Hot tearing and deformation in direct-chill casting of aluminum alloys

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Hot tearing and deformation

in direct-chill casting of aluminum alloys

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Hot tearing and deformation

in direct-chill casting of aluminum alloys

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 30 mei 2005 te 10.30 uur

door

Suyitno

bachelor of engineering in werktuigbouwkunde master of science in materialkunde

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Prof. ir. L. Katgerman

Toegevoed promotor: Dr. W.H. Kool

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof. ir. L. Katgerman, Technische Universiteit Delft, promotor

Dr. W.H. Kool, Technische Universiteit Delft, toegevoed promotor Prof. dr. R. Boom, Technische Universiteit Delft & CORUS

Prof. dr. I.M. Richardson, Technische Universiteit Delft Prof. dr. ir. S. van der Zwaag, Technische Universiteit Delft

Prof. dr. A. Mo, SINTEF Materials Technology, Oslo, Norway Dr. ir. R.N. Kieft, CORUS RD&T, Ijmuiden

This research was carried out as part of the strategic research program of the Netherlands Institute for Metals Research (NIMR) on the “Experimental description and process simulation of direct-chill (DC) casting of aluminum alloys” project number MP 97014.

ISBN 90-9019393-6

All right reserved. No part of the material protected by this copy right notice may be reproduced or utilized in any form or by any means, electronical or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the author.

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Chapter 1

Introduction

α

1.1 Direct-chill casting of aluminum alloys

Direct-chill (DC) casting, which was invented independently in the early 1930s [1] and 1940s [2], is a semi-continuous process for producing extrusion billets and rolling slabs. Although the process has been known for a relatively long time, the continuous technological development is still going on. This is due to the demand for more effectiveness and efficiency in this process with improving quality and productivity. Also more critical alloys are cast. Some comprehensive papers reviewing recent developments can be found in refs. [3-5].

The DC casting processes can be classified based on the applied casting orientation and are distinguished between vertical or horizontal direct-chill casting where the slab or billet moves in downward or horizontal direction, respectively.

α_{Adapted from the paper in: Progress in Materials Science, vol. 49, no. 5, 2004, pp.} 629-711

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For casting aluminum billets and slabs, which will be used in wrought alloy applications, DC casting is the standard industrial casting technique.

The vertical DC casting process is schematically shown in Fig. 1.1. Casting is performed by pouring the liquid metal from furnace (1) through the launder (2) inside a stationary mold (3) onto a moving bottom block (4), which is stationary in the beginning. The moving bottom block moves downwards and the billet or slab will moves with it. The mold is cooled by water flow that is called primary cooling. At the exit of the mold, the water flow directly impinges on the billet or slab that is called the secondary cooling. Because of the effective cooling, solidification proceeds rather fast. The casting process is stopped when the billet or slab has reached the desired length.

1

2

3

4 billet

3

Figure 1.1. Schematic diagram of DC casting apparatus. 1) tilting furnace, 2) launder, 3) hot-top round mold, and 4) bottom block.

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Introduction

3

Figure 1.2. Solidification front in DC casting.

The schematic view of the solidification front during DC casting is shown in Fig. 1.2. The solidification front forms a sump which its depth increases with distance from the surface of billet or slab. In the solidification front, three phase regions (solid, mushy and liquid) are developed. The vertical length of the regions is dependent on the casting parameters (i.e. casting speed, water flow rate, alloy composition, dimension of billet or slab). In DC casting, the name “mushy zone” is somewhat misleading, as its top part is actually a slurry, because the newly formed grains are still suspended in the liquid. Only after the temperature has dropped below the coherency temperature, a real mush is formed.

Commonly, three stages are known during the casting: start-up, steady state and ending stage. The position of the solidification front, the temperature field and stress and strain fields are the major characteristics in these stages. In the start-up and ending stages, they change with time. The variation of the solidification front, temperature field and stress and strain fields in the start-up phase leads to typical defects such as the formation of hot tears and deformation of the billet or slab.

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1.2 Hot tearing

1.2.1 Solidification behavior

Unlike pure metals, which solidify at one temperature, alloys transform gradually from liquid to solid over a (wide) temperature interval. During casting there is a considerable time during which the alloy consists of both solid and liquid. The material in this semi-solid state is divided into two classes: slurries and mushes. A slurry is defined as a liquid with suspended solid particles. At some temperature solid grains start to interact with each other and the material develops certain strength. Below this temperature, the material is called a mush, i.e. a solid network with liquid in between. The solid fraction at which this transition occurs varies between 0.25 and 0.6, depending on the morphology of the solid particles. Because of the strongly different mechanical behavior of these different morphologies, slurries are usually described by viscosity-based models and mushes are usually described by deformation-based models [6]. The viscosity-based models start from the liquid side and are modified to take into account the effect of the increasing amount of solid particles. The deformation-based models are based on models for hot working, which are modified to take into account the presence of liquid. The transition from a slurry to a mush remains complicated to model, and a satisfactory model, which describes the behavior for the complete solidification range, is yet to be developed.

The solidification process can be divided in four stages, based on the permeability of the solid network [10,13,16,7]: Mass feeding, in which both liquid and solid are free to move; Interdendritic feeding, in which the remaining liquid has to flow through the dendritic network. After the dendrites have formed a solid skeleton, the remaining liquid has to flow through the dendritic network. A pressure gradient may develop across the mushy zone by solidification shrinkage occurring deeper in the mushy zone. However at this stage the permeability of the network is still large enough to prevent pore formation; Interdendritic separation, in which the liquid network becomes fragmented and pore formation or hot tearing may occur. With increasing solid fraction, liquid is isolated in pockets or immobilised by surface tension. When the permeability of the solid network becomes too small for the liquid to flow, further thermal contraction of the solid will cause pore formation or hot tearing; Interdendritic bridging or solid feeding, in which the ingot has developed a considerable strength and solid-state creep compensates further contraction. At the final stage of solidification (fS > 0.9), only

isolated liquid pockets remain and the ingot has a considerable strength. Solid-state creep can now only compensate solidification shrinkage and thermal stresses.

In the casting practice of alloys one is only familiar with various defects occurring in the final product. One of the main defects is hot tearing (also

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Introduction

5 mentioned hot cracking, or hot shortness). From many studies [8-16] starting already in the fifties, and reviewed by Novikov [17] and Sigworth [18], it appears that hot tears initiate above the solidus temperature and propagate in the interdendritic liquid film. This result in a bumpy fracture surface covered with a smooth layer of liquid film and sometimes with solid bridges that connect or connected both sides of the crack [15,16,19-25]. During solidification, the liquid flow through the mushy zone decreases until it becomes insufficient to fill initiated cavities so that they can grow further. It is known that a fine grain structure and controlled casting (without large temperature and stress gradients) help to avoid hot cracking. The hot cracking susceptibility depends on the composition of an alloy and their connection is many times established [26].

During the DC casting of billet, the hot tear formation is critical during the start-up phase and at high casting speed in steady state phase. The ramping procedure is commonly applied in the start-up phase for avoiding the hot tear. The hot tear is developed in the center of a billet. An illustration of hot tears in the start-up phase and the steady-state phase due to increasing casting speed are shown in Figs. 1.3 and 1.4, respectively.

