tixoformabilidade geral de atkinson

8/13/2019 Tixoformabilidade Geral de Atkinson

1/72

Modelling the semisolid processingof metallic alloys

H.V. Atkinson *

Department of Engineering, University of Leicester, University Rd., Leicester LE1 7RH, UK Received 1 April 2004; accepted 19 April 2004

Abstract

Semisolid processing of metallic alloys and composites utilises the thixotropic behaviour of materials with non-dendritic microstructure in the semisolid state. The family of innovativemanufacturing methods based on this behaviour has been developing over the last 20 years orso and originates from scientic work at MIT in the early 1970s. Here, a summary is given of:

routes to spheroidal microstructures; types of semisolid processing; and advantages and dis-advantages of these routes. Background rheology and mathematical theories of thixotropy arethen covered as precursors to the main focus of the review on transient behaviour of semisolidalloy slurries and computational modelling. Computational uid dynamics (CFD) can be usedto predict die lling. However, some of the reported work has been based on rheological dataobtained in steady state experiments, where the semisolid material has been maintained at aparticular shear rate for some time. In reality, in thixoforming, the slurry undergoes a suddenincrease in shear rate from rest to 100 s 1 or more as it enters the die. This change takes placein less than a second. Hence, measuring the transient rheological response under rapid changesin shear rate is critical to the development of modelling of die lling and successful die designfor industrial processing.

The modelling can be categorised as one-phase or two-phase and as nite difference or niteelement. Recent work by Alexandrou and coworkers and, separately Modigell and coworkers,has led to the production of maps which, respectively summarise regions of stable/unstableow and regions of laminar/transient/turbulent ll. These maps are of great potential use forthe prediction of appropriate process parameters and avoidance of defects. A novel approachto modelling by Rouff and coworkers involves micro-modelling of the active zone aroundspheroidal particles. There is little quantitative data on the discrepancies or otherwise betweendie ll simulations and experimental results (usually obtained through interrupted lling).There are no direct comparisons of the capabilities of various software packages to model the

* Tel.: +44-116-2231019; fax: +44-116-2522525.E-mail address: [email protected] (H.V. Atkinson).

0079-6425/$ - see front matter 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.pmatsci.2004.04.003

www.elsevier.com/locate/pmatsci

Progress in Materials Science 50 (2005) 341412
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


2/72

lling of particular geometries accurately. In addition, the modelling depends on rheologicaldata and this is sparse, particularly for the increasingly complex two-phase models. Direct owvisualisation can provide useful insight and avoid the effects of inertia in interrupted lling

experiments. 2004 Elsevier Ltd. All rights reserved.

Contents

1. Introduction to semisolid processing...................................................................................... 3451.1. Routes to spheroidal microstructures ....................................................................... 3461.2. Types of semisolid processing...................................................................................... 3491.3. Advantages and disadvantages.................................................................................... 353

2. Background rheology ................................................................................................................... 355

3. Origins of thixotropy.................................................................................................................... 358

4. Mathematical theories of thixotropy ...................................................................................... 3624.1. Models based on a structural parameter ............................................................. 3624.2. Direct structure theories................................................................................................. 3634.3. Simple viscosity theories ................................................................................................ 364

5. Transient behaviour of semisolid alloys ................................................................................ 364

5.1. Rapid shear rate changes in rheometers .................................................................. 3645.2. Rapid compression........................................................................................................... 368

6. Modelling.......................................................................................................................................... 3716.1. Model of Brown and coworkers................................................................................. 3886.2. Finite difference modelling ............................................................................................ 388

6.2.1. One-phase nite difference based on the model of Brown et al...... 3886.2.2. One-phase nite difference based on FLOW3D .................................... 3896.2.3. One-phase nite difference based on MAGMAsoft............................. 3926.2.4. One-phase nite difference with Adstefan................................................ 3926.2.5. Two-phase nite difference............................................................................. 393

6.3. Finite element modelling................................................................................................ 3936.3.1. One-phase nite element................................................................................. 3936.3.2. Two-phase nite element ................................................................................ 4016.3.3. Micro-modelling................................................................................................. 402

7. Flow visualisation ........................................................................................................................... 404

8. Concluding remarks ....................................................................................................................... 406

Acknowledgements ........................................................................................................................................ 406

References ......................................................................................................................................................... 406

