageing treatments. Such predictions _{are sound} sLce thay in_

clude an intermediate calibration procedure of the modei pa_

rameters ^-based on experimental hardness measuredby Hardy in homogenised aluminium-copper alloys. Thus, tÉ

model establishes direct links between ine main process pa_

rameters ofcast alloys and its final yield stress.

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From the output of the phase transformation models, _the structural hardening model developed by Deschamps [3] ^is

used. It accounts for the volumetriò fraciion and toithè slze

of the precipitates, _as well as for the remaining solute con, tent of ^the matrix. The flow stress is thus catulated as a function of the interaction of the dislocations with the preci_

pitates, accounting for the transition occurring betweén the shearing and the Orowan mechanisms, u, *jl as for solid

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The authors would like to warmly acknowledge the Kórber Foundation and the Ecole polytechnique Fédèrale de Lau_

sanne for their financial supports.

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1) {..f. ^{Langer and A.} J. Schwarrz, phys. _Rev. A21 (1980) 948.

2) R. Kampmann _and R. Wagner, Decomposition _of Alloys:

th".:Tt staggs. Eds. p. _Haasen, V. Gerold, R. Wagner and M. F. Ashby, Pergamon press, Oxford (Igg4) 91.

3) A. Deschamps, PhD Thesis.INPG (F) (t992).

4) E. Hervé andZaoui, Inr. J. Engng Sci.31 (1gg3) l.

5) H. K. Hardy, _{J. Insr.} Merals ₇₉ (1951) 3Zt.

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E' stato svíluppato un modello per la previsione dell'evolu- zione della microstruttura di leghe per getti durante il pro- cesso industriale. I materiali presi in esame sono leghe bi- narie alluminio-rame. Il processo consiste in colata, omoge- neizzazione ed invecchiamento. E' stato costruito un model- lo, per il successivo calcolo delle microsegregazioni durante solidificazione e omogeneizzazione e della precipitazione durante l' invecchiamento della le ga.

Il modello ha consentito di calcolare la tensione di snerva- mento finale del materiale infunzione dei parametri di pro- cesso (velocità di raffreddamento della colata, inoculazione della lega, durata e temperatura di trattamenti termici, omogenizzazione ed invecchiamento). Ciò è stato possibile mediante l'introduzione di un modello d' indurimento strut- turale fra ^{i modelli di} trasfurmazione di fase indotta dalla dffisione e un modello meccanico finale per ⁱ materiali ete- rogenei.

I risultati principali del modello sono icapitolati nellafigu- ra 2.

Per una certa composizione della lega, con diverse combi-

nazioni dei parametri di processo (si veda didascalia della figura per la ^{lista dei} parametri studiati, e dell'ampiezza delle loro variaaioni), si è osservato che la variazione della tensione di snervamento della lega è una funzione ^diretta della frazione eutettica residua. _Questo semplice risultato finale dedotto da diversi calcoli è coerente alle normali pra- tiche industriali. Per ottimizzare il carico di snervamento delle leghe fuse, i ^parameîri trattati sono stati regolati in modo da mnssimizzare il contenuto di soluto della soluz.ione solida u-Al. Inoltre, questa pratica corrisponde ad una mi- nimiuazione de lla frazione ^{eutettic a re sidua.}

Tutti questi modelli sono rAggruppati in un singolo modello integrato che descrive il cambiamento microstrutturale del-

le leghe di alluminio-rame durante iI relativo processo ed effetto sulla tensione di snervamento finale. Il modello cosi' completato è stato utilizzato per calcolare l' effetto con- giunto di inoculazione, velocità di raffreddamento, tempera- tura e tempo dei trattamenti di omogenizzazione ed invec- chiamento. Tali previsioni sono attendibili poiché includono una procedura intermedia di calibratura dei parametri di modello basati su durezze sperimentali misurate da Hardy in leghe di alluminio-rame omogeneizzate. Quindi, il model- lo stabilisce collegamenti diretti fra ⁱ principali parametri di

processo delle leghe per getti e i relativi carichi di snerva- mento.finale.

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