In the first two stages of solidification, feeding is usually sufficient to avoid any casting defects. It is mainly the “interdendritic separation” stage in which the ingot is vulnerable to pore formation and hot tearing.

A large freezing range alloy promotes hot tearing since such an alloys spends a longer time in the vulnerable state in which thin liquid films exist between the dendrites. The liquid film distribution is determined by the dihedral angle θ. With a low dihedral angle, the liquid will tend to spread out over the grain boundary surface, which strongly reduces the dendrite coherency. With a high dihedral angle the liquid will remain as droplets at the triple points so that the solid network holds its strength.

Apart from these intrinsic factors, the solidification shrinkage and thermal contraction impose strains and stresses on the solid network, which are required for hot tearing. It is argued that it is mainly the strain and the strain rate, which are critical for hot tearing [9,16]. Stresses do not seem critical as the forces available during solidification are very high compared to the stresses a semi-solid network can resist [16].

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a.

b.

Figure 1.3. The hot tear in the start-up phase of 20 cm radius billet. a) radial and b) longitudinal cross section view.

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Introduction

7

1.2.2 Hot tearing theories

A lot of efforts have been devoted to understand the hot tearing phenomenon. The compilation of research in this field has been done by Novikov [17], Sigworth [18] and Eskin et al. [26]. Zheng et al. [27] reviewed the possible causes of hot tearing. Some of the hot tearing mechanisms, suggested in literature, are briefly described below.

Novikov and Novik [28] have reported that at low strain rates grain boundary sliding is the main mechanism of deformation of a semi-solid body. The load applied to the semi-solid body will be accommodated by a grain boundary displacement that is lubricated by the liquid film surrounding the grain. Prokhorov [29] proposed a model for deformation of the semi-solid body. If two tangential forces τ1 and τ2 are applied to the semi-solid body in equilibrium, the response of

the body manifests itself as grain movement and at some point the grain will touch each other. The liquid covering the grain will circulate to the lowest pressure point. Further deformation will be possible if the surface tension and resistance to liquid flow are sufficient to accommodate the stress imposed. If not, a brittle intergranular fracture or hot tearing will occur. In relation to this theory, Prokhorov postulated that: (1) an increase in film thickness increases the fracture strain, (2) a decrease in grain size increases the fracture strain, (3) any non-uniformity of grain size decreases the fracture strain. Based on this theory, the main measure for hot tearing is the ductility of the semi-solid body. A hot tear will occur if the strain of the body exceeds its ductility.

A theory of shrinkage-related brittleness divides the solidification range into two parts. In the upper part the coherent solid-phase network does not exist. Cracks or defects occurring in this stage can be healed by liquid flow. As the solidification progresses and the solid fraction further increases, at a certain stage or a certain solid fraction a coherent network is formed. This stage is considered as the start of linear shrinkage. At the coherence point temperature and below, the shrinkage stress is imposed onto the semi-solid body. Fracture or hot tearing occurs if the shrinkage stress exceeds the rupture stress [30,31].

Pellini [9] suggested a hot tearing theory based on the strain accumulation with the following main features: (1) cracking occurs in a hot spot region, (2) hot tearing is a strain-controlled phenomenon which occurs if the accumulated strain of the hot spot reaches a certain critical value, and (3) the strain accumulated at the hot spot depends on the strain rate and time required for a sample to pass through a the stage where the interdendritic liquid exists as thin layer. The most important factor of hot tearing based on this theory is the total strain at the hot spot region. The total strain is the additive of strain over a period within which the hot spot exists. With taking into account that the highest strain accumulates in the liquid

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film, Dodd [32] and Metz and Flemings [33] explain the increase of hot tearing caused by the segregation of a low-melting component from the viewpoint that this addition increases the time of the liquid film existence. Although Pellini mentions a critical value of the accumulated strain, it is not clear whether it is ductility or another entity. The Pellini theory is a basis for a hot tearing criterion proposed by Clyne and Davies [34].

Some authors suggest that not the strain but the strain rate is the critical parameter for hot cracking. The physical explanation of this approach is that the strain rate during solidification is limited by the minimum strain rate at which the material will fracture. Prokhorov [29] is the first who suggested a criterion based on this approach. More recently, a strain-rate based hot tearing criterion is proposed by Rappaz et al. [35].

Yet another approach to the hot tearing phenomenon is the assumption that failure take place at a critical stress. The liquid surrounding the grain is considered as a stress riser of the semi-solid body [36,37]. In this theory, a liquid-filled crack is considered as a possible crack initiation site. The propagation of the crack is determined by the critical stress [37]. The critical stress is mostly determined using the modified Griffith energy balance approach. The modification of the Griffith approach particularly accounts for the effect of plasticity as proposed by Gilman [38] and Orowan [39]. Another approach within the fracture mechanics theory is proposed by Sigworth [18] who considers a possibility of applying a liquid-metal-embrittlement concept to the hot tearing case.

There is also a group of hot tearing theories that consider the hindered feeding of the solid phase by the liquid as the main cause of hot tearing. Niyama [40] and Feurer [41] use this approach to derive the hot tearing criterion. Based on this theory, hot tearing will not occur as long as there is no lack of feeding during solidification. Clyne and Davies [13,34,42] give more attention to the time spent in the mushy state. The last stage of solidification is considered as most susceptible to hot tearing. However, during the further decrease of the liquid fraction, bridging between adjacent dendrites is established so that the interdendritic separation is prevented.

Based on this models, several hot tearing criteria have been postulated in the past decades. Feurer [12] used the fluid flow through a porous network to calculate the afterfeeding by liquid metal. Hot tears will initiate when this afterfeeding cannot compensate the solidification shrinkage. Clyne and Davies [13] defined a cracking susceptibility coefficient (CSC) as the ratio between the time tV during

which the alloy is prone to hot tearing and the time tR during which stress

relaxation and afterfeeding can take place. These times are defined as the periods during which the fraction liquid is between 0.1 and 0.01 and between 0.1 and 0.6, respectively. These criteria were combined with a heat flow model describing the

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Introduction

9 DC casting process by Katgerman [43]. This enabled the determination of the cracking susceptibility coefficient as a function of the casting parameters. Unfortunately, the above criteria are restricted in their use because they give only a qualitative indication for the hot tearing susceptibility.

The ductility of semi-solid non-ferrous (including aluminium) alloys was used as a basis for a hot shortness criterion suggested by Novikov [17]. A characteristic called a “reserve of plasticity in the solidification range” is proposed which is the difference between the average integrated value of the elongation to failure and the linear shrinkage in the brittle (or effective, or vulnerable) temperature range. Prokhorov [29] formulated a mechanical criterion, in which the hot cracking sensitivity is determined by the shrinkage and apparent strain rate in the mush in relation to the fracture strain rate of the mush. In this approach, effects of the surrounding configuration are accounted for by the apparent strain rate.