342 H.V. Atkinson / Progress in Materials Science 50 (2005) 341412


3/72

Nomenclature (in the order in which the symbols appear in the paper)

s shear stress_c shear rateg viscositysy yield stressk constant related to the viscosity in Eqs. (1)(3)n shear rate exponent in Eqs. (2) and (3)g1 viscosity as the shear rate tends to innityg0 viscosity as the shear rate tends to zero f s fraction solidk structural parameter varying between 1 for completely built up and 0

for completely broken downt timea ; b; c; d constants in Eq. (5)k 1; k 2 constants for breakdown and buildup in Eq. (6) p ; q exponents in Eq. (6) N average number of links per chain in Eq. (7)k 0; k 1; k 2 rate constants in Eq. (7) P number of single particles per unit volume in Eq. (7) N e average number of links per chain at equilibrium in Eq. (8)ge equilibrium viscosity K ; r constant and exponent in Eq. (10)k ; m material constants in Eq. (11)gp peak-stress viscosity in Table 2gss rst steady-state viscosity in Table 2, i.e. after the fast breakdown

process as opposed to the slowsb rst breakdown time in Table 2 i.e. for the fast breakdown process Ak hydrodynamic coefficient as a function of k in Eq. (12), Table 4 and also

in Eqs. (25) and (29), Table 5c effective volume packing fraction solid in Eq. (12), Table 4 and in Eq.

(25) and (29), Table 5cmax maximum effective volume packing fraction solid in Eq. (12), Table 4and in Eq. (25) and (29), Table 5

gf viscosity of uidC T exponential function of temperature in Eq. (12), Table 4 and in Eq. (25)

and (29), Table 5 H agglomeration function in Eq. (13), Table 4 and in Eq. (26), Table 5G disagglomeration function in Eq. (13), Table 4 and in Eq. (26), Table 5u velocity vectorx inverse of the relaxation time in Eq. (14), Table 4

b1; b2 constants in Eq. (15), Table 4c constant in Eq. (16), Table 4

H.V. Atkinson / Progress in Materials Science 50 (2005) 341412 343


4/72

a rate constant for thinning in Eq. (18), Table 4b rate constant for thickening in Eq. (18), Table 4 and also in Eq. (24),

Table 4 B; k coefficients in Eq. (19), Table 4j structural parameter in Eq. (19), Table 4, varying between zero (fully

broken down) and innity (fully built up)j e equilibrium value of ja ; b constants in Eq. (20), Table 4 A; B coefficients in Eq. (23), Table 4s shear stress tensor in equations in Table 5Du rate of deformation tensor in equations in Table 5 DII second invariant of the rate of strain tensor in equations in Table 5m coefficient in Eq. (27), Table 5u velocity scalarq density K ; G coefficients in Eq. (31), Table 5s ij viscous stress tensor in Eq. (33), Table 5 Dij uij u ji in Eq. (33), Table 5m coefficient controlling the exponential rise in stress in Eq. (33), Table 5 K coefficient in Eq. (34), Table 5a ; b; c coefficients in Eq. (40), Table 5ke equilibrium value of the structural parameter in Eq. (41), Table 5 D rate of strain tensor in Eq. (42), Table 5r ow stress in Eq. (45), Table 5 and subsequent equationse strain in Eq. (45), Table 5_e strain ratea ; b; m; n coefficients and exponents in Eq. (45), Table 5T temperature_c0 strain rate cut-off in the power law cut-off model, Eqs. (47) and (54),

Table 5 K current yield stress in Eq. (48), Table 5

K 0 yield stress in Eq. (48), Table 5e effective strain in Eq. (48), Table 5r 0 true stress at the solidus temperature and a strain rate _e0 of 1 s 1 in Eq.

(50), Table 5sM Maxwell time g= E where E is the Youngs modulus at the temperature

under investigation (Eq. (50), Table 5) Rb bond radius in Eq. (50), Table 5 R average radius of the primary particles in Eq. (50), Table 5

Rb= R 0:25Q activation energy for self-diffusion in Eq. (50), Table 5

R ideal gas constant in the exponential term in Eq. (50), Table 5m material exponent in Eq. (50), Table 5



5/72

1. Introduction to semisolid processing

Mascara, honey and certain kinds of paint are all thixotropic. When they aresheared they ow, when allowed to stand they thicken up again; their viscosity isshear rate and time dependent. Spencer et al. [1] rst discovered such behaviour insemisolid metallic alloys in the early 1970s when investigating hot tearing with a

rheometer. If the material was stirred continuously during cooling from the fullyliquid state to the semisolid state the viscosity was signicantly lower than if thematerial was cooled into the semisolid state without stirring. Stirring breaks up thedendrites which would normally be present so that the microstructure in the semi-solid state consists of spheroids of solid surrounded by liquid (Fig. 1). It is thismicrostructure which is a requirement for thixotropic behaviour and for semisolidprocessing. When such a semisolid microstructure is allowed to stand, the spheroidsagglomerate and the viscosity increases with time. If the material is sheared, theagglomerates are broken up and the viscosity falls. In the semisolid state, with be-tween 30% and 50% liquid, if the alloy is allowed to stand it will support its own

weight and can be handled like a solid. As soon as it is sheared, it ows with aviscosity similar to that of heavy machine oil. This is the behaviour which isexploited in semisolid processing [2] and which is illustrated in Fig. 2, where the alloycan be cut and spread like butter.

Nearly 30 years of work and effort have been invested in the eld of semisolidprocessing and the increase in interest in this eld has been marked by seven inter-national conferences [39] with an eighth planned in Cyprus in 2004. Semisolidprocessing is rivalling other manufacturing routes for military, aerospace and mostnotably automotive components [1012]. In Europe, suspension parts, enginebrackets and fuel rails for automotives are being produced. In the USA, examples

include mechanical parts for mountain bikes and snowmobiles [13], while in Asiathere is concentration on the production of electronic components such as computernotebook cases and electrical housing components, particularly in magnesium alloys

_c strain rate tensorg0 viscosity at a characteristic shear strain rate _cc in Eq. (54), Table 5_cc characteristic shear strain rate in Eq. (54), Table 5r effective stress in Eq. (56), Table 6S separation coefficient in Eq. (56), Table 6 K ; m; b coefficients in Eq. (56), Table 6ecr ; ecr1 critical strain I and critical strain II in Eq. (56), Table 6b breakage ratio in Eq. (58), Table 6 f sA volume solid fraction of the active zone in Eq. (60), Table 6 f c critical fraction solid in Eq. (60), Table 6 D; n material parameters in Eq. (60), Table 6 Bi Bingham number Re Reynolds number



6/72

via thixomolding e.g. [12]. Fig. 3 shows some of the components produced by Thixo-forming at Stampal for an Alfa Romeo car.

1.1. Routes to spheroidal microstructures

There are many different routes for obtaining non-dendritic microstructures. Themain ones are described here.

(1) Magnetohydrodynamic (MHD) stirring: This involves stirring electromagneti-cally (and hence without the contamination, gas entrapment and stirrer erosioninvolved in mechanical stirring) in the semisolid state to break up the dendritese.g. [14]. Much of the commercial production of aluminium alloy components todate has been based on MHD material supplied by Pechiney and SAG. There aresome problems associated with this route including lack of uniformity and thefact that the spheroids are not completely round with some rosette characterremaining.

(2) Sprayforming: Sprayforming is a relatively expensive route but one which can beused to produce alloys, which cannot be produced in any other way, such as alu-

miniumsilicon alloys with greater than 20 wt.% silicon e.g. [15]. Sprayformingessentially involves the atomisation of a liquid metal stream and collection of the droplets on a former. The resulting microstructure is ne and equiaxed.When heated into the semisolid state it is ideal for thixoforming [16].

(3) Strain induced melt activated (SIMA)/recrystallisation and partial melting(RAP): These routes are similar but distinct. The material is worked, e.g. byextrusion. On reheating into the semisolid state, recrystallisation occurs andthe liquid penetrates the recrystallised boundaries so resulting in spheroids sur-rounded by liquid. The SIMA route [17] involves hot working and the RAProute [18] warm working. The advantages of these routes are that some alloys

are supplied in the extruded state in any case and the spheroids are more fullyrounded than those from the MHD route. The main disadvantages are that theremay be variation in the amount of stored work across the section, resulting in

Fig. 1. Micrograph of a typical (a) dendritic microstructure in an as-cast sample and (b) globularmicrostructure in a semisolid alloy sample.



7/72

inhomogeneity, and extrusion can be difficult and expensive with wider billetdiameters.