The first two-phase model, which takes into account both fluid flow and deformation of the solid network, is the Rappaz–Drezet–Gremaud (RDG) hot tearing criterion [35]. The RDG criterion is formulated on the basis of afterfeeding, which is limited by the permeability of the mushy zone. At the solidification front the permeability is high but deeper in the mushy zone the permeability is restricted. A pressure drop along the mushy zone exists which is a function of this permeability and the strain rate. If the local pressure becomes lower than a critical pressure, a cavity is initiated. The model is implemented in a thermomechanical model for DC casting by Drezet et al. [44] to predict hot tearing during billet casting. The hot tearing susceptibility is found higher during start-up of the casting and in the centre of the billet, which agrees with general casting practice.

A further development of the RDG criterion is carried out by Braccini et al. [45]. They included plastic deformation of the solid phase and a criterion for the growth of a cavity. They base their model on two simplified geometric models, one for a columnar dendritic and one for an equiaxed dendritic structure. Explicit relations are developed for critical strain rates and they indicate that the critical strain rate decreases with increasing solid fraction.

Many studies have used tensile testing at semi-solid temperatures to study hot tearing either by in-situ solidification experiments [19-21,7,46-48] or by reheating specimens from room temperature [23,24,37,47,49,50]. Both techniques led to the following general results. In several aluminum alloys it is observed that both strength and ductility strongly decrease from just below the solidus temperature to above solidus temperature, while the fracture surface changes from rough, related to the ductile behavior, to smooth, related to the presence of a liquid film. Further, cracks initiate at micro-pores or molten inclusions and continue along grain boundaries.

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1.3 Outline of this thesis

This thesis deals with hot cracking during direct-chill casting of aluminum alloys. Issues addressed are the thermomechanical modeling of direct-chill casting, the implementation and evaluation of existing hot tearing criteria, experimental observation the microstructure and hot tearing at various parameters during DC casting, proposing a route for hot tearing prediction and thermal contraction during and after solidification enabling the prediction of ingot distortion.

In Chapter 2, the finite element simulation of DC casting is described. The emphasis is put on the mushy zone and close surroundings. The effect of casting speed and start-up conditions on the stresses, strains, depth of sump and length of mush is investigated. The computation described in this chapter will serve as a basis for understanding the computed hot cracking tendencies using various hot tearing criteria that are investigated in the next chapter.

In Chapter 3, various hot tearing criteria are implemented and evaluated. Calculations of hot cracking tendency using these criteria are performed at various casting speed and casting conditions. The results are compared and critically examined.

In Chapter 4, experimental observation after the direct-chill casting of Al-Cu alloys are reported. The basic aim of this chapter is to analyze the effects of alloy composition and casting speed on structure formation and hot tearing of Al–Cu alloys. The analysis is based on systematic examination of billets of binary Al–(1– 5)% Cu alloys cast at different casting speeds. Experimental results on structure and hot tearing are correlated to computer simulated solidification and hot tearing patterns. Thermomechanical behavior of a solidifying billet is combined with available hot tearing criteria and the results are compared to the experiment.

In chapter 5, observation of the hot tear surface and micro-porosity are reported. The aim of this work is to investigate the hot tear surface of Al–Cu alloys with different copper concentrations produced by DC casting. The billets of Al–(1-4.5%)Cu alloys were cast with a ramping casting speed (i.e. steadily increase followed by a steadily decrease). The microstructure and porosity fraction are examined. The possible interaction between the hot tear and porosity is addressed.

In chapter 6, a hot tearing model is derived based on cavity formation when there is insufficient feeding during solidification. The feeding during solidification is incorporated using a transient mass balance equation. The cavity formed becomes a hot tear in case a critical dimension is achieved. The criterion for deciding whether the cavity becomes a hot tear or micro-porosity, which is not found in other models is included. Also, the flow behavior of the semi-solid state has been included in order to model the mechanical response of the semi-solid body.

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Introduction

11 In Chapter 7, experimental measurements of linear contraction during and after solidification of an AA5182 alloy are reported. These experimental results are compared to the finite element method (FEM) simulation of thermal contraction. Implementation for the prediction of ingot distortion during the start-up phase of direct-chill casting is also presented.

References

[1] Roth, W. (1936) Deutches Reichs patent No. 974203, date of registration 8 August 1936.

[2] Ennor, W.T. (1942) US patent No. 2301027, date of registration 3 November 1942.

[3] Emley E.F. (1976) International Metals Reviews, Review 206, p. 75. [4] Katgerman, L (1991) Cast Metals, 4, 3, p. 133.

[5] Grandfield, J. and McGlade, P.T. (1996) Materials Forum, 20, p. 29.

[6] Quaak, C.J. (1996) PhD Thesis, Delft University of Technology, Delft, The Netherlands.

[7] Forest, B. and Bercovici, S. (1980) Proc. Conf. Solidification Technology in Foundry and Casthouse, University of Warwick, Coventry, Metals Society, UK, p. 607.

[8] Bishop, H.F., Ackerlind, C.G. and Pellini, W.S. (1952) AFS Transactions, 60, p. 818.

[9] Pellini, W.S. (1952) Foundry, 80, p. 124.

[10] Borland, J.C. (1960) British Welding Journal, 7, p. 508.

[11] Metz, S.A. and Flemings, M.C. (1970) AFS Transactions, 78, p. 453. [12] Feurer, U. (1976) Giessereiforschung , 28, p. 75.

[13] Clyne T.W. and Davies G.J (1979) Proc. Conference on Solidification and Castings of Metals, Metals Society, London, UK, p. 275.

[14] Matsuda, F., Nakagawa, H., Katayama, S. and Arata,Y. (1982) Transaction of Japan Welding Society, 13, p. 41.

[15] Rogberg, B. (1983) Scandinavian Journal of Metallurgy, 12, p. 51. [16] Campbell, J. (1991) Castings, Butterworth Heinemann, Oxford, UK.

[17] Novikov, I.I. (1966) Goryachelomkost tsvetnykh metallov i splavov (Hot Shortness of Non-Ferrous Metals and Alloys), Moscow: Nauka (in Russian). [18] Sigworth, G.K. (1996) AFS Transactions, 104, p. 1053.

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[20] Ohm, L. and Engler, S. (1990) Giessereiforschung, 42, 4, p. 149. [21] Nedreberg, M.L. (1991) PhD Thesis, University of Oslo, Oslo, Norway. [22] Boyle, J.P., Mannas, D.A. and Walsh, D.W. (1992) International

Symposium on Physical Simulation, eds. D. Ferguson and J. Jacon, Dynamic Systems Inc, Poestenkill, NY, USA, p. 165

[23] Spittle, J.A., Brown, S.G.R., James, J.D. and Evans, R.W. (1997) Proceedings of 7th_{Intern. Symposium on Physical Simulation, Tsukuba,}

Japan, p. 81.

[24] Van Haaften, W.-M., Kool, W.H. and Katgerman, L. (2000) Continuous Casting, Ed. K. Ehrke and W. Schneider, Wiley-VCH, Germany, p. 239. [25] Farup, I., Drezet J-.M. and Rappaz, M. (2001) Acta Materialia, 49, p. 1261. [26] Eskin, D.G., Suyitno and Katgerman, L. (2004) Progress in Materials

Science, 49, 5, p. 629.