(4) Liquidus/near-liquidus casting: There have been recent developments in produc-ing feedstock by manipulating the solidication conditions. The UBE new rheo-casting (NRC) process [19,20] is based on this principle with the molten metal at

Fig. 2. A photographic sequence illustrating the thixotropic behaviour of semisolid alloy slugs (courtesy:University of Sheffield).

Fig. 3. Automotive components produced by STAMPAL for the Alfa Romeo car: (a) multi-link, rearsuspension support 8.5 kg, A357, T5; (b) steering knuckle A357, T5 substitution of cast iron part.



8/72

near-liquidus temperature poured into a tilted crucible and grain nucleationoccurring on the side of the crucible. The grain size is ne because the tempera-ture is near liquidus. An allied technique is the direct thermal method [21]. In thecooling slope method [22], liquid metal is poured down a cooled slope and col-lects in a mould. Nucleation on the slope ensures the spheroid size is ne. Withliquidus casting, a high rate of nucleation can be achieved within the entire vol-ume of undercooled melt [23,24].

(5) New MIT Process: This is a recently developed hybrid of stirring and near liq-uidus casting [25] (Fig. 4). A stirrer that also provides the cooling action is in-serted into an alloy melt held a few degrees above the liquidus. After someseconds of stirring, the melt temperature decreases to a value which correspondsto a fraction of solid of only a few percent and the stirrer is withdrawn.

(6) Grain renement: Grain rened alloys can give equiaxed microstructures e.g. [26]but it is difficult to ensure the grain structure is uniformly spheroidal and neand the volume of liquid entrapped in spheroids tends to be relatively high.

(7) Semi-solid thermal transformation: Spheroidal structure can also be produced byheating a dendritic structure to the semisolid temperature range for a period of time. This is known as semisolid thermal transformation, or SSTT [27]. Thestructures produced by this route tend to be relatively coarse (around 100 l mdiameter particles).

Other methods are summarised in [2,12,28].

Fig. 4. New MIT process (courtesy: Prof. M.C. Flemings, MIT).



9/72

1.2. Types of semisolid processing

Semisolid processing now covers a whole family of processes. The terminology isas follows.

Rheocasting refers to the process where the alloy is cooled into the semisolidstate and injected into a die without an intermediate solidication step. A typicalrheocaster is shown in Fig. 5. The non-dendritic microstructure can be obtained by avariety of means during cooling (e.g. by mechanical stirring, by stimulated nucle-ation of solid particles as in the new rheocasting NRC process recently patented byUBE [19,20] (see Fig. 6), or by electromagnetic stirring in the shot sleeve as in thenew semi-solid metal casting process from Hitachi [30] (see Fig. 7). The NRC processinvolves pouring molten alloy, at a temperature slightly above the liquidus, into asteel crucible and then controlled cooling to achieve a spheroidal microstructurebefore transfer to a forming machine. There is no need for specially treated thixo-formable feedstock and scrap can be readily recycled within the plant. Hall et al. [20]

Fig. 5. Continuous rheocaster (after [29]).



10/72


11/72

Thixoforming is the process where suitable material is heated into the semisolidstate and injected into a die. Usually, the liquid content is between 30 and 50% priorto forming. This is the type of process used by Stampal in Italy to produce the AlfaRomeo suspension component and a number of other automotive components [39].It is also the process used by Vforge in the US to produce master cylinders, anti-lockbrake system valves and automotive steering pumps amongst others. A thixoformingpress is shown in Fig. 10 and illustrates the steps in the process, although it is a

vertically upwards acting press whereas most commercial presses are horizontal. Thespecimen is induction heated into the semisolid state. When it has reached theappropriate proportion of liquid it is forced into the die. Usually in commercial

Fig. 7. Electromagnetic stirring in the shot sleeve in the new semisolid metal casting process from Hitachi(after [30]).



12/72

thixoforming, the slugs are heated on a carousel. Cycle times are then very com-parable with die casting, if not faster because the full solidication range does nothave to be gone through.

The distinctions between rheocasting, thixocasting and thixoforging are illus-trated in Fig. 11 [40].

Other processes include the shear-cooling roll (SCR) process [41,42] and thecooling slope process [43]. In addition, there is the possibility of using semisolidslurries in solid freeform (SSF) technology [44]. This method deposits a stream of

Fig. 8. Schematic diagram of a rheomolder using a twin screw process [34]. Liquid metal is used instead of the solid chips used in the Thixomolder e (see Fig. 9).

Fig. 9. Thixomolding process [35].



13/72

slurry through a nozzle that moves relative to a substrate. Components are built bybuilding up successive layers so as to rapidly fabricate dense metal structures.

1.3. Advantages and disadvantages

As with any manufacturing process, there are certain advantages and disadvan-tages in semisolid processing. They are [2,4547]:

AdvantagesThe main advantages of semisolid processing, relative to die casting, are as fol-

lows.

Fig. 10. Thixoforming press (courtesy: University of Sheffield).

Fig. 11. Schematic illustration of different routes for semisolid metal processing [40].



14/72

(1) Energy efficiency: Metal is not being held in the liquid state over long periods of time.

(2) Production rates are similar to pressure die casting or better.(3) Smooth lling of the die with no air entrapment and low shrinkage porosity gives

parts of high integrity (including thin-walled sections) and allows application of the process to higher-strength heat-treatable alloys.

(4) Lower processing temperatures reduce the thermal shock on the die, promotingdie life and allowing the use of non-traditional die materials e.g. [48] and process-ing of high melting point alloys such as tool steels and stellites [49] that are dif-cult to form by other means.

(5) Lower impact on the die also introduces the possibility of rapid prototyping dies[48].

(6) Fine, uniform microstructures give enhanced properties.(7) Reduced solidication shrinkage gives dimensions closer to near net shape and

justies the removal of machining steps; the near net shape capability (quanti-ed, for example, in [50]) reduces machining costs and material losses.

(8) Surface quality is suitable for plating.

Disadvantages

(1) The cost of raw material can be high and the number of suppliers small.(2) Process knowledge and experience has to be continually built up in order to facil-

itate application of the process to new components.(3) This leads to potentially higher die development costs.(4) Initially at least, personnel require a higher level of training and skill than with

more traditional processes.(5) Temperature control. Fraction solid and viscosity in the semisolid state are very

dependent on temperature. Alloys with a narrow temperature range in the semi-solid region require accurate control of the temperature.

(6) Liquid segregation due to non-uniform heating can result in non-uniform com-position in the component.

The economic advantages of thixoforming have been discussed [51,52], includingthe use of quality function deployment (QFD) to evaluate the interrelationshipsbetween thixoforming characteristics (energy usage, near net shape capability,mechanical integrity of product, short cycle time, reduced die wear, raw materialcost, process development, skills/wages of work force) and product characteristics(weight, strength, geometry, tolerances, price premium, lead time, exibility, n-ishing operations) and quantied in software ( www.shef.ac.uk/~ibberson/thixo.html ). The economics of the NRC process have been analysed [20]. The NRCprocess does not suffer from the disadvantage of (1). Such analysis is important forindustries adopting novel manufacturing methods where the cost base is not yet

established through custom and practice.Much of the work on semisolid processing has been reported in the major series of

conferences [39]. Some of these conferences have been refereed whilst others have

http://www.shef.ac.uk/~ibberson/thixo.htmlhttp://www.shef.ac.uk/~ibberson/thixo.htmlhttp://www.shef.ac.uk/~ibberson/thixo.htmlhttp://www.shef.ac.uk/~ibberson/thixo.html


15/72

not. It is therefore the policy in this review to give references as far as possible torefereed journal papers if the work has subsequently been published in that form.Previous reviews include those by Kenney et al. [14], Flemings [45], Kirkwood [2],Quaak [47], Collot [53] and Fan [12]. In addition, a book has recently been publishededited by Figueredo [29]. The recent Fan review is comprehensive and the aim here isto complement that by providing a more detailed review of modelling of semisolidprocessing and the transient rheological experiments required to provide data forthat.

Millions of components are now made annually by semisolid processing. Alu-minium alloy components produced by thixoforming and the NRC process aresupplied to the automotive industry. Thixomolding is widely used, particularly inJapan, to produce lightweight magnesium components for mobile phones, laptopcomputers and cameras. New variants are still emerging (e.g. the new MIT process[25]). The cutting edge research issues are now in developing the potential for pro-ducing high performance alloys [5460] and in modelling die ll with its concomitantrequirement to obtain the experimental data which can support the modelling. Thisreview focuses on the latter two areas.