[27] Zheng, M., Suyitno and Katgerman, L. (2001) Proc. 22nd Risø International Symposium on Materials Science, Edts:Dinesen AR, Eldrup M, Juul Jensen D, Linderoth S, Pedersen TB, Pryds NH, Schrøder Pedersen A, Wert JA, Risø National Laboratory, Denmark, p. 455.

[28] Novikov, I.I. and Novik (1963) Doklady Akad. Nauk SSSR, Ser. Fiz., 7, p. 1153.

[29] Prokhorov, N.N. (1962) Russian Castings Production, no. 2, p. 172. [30] Lees, D.C.G. (1946) Journal Institute of Metals, 72, p. 343.

[31] Korol’kov, A.M. (1963) Casting properties of metals and alloys, Consultants Bureau, New York, USA.

[32] Dodd, R.A. (1956) Foundry Trade Journal, p. 321.

[33] Metz, S.A. and Flemings, M.C. (1969) AFS Transactions, 77, p. 329. [34] Clyne, T.W. and Davies, G.J (1975) British Foundrymen, 68, p. 238.

[35] Rappaz, M., Drezet, J.M. and Gremaud, M. (1999) Metallurgical and Materials Transactions A, 30A, p. 449.

[36] Patterson, K. (1953) Giesserei, 40, p. 597.

[37] Williams, J.A and Singer, A.R.E. (1968) Journal Institute of Metals, 96, p. 5. [38] Gilman, J.J. (1960) Proceedings Conference of Plasticity; 2nd Symposium on

Naval Structural Mechanics, Rhode Island, RI, USA, 1960, p. 43.

[39] Orowan, E. (1950) Fatigue and Fracture of Metals, ed. W.M. Murray, Technology Press of MIT, Cambridge, Massachussets, USA, p. 139.

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Introduction

13 [40] Niyama, E. (1977) Japan–US Joint Seminar on Solidification of Metals and

Alloys, p. 271.

[41] Feurer, U. (1977) Quality Control of Engineering Alloys and the Role of Metals Science, Delft University of Technology, p. 131.

[42] Clyne, T.W. and Davies, G.J. (1981) British Foundryman, 74, p. 65. [43] Katgerman, L. (1982) Journal of Metals, 34, 20, p. 46.

[44] Drezet, J-.M. and Rappaz, M. (2001) Light Metals 2001, ed. J.L. Anjier, TMS, Warrendale, PA, USA, p. 887.

[45] Braccini, M., Martin, C.L. and Suery, M. (2000) Modeling of Casting Welding and Advanced Solidification Processes IX, eds. P.R. Sahm, P.N. Hansen, and J.G. Conley, Shaker Verlag, Aachen, p.18.

[46] Lankford, W.T. (1972) Metallurgical Transactions, 3, p. 1331.

[47] Magnin, B., Maenner, L., Katgerman, L. and Engler, S. (1996) Materials Science Forum, 217–222, p. 1209.

[48] Instone, S., StJohn, D. and Grandfield, J. (2000) International Journal of Cast Metals Research, 12, p. 441.

[49] Martin, C.L., Favier, D. and Suéry, M. (1999) International Journal of Plasticity, 15, p. 981.

[50] Fredriksson, H. and Lehtinen, B. (1979) Proceedings Conference on Solidification and Casting of Metals, University of Sheffield, Sheffield, UK, p. 260.

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Chapter 2

FEM simulation of mushy zone behavior during

direct-chill casting of an Al−4.5%Cu alloy

β

In this chapter, characteristic parameters such as stresses, strains, sump depth, mushy zone length and temperature fields are calculated through the simulation of the DC casting process for a round billet by using a finite element method. Focus is put on the mushy zone and solid region close to it. In the center of the billet, circumferential stresses and strains (which play a main role in hot cracking) are tensile at temperatures close to the solidus temperature, whereas they are compressive near the surface of the billet. The stresses, strains, depth of sump and length of mushy zone increase with increasing casting speed. They are maximum in the up phase and are reduced by applying a ramping procedure in the up phase (application of a lower but slowly increasing casting speed during start-up). Stresses, strains, depth of sump and length of mushy zone are highest in the center of the billet for all casting conditions considered.

β_{Based on the paper in: Metallurgical and Materials Transactions A, vol. 35A, no. 9,} 2004, pp. 2917-2926.

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2.1 Introduction

Direct-chill (DC) casting is a semi-continuous process for producing extrusion billets and rolling slabs. In this casting process, liquid metal is poured onto a moving bottom block inside a mold. The mold is cooled by water flow that is called primary cooling. At the exit of the mold, water flow directly impinges on the billet or slab, which is called secondary cooling. During the casting process, metal will pass through a mushy state that is critical for the occurrence of some defects such as hot tearing and micro-porosity. The defects are related to the mechanical behavior of the mush, in combination with the feeding possibilities. Understanding of the behavior of the mush during DC casting is not an easy task because of the complex phenomena occurring during the solidification.

Modeling of stresses, strains and temperatures during DC casting is generally done by using a finite element method [1-8]. Most researchers were working on the simulation of the thermomechanical behavior of a DC cast slab, and most attention was devoted to the thermomechanical behavior at temperatures lower than the solidus temperature [1-4]. The stresses and strains in the center of a billet at temperatures lower than the solidus temperature are tensile except for the axial strain that is compressive [1,2]. Also the stresses and strains in the center of a slab are found tensile [3,4]

Recently, simulation of the thermomechanical behavior of a billet and/or slab at temperatures above the solidus temperature gains significant attention [5-8]. Several constitutive models of the mushy state are used for simulation of the thermomechanical behavior such as an elastoviscoplastic law [5,6] and Garofalo’s law [7]. All of these models have been fitted to experimental data. In another study [8] solid data are simply extrapolated to temperatures corresponding to the mushy zone up to the coherency point.

By applying an elastoviscoplastic law, a tensile circumferential stress in the center of billet and compressive circumferential stress at the billet surface was reported [5,6]. In these studies an increase of the casting speed caused an increase of the principal plastic strain. Another study [7] showed that there is no significant effect of applying Garofalo’s law for the mushy zone on the computed stresses and strains in the a slab, instead of extrapolating the solid data to the mushy range. It is reported [8] that in the start-up phase stresses, strains and strains rates in the mushy zone at 0.95 fraction of solid increase with increasing casting speed.

The prediction of the occurrence of defects such as hot tears by using certain criteria requires the computation of the mechanical and physical behavior of the mush. To that purpose, extensive modeling is needed based on thermophysical and thermomechanical data. In this work, the stresses, strains, sump depth, mushy zone length and temperatures fields are calculated through the simulation of the DC

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FEM simulation of mushy zone behavior during direct-chill casting

17 casting process for a round billet. Focus is put on the mushy zone and its surroundings. The mechanical behavior is explored by means of FEM simulations that include a solidification and constitutive model of the solid and the mushy. The effects of various casting conditions are calculated.

2.2 Modeling

2.2.1 General computation procedure

DC casting of a billet, 100 mm radius and 1000 mm length, is simulated. An axi-symmetric model is used in this work. Due to the symmetry, only a half section of the billet and bottom block needs to be modeled. For the simulation, a coupled computation of stress and temperature fields is applied using 4-node rectangular elements with 4 Gaussian integration points.