2. Background rheology

In a Newtonian uid, the shear stress, s is proportional to the shear rate, _c, and

the constant of proportionality is the viscosity, g. Thixotropic uids are non-New-tonian i.e. the shear stress is not proportional to the shear rate. The viscosity is thentermed the apparent viscosity and is dependent on shear rate, pressure, temperatureand time. Some non-linear uids also show viscoelasticity i.e. they store some of themechanical energy as elastic energy. Thixotropic materials do not store energyelastically and show no elastic recovery when the stress is removed.

If a uid exhibits a yield stress and then gives a linear relationship between shearstress and shear rate, it is termed a Bingham material (Fig. 12). Then

s sy k _c 1

where k is a constant related to the viscosity. The HerschelBulkley model is wherebehaviour is non-linear after yield i.e.:

s sy k _cn 2

There is dispute over whether thixotropic semisolid alloys display yield e.g. [61] andwhether they should be modelled as such (e.g. [62]). Barnes et al. [6365] concludedthat the presence of a yield stress as reported by some workers for thixotropicmaterials (but not semisolid alloys) is probably due to the limitations of theirexperimental apparatus in not being able to measure shear stresses at very low shearrates. Koke and Modigell [66] have used a shear stress controlled rheometer to

measure yield stress directly on Sn 15%Pb. They distinguish between a static yieldstress where the uid is at rest prior to the application of a shear stress, and a dy-namic yield stress where the uid is being continuously sheared. Their results are



16/72

shown in Fig. 13. The yield stress increases with rest time prior to deformationbecause of the increasing degree of agglomeration. In terms of modelling semisolidalloy die ll, the use of a yield stress may be appropriate because a vertical billet does

Shear Rate (s -1)

Shear Stress (Pa)

y

Newtonian

BinghamHerschel-Bulkley

Shear Thinning

Shear Thickening

Shear Rate (s -1)

Viscosity (Pas)

Newtonian

Shear Thickening

Shear Thinning

Herschel-Bulkley

Fig. 12. Shear stress versus shear rate and associated viscosity versus shear rate curves for a variety of types of rheological behaviour.

Fig. 13. Shear-stress ramp experiments after different rest times ( t r ) for Sn15%Pb [66]. (a) Shear stressversus deformation angle; (b) yield stress versus rest time. Temperature 195 C, fraction solid 0.5, globularstructure prepared by shearing at 100 s 1 at a cooling rate of 1 C/min.



17/72

not collapse under its own weight unless the liquid fraction is too high. In addition,in rapid compression experiments to be described later (in Section 5.2) an initial peakin the load versus displacement curve is detected. Contrary to this though is the factthat at the thixoforming temperature the initial peak is so small as to be unde-tectable.

The Ostwald-de-Waele relationship:

s k _cn 3

is used to describe uids which do not have a yield point and where there is a powerlaw relationship between the shear stress s and the shear rate _c. If the exponentn 1, this reduces to the expression for a Newtonian uid with the constant k equalto the viscosity g. In Fig. 12, the shear thinning material (whose viscosity decreasesas the shear rate increases) would have a value of n of less than 1 and the shearthickening material would have n greater than one. Thixotropic materials areessentially shear thinning but also thicken again when allowed to rest (i.e. allthixotropic materials are shear thinning but not all shear thinning uids are thixo-tropic).

It is thought that at very high shear rates and at very low shear rates, thixotropicuids effectively become Newtonian. This is expressed in the Cross model [67]:

g g1 g0 g1

1 k _cn 4where as the shear rate

_c ! 0, g ! g0 and as

_c ! 1 , g ! g1 . Fig. 14 shows datafrom a number of studies [6770] for Sn15%Pb alloys with various fractions of solid

-2

-1

0

1

2

3

4

5

6

7

8

9

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4

Log 10 Shear rate (s-1 )

L o g 1 0

v i s c o s i t y

( P a s )

Lax & Flem (Fs=0.5) [68]

Turng & Wang (Fs=0.5) [69]

McLelland (Fs=0.5) [70]

Liu (Fs=0.5) [71]

Cross Model at Fs=0.5 [67]

Laxmanan (Fs=0.36) [68]


McLelland (Fs=0.36) [70]Liu (Fs=0.36) [71]

Cross Model (Fs=0.36) [67]

Lax & Flem (Fs=0.3) [68]

Liu (Fs=0.3) [71]


Lax & Flem (Fs=0.2) [68]


McLelland (Fs=0.2) [70]

Liu (Fs=0.2) [71]


Fraction solid at 0.5

Fraction Solid at 0.36



-

Fig. 14. Cross model tted to apparent viscosities obtained from various works on Sn15%Pb alloys [71].