In the simulation, the ingot remains in a stationary position, while the mould and the impingement point of the water flow move upwards with a velocity equal to the casting speed. Continuous feeding of the liquid metal is implemented by activating horizontal layers of elements incrementally. The computational domain is shown in Fig. 2.1.

Stresses, strains and temperatures are calculated as a function of position in the billet and time. Equations are solved iteratively using certain time steps. The computation continues until a preset time is reached. In the calculation a solidification model and a constitutive model are used, which are incorporated in the finite element iteration.

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bottom block Γ 1 Γ 2 Γ 4 Γ 5 Γ 3 mold Γ 6 Γ 7 billet r z

Figure 2.1. Computational domain of the DC cast billet. Γ1 - Γ7 correspond with boundary

conditions defined in Section 2.2.3.

2.2.2 FEM formulation

The FEM model of the billet is derived based on the coupling of the heat flow and mechanical equilibrium equations.

Finite element equation of the heat flow equations

For the non-stationary condition, the thermal balance equation is written as:

t T c q T c k p v p ∂ ∂ = + ∇

ρ

( 2.1 )

where T is the temperature, k is the thermal conductivity, qv is the heat source, cp is

the specific heat, ρ is the density, and t is the time. In case of solidification, the heat source is the latent heat released during the solidification. The boundary conditions imposed to the thermal balance equation are:

c T T = ( 2.2a ) c q n T k = ∂ ∂ ( 2.2b )

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FEM simulation of mushy zone behavior during direct-chill casting 19

(

f

)

cT T n T k = − ∂ ∂

_α

( 2.2c )

where Tc is the temperature of the body surface, qc is the heat flux normal to the

body surface, αc is the heat convection coefficient and Tf is the temperature of the

liquid. S is the total surface of the body. The finite element formula is obtained by applying the Galerkin method [9] to Eq.2.1 with the boundary conditions Eqs. 2.2a-c and assuming that the temperature T in the element can be approximated by the temperature vector {T}e on the nodes by an interpolation function [H] as

[ ]{ }

H T e

T = , and [G] is the matrix of the gradient operator [N];

[ ]{ } [ ]

K_I T + C

{ }

T& =

{ }

F ( 2.3 ) where:

[ ]

_{∑ ∫}

[ ] [ ]

_{∑ ∫}

[ ] [ ]

_{∑ ∫}

[ ]

∂ ∂ − + = e S T e S c T e V T I G k G dV H H dS H k _n H dS K

α

( 2.4 )

[ ]

=

_{∑ ∫}

[ ]

e V p T _c _H _dV H C

ρ

( 2.5 )

{ }

=

_{∑ ∫}

[ ]

−

_{∑ ∫}

[ ]

−

_{∑ ∫}

[ ]

e S c f T e S c T e V v T_q _dV _H _q _dS _H _T _dS H F α ( 2.6 )

The following notations are used here: [KI] is the global heat capacity

matrix, [C] is the global conductivity matrix and {F} is the global load vector. Various time-integration schemes are available for solving Eq. 2.3. The computed temperature is used for updating the temperature dependent material properties and computing thermal stresses in the mechanical computation.

Finite element equation of the mechanical equilibrium equations

The statement of virtual work is as follows:

0 = Γ − −

_∫

∫

Γ d t u dV f u dV T _i i V i T i V ij T ij

σ

δ

δε

( 2.7 )

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where σij is the internal stress tensor, fi are the components of externally applied

forces per unit volume, ti are the components of the applied surface traction per

unit length acting on the boundary Γ, δui are the components of the virtual

displacement vector, δεij is the strain tensor corresponding to the virtual

displacement, and V is the volume element.

The additive decomposition for the rate of the strain tensor in a thermo-elasto-viscoplastic deformation in a solid body has the form:

vp ij T ij e ij ij

ε

& = & + & + & ( 2.8 )

where

ε

_&_ije are the components of the elastic strain rate tensor,

ε

_&_ijTare the components of the thermal strain rate tensor and

ε

_&_ijvp are the components of the viscoplastic strain rate tensor.

The linear elastic deformations rate are related to the stress rates as follows:

kl ijkl e

ij E

σ

ε

& = _& ( 2.9 )

where

E

_ijkl denote the material coefficients that are defined by Young’s modulus and Poisson’s ratio. The thermal strain rate tensor is a function of the temperature variation and reads:

T ij T ij & &

α

ε

= ( 2.10 )

where

α

_ijis the linear thermal expansion tensor component. The viscoplastic strain rate tensor is formulated as follows:

(

)

(

)

   = T , f , f T , f S vp ij

_ϕ

_σ

σ

ϕ

ε

& sol sol T T T T 〉 ≤ ( 2.11 )    = 1 0

ϕ

Y Y

σ

〉 ≤

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21 where σ is the effective stress determined by the constitutive models,

σ

_Y is the yield stress and Tsol is solidus temperature. Separate constitutive models are applied

for the solid region and the semi-solid region as described in Section 2.2.5.

Spatial discretization of the mechanical balance equation is done by adopting a purely displacement-based formulation [9]. In the spatial discretization, the incremental displacement vector {∆u} and the incremental strain vector {∆ε} can be related using the strain displacement matrix [B], and interpolated by the nodal displacement vector in the form:

{ }

∆u =

[ ]{ }

N ∆u e ( 2.12 )

{ }

∆

ε

=

[ ]{ }

B ∆u e ( 2.13 )

The discrete forms of the mechanical balance equations are:

[ ]{ } { }

K ∆u = ∆L ( 2.14 ) where:

[ ]

K

[ ] [ ][ ]

B D BdV e V T

∑ ∫

= ( 2.15 )

{ } { }

L L

[ ] [ ]

B D

{ }

dV

[ ] [ ]

B D

{ }

vp dV e V T T e V T _∆

_ε

₊ _∆

_ε

+ ∆ = ∆ ₁

_{∑ ∫}

( 2.16 )

where [K] is the incremental stiffness matrix, {∆L} is the total force increment

vector, {∆L1} is the external force increment, [D] is the elastic stress-strain matrix,

{∆εT_{} is the thermal strain increment vector, and {}_∆εvp_{} is the elasto-viscoplastic}

strain increment vector. The unknown quantity i.e. the deformation increment {∆u}, can be solved by using Eq. 2.14.

A full Newton-Raphson algorithm is applied to treat the set of non-linear algebraic equations that arise after discretization of the non-linear continuum equations. The computation is started with given boundary conditions, initial conditions and a given time step. Eqs. 2.3 and 2.14 are solved iteratively and when values converge for a certain time step, they are then used as initial values for the next time step. The computation continues until the preset time is reached.

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2.2.3 Boundary conditions

Different boundary conditions are applied to the different boundaries Γ of the billet (Fig. 2.1). At the bottom, Γ1, it is assumed that the convective heat

transfer to the environment is constant, and that the position z is constant. At the interface of billet and bottom block, Γ2, the heat transfer is determined by

conditions of either contact, non-contact (open gap) or water intrusion, between billet and bottom block. These conditions depend on the gap distance and correspond with different values of the heat transfer coefficient. The values of the heat transfer coefficients depend on temperature (contact and water intrusion situation) or gap distance (non-contact situation). At boundary condition Γ2, the z

position of the bottom block is fixed but the billet can freely move in axial and radial direction.

The boundary condition, Γ3, corresponds with either contact or non-contact

between billet and mould, and it uses similar criteria and values as for the boundary conditions at Γ2. The billet may freely move in the negative radial direction. The

boundary condition Γ3 moves in axial direction with the casting speed. Boundary

conditions related to secondary cooling (water flow) are defined in two regions: water impingement zone, Γ4 and downstream zone, Γ5. With the boundary

conditions Γ4 and Γ5, the billet may move freely in radial and axial direction.

At boundary condition Γ6 temperature is constant (casting temperature) and

the boundary moves in axial direction with the casting speed. At the boundary Γ7

heat flux is zero, due to the axial symmetry.

All boundary condition data are summarized in Table 2.1. In this table the following parameters are defined: hbc is the heat transfer coefficient from bottom to

the surrounding, htc htnc and htwi are the heat transfer coefficients between bottom

block and billet for contact, non-contact and water intrusion situation respectively,

hmc and hnc are the heat transfer coefficients of mold and billet interface in contact

and non-contact situations respectively, q is heat transferred, A is area, Qw is water

flow rate, Tbulk is temperatureof cooling water, Tsat is saturation temperature i.e. the

temperature where the water will boil off the surface of the billet, and Vc is the

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23 Table 2.1. Data of boundary conditions for simulation.

Γ1 h = hbc z = constant hbc: 50 W/m2K Tsurroundings: 20 oC [10] Γ2      = twi tnc tc h h h h

if

m d m d m m d t t t 3 3 5 5 10 10 10 10 − − − − > < < <

zbottom block = constant

tnc tc,h h and h_twi see Tables 2.2-2.4 [10] Γ3







=

tnc mc

h

if

m

10 d

m

10 d

5 t 5 t − −

≥

<

Tmold = constant t V z= _c hmc: 1500W/m2K htnc: see Table 2.2 Tmold: 100 oC [10] Γ4 A q = 27300 T – 1273088.915 if T < 120 o A q = 94252.48 T – 9240434.453 if 120 oC ≤ T < 150 oC A q = 12259.18 T – 3058560.867 if T ≥ 150 oC t V z= _c [11] Γ5 A q =(-167000+cTbar)(Qw/60000)1/3 ∆T + 100(∆Tx)3 c = -21.2035Qw2 + 1.1508Qw + 62794 Tbar = ( T + Tbulk)/2 ∆T = T + Tbulk ∆Tx = T + Tsat

t

V

z

=

_c Qw: 150 l/min Tbulk: 10 oC Tsat: 90 oC [11] Γ6

T

=

T

_in

t

V

z

=

_c Tin: 700 oC

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0 A

q =

Γ7

0 A

q =

Table 2.2. Heat transfer coefficient in contact situation.

T (oC) htc (W/m2K) 0 200 500 600 680 700 800 1000 1000 1000 1000 1500 1500 1500

Table 2.3. Heat transfer coefficient in non-contact situation. Gap (m) htnc (W/m2K)

0.000 0.015

450 300

Table 2.4. Heat transfer coefficient during water intrusion.

T (oC) htwi (W/m2K) 0 100 150 200 400 600 700 1000 1000 2000 3000 1000 400 400

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25

2.2.4 Materials data

An Al–4.5%Cu billet was simulated. The material of the bottom block was taken as AA6063. Temperature dependent data for thermal conductivity, specific heat, thermal expansion coefficient and Youngs’ modulus were taken from literature [12-15].

2.2.5 Solidification and constitutive model

The simulation uses a solidification model, which accounts for back-diffusion [16]. The fraction of liquid fl is given by:

                − − − = − − ∗ ∗ 1 2 1 2 1 1 k k l m m s l s T T T T k f α

α

( 2.17 ) with

( )

[

exp _s

]

exp

( )

_s s * α α

α

₂1 2 1 1 1− − − − = ( 2.18 )

where Tm is melting temperature of the pure metal, T is temperature, k is partition

coefficient, αs is back diffusion coefficient and α*s is modified dimensionless solid

state back-diffusion parameter.

For the constitutive model of the solid metal, plastic deformation is described by a viscoplastic equation [5].

(

) (

)

n po p u po p K

ε

σ

= & + & + ( 2.19 )

where

σ

_{is true stress (MPa); K is stress at a strain and strain rate of unity (MPa);}

p

ε& is strain rate (s-1_); po

ε& is a small constant plastic strain rate (10-4_s-1_); p

ε is plastic strain; ε is a small constant plastic strain (10_po -2_{); u is strain rate}

sensitivity coefficient; n is strain hardening coefficient. The parameters K, u and n in this equation were fitted to the experimental data described in ref. [5]. They are temperature dependent. The bottom block is assumed to be rigid, so its thermomechanical behavior is not calculated.

For the constitutive equation of the mush we use the following expression:

( )

m s o _RT mQ exp f exp

α

ε

σ

 &      = ( 2.20 )

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where Q is the activation energy which is given by the solid phase deformation behaviour, m is the strain rate sensivity coefficient, R is the gas constant,

σ

_oand α are material constants andε& is the strain rate. The values for Q, σo, α and m were

fitted from experimental data [17] and were Q: 160 kJ/mol;

σ

_o: 4.5 Pa; α: 10.2 and m: 0.26.

2.2.6 Computation

Computations were performed for both the start-up phase and the steady state phase. For the start-up phase, four casting conditions denoted 1 to 4 were applied in the computation to calculate the stresses, strains, depth of sump and length of mush as a function of the axial position. The casting modes are shown in Fig. 2.2. The stresses, strains, depth of sump and length of mush were also calculated at a distance of 750 mm from the beginning of the billet, which were considered representative for the steady state phase. The casting speeds selected were constant and equal to 120, 150 and 180 mm/min. Here, the stresses, strains, depth of sump and length of mush were calculated as a function of the radial position in the billet.

80 90 100 110 120 130 140 150 160 0 100 200 300 400 500 600 Length of billet (mm) C

asting speed (mm/min)

Figure 2.2. The casting modes applied for simulation of the start-up phase of DC casting. Casting conditions: () 1, ( ) 2, (∆) 3 and (x) 4.

2.3 Experiment

Although this chapter concentrates on modeling only, one validation experiment was carried out. A billet of an Al−4.5 %Cu alloy was hot top cast in a

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27 DC casting pit, situated in our laboratory, under the following conditions: casting speed was 120 mm/min, melt temperature at the launder was 700 o_{C, and water}

flow rate was 118 l/min. Billet diameter was 20 cm and billet length was 150 cm. A temperature measurement was performed as a function of time a radial distance of 90 mm. A chromel-alumel thermocouple was used, connected to an acquisition data system. The thermocouple was inserted in the molten metal during steady state casting, fixed at a certain length of the billet and frozen in during solidification.

2.4 Results

2.4.1 General overview of stresses and strains in DC casting

Axial (σzz), radial (σrr) and circumferential (σθθ) stresses during steady state

casting are shown in Fig. 2.3, together with the solidus (551.2 oC) and liquidus (654.1 o_{C) lines, at a casting speed of 120 mm/min. It is seen that stresses develop}

in the mushy zone in the vicinity of the solidus. It is found that for solid material far from the mushy zone the axial stress is negative (compressive) close to the edge of the billet and positive (tensile) in the center of the billet (Fig. 2.3a). The radial stress is tensile at all positions (Fig. 2.3b). The circumferential stress is compressive close to the edge and tensile in the center of billet (Fig. 2.3c). The stress state in the mushy zone is tensile in the vicinity of the solidus line.

Axial (εzz), radial (εrr) and circumferential (εθθ) viscoplastic strains during

steady state casting are shown in Fig. 2.4 for the same casting speed. It is found that for the solid material far from the mushy zone, in contrast with the axial stress, the axial viscoplastic strain is positive (tensile) close to the edge of the billet and negative (compressive) in the center of the billet (Fig. 2.4a). The radial viscoplastic strain is tensile for all locations in the billet (Fig. 2.4b). The circumferential viscoplastic strain is compressive near the edge of the billet and tensile in the center of the billet (Fig. 2.4c). The radial and circumferential viscoplastic strains in the mushy state are tensile in the vicinity of the solidus. The axial viscoplastic strain in the mushy state is compressive.

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551.2 CL 551.2 CL 551.2 CL a. b. c.

Figure 2.3. Axial (a), radial (b) and circumferential (c) stresses (MPa) during steady state casting. Solidus (654.1 o_{C) and liquidus (551.2}o_{C) are indicated. Casting speed: 120}

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FEM simulation of mushy zone behavior during direct-chill casting 29 551.2 -0 .0 1 2 -0 .0 0 9 -0 .0 0 6 -0 .0 0 3 0. 000 CL 0 .0 0 0 0 .0 0 3 0 .0 0 6 0 .0 0 3 0 .0 0 6 551.2 CL 0 .0 0 0 0 .0 0 3 0 .0 0 6 551.2 CL a. b. c.

Figure 2.4. Axial (a), radial (b) and circumferential (c) strains during steady state casting in MPa. Solidus (654.1) and liquidus (551.2) are indicated. Casting speed: 120 mm/min. CL denotes billet center.

2.4.2 Time evolution of temperature and stresses

Fig. 2.5 shows the calculated temperatures at 0, 50, 70 and 90 mm distance from the center of the billet as a function of time, during steady state casting with a casting speed 120 mm/min. The characteristic times, corresponding with the mushy state are 22 s (0 mm), 19 s (50 mm), 13 s (70 mm) and 10 s (90 mm), which correspond with mush lengths of approximately 44, 38, 26 and 20 mm, respectively. It is clear that cooling is more effective near the edge of the billet (curve 5) than in the center of the billet (curve 1). In the figure, also the experimental data of the temperature evolution at a radial distance of 90 mm are given. It is found that the computed result is in excellent agreement with the experimental result.

In Fig. 2.6 the time evolution of the axial, radial and circumferential stresses are given both at the center and at a radial distance of 90 mm. Conditions are identical to those of Fig. 2.5. In the center, radial and circumferential stresses are

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identical. It can be seen that stresses develop during solidification and further cooling till at a temperature of approximately 200 o_{C stresses become constant.}

In the center all stresses are tensile over the whole solid region. In the region close to the edge of billet, situation is more complicated. There, the axial and circumferential stresses are tensile at the mushy zone boundary and become compressive at certain distances. The radial stress is tensile over the whole solid region. 0 100 200 300 400 500 600 700 800 0 50 100 150 200 Time (s) Te m p er at ur e ( oC) ₁ 3 2 5 4 -150 -100 -50 0 50 100 150 200 0 50 100 150 200 Time (s) S tr ess ( M P a ) 1 6 5 4 2,3

Figure 2.5. Time evolution of the temperature. Casting speed: 120 mm/min. Steady state. Calculated temperatures at radial distance of 1) 0 mm, 2) 50 mm, 3) 70 mm and 5) 90 mm and measured temperature at radial distance of 4) 90 mm.

Figure 2.6. Time evolution of the stress. Casting speed: 120 mm/min. Steady state. 1) axial, 2) radial and 3) circumferential stress at the center of the billet, and 4) axial, 5) radial and 6) circumferential stress at radial distance of 90 mm.

2.4.3 Development of stresses and strains in and near the mush

Fig. 2.7 shows the circumferential stress and circumferential viscoplastic strain for temperatures, which correspond to the mushy state or to the solid region close to the mush, for various locations of the mush. In general these stresses and strains are tensile. At 50 mm from the bottom, which corresponds with the start-up phase of billet casting, stress and strain are highest. They become lower at larger distance from the bottom, when a steady state condition is reached. Near the edge of the billet, stress and strain become lower than in the center of the billet.

Fig. 2.8 presents the circumferential stress and circumferential viscoplastic strain during the start-up phase for the various casting modes applied (see Section 2.2.6). The results show that stresses and strains become lower in the order (of casting modes) 3, 1, 4 and 2, which is the same order as found in Fig. 2.2 at 50 mm

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31 for the momentary casting speed. Clearly casting speed and the start-up conditions influence the stress and strain. The ramping procedure in the start-up phase will reduce the stress and strain.

The effect of casting speed on the circumferential stress and viscoplastic strain during steady state casting is shown in Fig. 2.9. The figure clearly shows that an increasing casting speed results in an increase of stress and strain in the mush.

The circumferential stress and circumferential viscoplastic strain at solidus temperature are shown in Fig. 2.10, for the center of the billet during the start-up phase. The stresses and strains increase to a maximum at about 50 mm and 40 mm respectively, and then decrease till the steady state value is reached (casting conditions 1, 3) or decrease and then increase again till the steady state value is reached (casting conditions 2, 4). Steady state values are reached at about 140 mm for casting conditions 1 and 3, and at about 390 mm for casting conditions 2 and 4.

-5 0 5 10 15 20 25 30 500 520 540 560 580 600 620 640 660 Temperature (o_C) C ir cu m fer en ti al st ress ( M P a) Tsol Tliq 1 3 2 -0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 500 520 540 560 580 600 620 640 660 Temperature (o_C) C irc um fe re nt ia l v is co pl as ti c st ra in 1 2 3 Tsol Tliq a. b.

Figure 2.7. Circumferential stress (a) and circumferential viscoplastic strain (b) in and near the mush. 1) center of the billet at 50 mm from the bottom, 2) center of the billet at steady state and 3) at radial distance of 90 mm at steady state. Casting speed: 120 mm/min.

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-5 0 5 10 15 20 25 30 500 520 540 560 580 600 620 640 660 Temperature (o_C) C ir cu m fe re n ti al st ress ( M P a) Tsol 3 4 2 1 Tliq -0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 500 520 540 560 580 600 620 640 660 Temperature (o_C) C ir rc u m fe re n tia l v is co p la st ic s tr ain Tsol Tliq 1 3 4 2 a. b.

Figure 2.8. Circumferential stress (a) and circumferential viscoplastic strain (b) in the center of the billet at a distance of 50 mm from the bottom for different start-up casting conditions. Conditions 1-4: see Fig. 2.2.

-5 0 5 10 15 20 25 30 500 520 540 560 580 600 620 640 660 Temperature (o_C) C irc u m fe re n ti a l s tr es s ( M P a ) 1 3 2 Tsol Tliq -0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 500 520 540 560 580 600 620 640 660 Temperature (o_C) C ir cu m fe re n ti al vi sco p last ic st ra in 1 2 3 Tsol Tliq a. b.

Figure 2.9. Circumferential stress (a) and circumferential viscoplastic strain (b) in the center of the billet. Steady state. Casting speed is 1) 120 mm/min, 2) 150 mm/min and 3) 180 mm/min.

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FEM simulation of mushy zone behavior during direct-chill casting 33 0 5 10 15 20 25 0 50 100 150 200 250 300 350 400 450

Distance from bottom (mm)

C ir cu m fe re n ti al st ress ( M P a) 1 3 4 2 0.000 0.002 0.004 0.006 0.008 0.010 0 50 100 150 200 250 300 350 400 450

Distance from bottom (mm)

C ir cu m fe re n ti al vi sco p last ic st ra in 1 4 2 3 a. b.

Figure 2.10. Circumferential stresses (a) and circumferential viscoplastic strain (b) at solidus temperature as a function of distance from the bottom of the billet for different start-up casting conditions. Casting condition 1-4: see Fig. 2.2.

2.4.4 Mush dimension

Fig. 2.11 shows the evolution of the depth of sump, defined by the distance between the location of the solidus and the melt level, and the length of the mushy zone during the start-up phase for the various casting conditions applied (see Section 2.2.6). The results show that the depth of sump and the length of the mushy zone increase to a maximum at about 80 s and then decrease somewhat (casting conditions 1, 3) or decrease somewhat and increase again (casting conditions 2, 4). Steady state is obtained for condition 1 at 145 s (135 mm), 2 at 240 s (385 mm), 3 at 165 s (150 mm) and 4 at 215 s (400 mm).

Comparing condition 1 with 3 it is seen that for the higher casting speed steady state is attained after a slightly longer time, and consequently after a longer cast length. Use of ramping delays the time and increase the length for which steady state is reached.

The effect of casting speed during steady state on the depth of sump and the length of the mushy zone is presented in Fig. 2.12. In the center of the billet, depth of sump and length of mushy zone are higher than near the edge of the billet. Depth of sump and length of mushy zone increase with increasing casting speed.

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0 20 40 60 80 100 120 140 160 180 200 0 50 100 150 200 250 300 Time (s) D e pth of s u m p ( m m ) 1 4 3 2 0 10 20 30 40 50 60 70 80 0 50 100 150 200 250 300 Time (s) Le ngt h o f m u sh y z o ne ( m m ) 1 2 3 4 a. b.

Figure 2.11. Depth of sump (a) and length of mushy zone (b) in the center of billet as a function of time for different start-up casting conditions as defined in Fig. 2.2.

0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100

Distance from center (mm)

D ept h of s u m p ( m m ) 1 3 2 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100

Distance from center (mm)

L eng th of m u sh y z one ( m m ) 1 3 2 a. b.

Figure 2.12. Depth of sump (a) and length of mushy zone (b) as a function of radial distance. Steady state. Casting speed: (1) 120 mm/min, (2) 150 mm/min and (3) 180 mm/min.

2.5 Discussion

2.5.1 Stresses and strains during steady state casting

Stresses and strains developed in DC cast billets or slabs have been reported by others [1-8]. In general these stresses and strains were calculated using a FEM approach and using certain constitutive models for the solid and/or semi-solid

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35 metal. However, papers dealing with stresses and strains in round billets are limited [1,2,5,6,8] and they only report on the solid part of the billet. In addition to this, the present paper also reports on stresses and strains in the mush. This study not only uses a constitutive model for the solid state, but also incorporates a constitutive model for the mush. This not only provides data for the mush itself, but it will also influence the results in the solid part.

Fracture during solidification of a DC cast billet is characterized by a granular fracture that has a tendency to be brittle [18,19]. In case of brittle fracture, one- directional stress/strain is more important than the three-directional stress/strain. For a billet, axial, radial and circumferential stresses and strains are then the principal stress and strain components in assessing the tendency for fracture and its direction.

In general, hot tears in a billet initiated in the mush are found in the center of the billet and have a star-like shape [20-22], which shows that the crack grows in radial directions. Under such conditions, circumferential tensile stresses and strains are determining factors for the hot tear development in the billet. Therefore, this study concentrates on the circumferential stresses and strains, developing in the region around the solidus temperature.

In this study it is found that local stresses and strains in the region around the solidus temperature are tensile in the center of billet for all stress/strain components except for the axial strain, which is compressive in the mush. This is in agreement with the results reported in refs. [1,2,5,6]. The stress evolution in the center and near the surface of the billet, as depicted in Fig. 2.6 is also supported by refs. [5] and [6]. In some cases, an axial tensile stress is found with an axial compressive strain. This is also reported in ref. [1], where this phenomenon is attributed to the consequence of a large compressive axial viscoplastic strain rate near the solidification front.

Although the present general findings and trends are supported qualitatively [1,2,5,6,8], quoted values of the various stress/strain components are generally different since they strongly depend on the constitutive material model and on the boundary conditions applied. The value of such a comparison is however limited since none of the simulations in literature could be validated so far with experimental data. The material model we used consists of a model for the solid and a model for the semi-solid. Using this approach, a better description of stresses and strains in and near the mush should be found. Validating the temperature evolution as a function of time for a particular position (see Fig. 2.5), excellent agreement is found which strongly indicates that the right boundary conditions were applied in this study.

Close to the edge of the billet circumferential stress and strains components are highest and tensile, but the circumferential stress and strain are considerably

Hot tearing and deformation in direct-chill casting of aluminum alloys

Hot tearing and deformation

in direct-chill casting of aluminum alloys

Hot tearing and deformation

in direct-chill casting of aluminum alloys

Suyitno

Table of Contents

Chapter 1

Introduction

1

2

3

4

billet

3

1.2 Hot tearing

1.3 Outline of this thesis

References

Chapter 2

FEM simulation of mushy zone behavior during

direct-chill casting of an Al−4.5%Cu alloy

2.1 Introduction

2.2 Modeling

ρ

ρ

(

)

α

[ ]{ }

[ ]{ } [ ]

{ }

{ }

[ ]

∑ ∫

[ ] [ ]

∑ ∫

[ ] [ ]

∑ ∫

[ ]

[ ]

α

[ ]

∑ ∫

[ ]

[ ]

ρ

{ }

∑ ∫

[ ]

∑ ∫

[ ]

∑ ∫

[ ]

∫

∫

∫

σ

δ

δ

δε

ε

ε

ε

ε

ε

ε

ε

σ

ε

E

α

ε

α

(

)

(

)

ϕ

σ

σ

_α

_{∑ ∫}

_{∑ ∫}

_{∑ ∫}

_{∑ ∫}

_{∑ ∫}

_{∑ ∫}

_{∑ ∫}

_∫

_∫

_ϕ

_σ

_ε

_ε

_{∑ ∫}

_{∑ ∫}