18/72

f s. The data obey the Cross model, but information on the extremes is sparse. Thesedata are for steady-state viscosities and, as will be discussed below, it is the transientbehaviour which is of importance for the modelling of thixotropic die ll.

Viscosity is highly dependent on temperature. For a Newtonian uid (e.g. theliquid matrix in a semisolid slurry), the viscosity decreases with increase in tem-perature. Temperature also affects the microstructure. Thus in semisolid slurries, thefraction solid decreases with increase in temperature, with a consequent effect onviscosity (see Fig. 14). In addition, over time, the microstructure will coarsen bydiffusion and this will be accelerated as the temperature increases. Fig. 14 is for Sn15%Pb alloy. There is little data on aluminium alloys because there are few com-mercially available rheometers that operate above about 500 C.

For a thixotropic material at rest, when a step increase in shear rate is imposed,the shear stress will peak and then gradually decrease until it reaches an equilibriumvalue for the shear rate over time (Fig. 15). The higher the shear rate after the step,the lower the equilibrium viscosity. The peak viscosity encountered will increase withincreasing rest time before it recovers back to the equilibrium viscosity of the shearrate specied.

3. Origins of thixotropy

What is the microstructural origin of thixotropic behaviour? The importance of the spheroidal microstructure which results on stirring has already been mentioned.The semisolid metallic systems have much in common with occulated suspensions

Fig. 15. Response of an inelastic thixotropic material to rstly a step-change up in shear rate and then astep-change down (after [64]).



19/72

(Fig. 16). At the shear rate _c1 corresponding to point a, the microstructure consistsof a series of large ocs. If the shear rate is increased from _c1 to _c2, the ocs break up

until the size corresponds to the ow curve which passes through point b. If theshear rate is then reduced back to _c1, the individual particles begin to collide andagglomerate until an equilibrium size is reached appropriate to the lower shear rate.In semisolid metallic systems, the agglomeration occurs because particles are col-liding (either because the shear brings them into contact or, if at rest, because of sintering) and, if favourably oriented, form a boundary. By favourable orientationis meant the fact that if the particles are oriented in such a way that a low energyboundary is formed, it will be more energetically favourable for the agglomeration tooccur than if a high energy boundary is formed. If a 3-D network builds upthroughout the material, the semisolid will support its own weight and can be

handled like a solid. As the shear rate is increased, these bonds between particles arebroken up and the average agglomerate size decreases. Once the bonds are formed,the agglomerated particles sinter, with the neck size increasing with time.

The viscosity in the steady state depends on the balance between the rate of structure buildup and the rate of breakdown. It also depends on the particle mor-phology. The closer the shape to that of a pure sphere, the lower the steady stateviscosity [45]. In addition, if liquid is entrapped within particles, it does not con-tribute to ow. Thus, although the fraction liquid may take a certain value, governedby the temperature (and indeed kinetics as the thermodynamically predicted fractionliquid is not achieved instantaneously on reheating from the solid state), in practice,

the effective fraction liquid may be less as some is entrapped within spheroids.There are similarities and differences between thixotropy in semisolid metallic

systems and that in other thixotropic systems. These are associated with the nature

Fig. 16. Flow curves of a occulated suspension (after [65]).



20/72

of the forces between the particles. Table 1 summarises the phenomena which are

occurring during structural buildup and structural breakdown in a variety of sys-tems. In general, the forces between particles include: Van der Waals attraction;steric repulsion due to adsorbed macromolecules; electrostatic repulsion due to thepresence of like charges on the particles and a dielectric medium; electrostatic at-traction between unlike charges on different parts of the particle (e.g. edge/face at-traction between clay particles). In semisolid metallic slurries, none of these forcesapply. What must actually be occurring in structural buildup is a process akin toadhesion in wear. As shear occurs, particles are forced into contact with each other.If it is energetically favourable for a solidsolid boundary to be formed, the twoparticles will stay in contact. If not, they will separate again. The process will be

inuenced by the rate of shear in two opposing ways. Increasing the rate of shear willincrease the possibility of particleparticle contact but it will decrease the time of contact and the formation of a new solidsolid boundary is a time dependent pro-

Table 1Examples of thixotropic materials with the mechanisms of recovery (after [64])

Materials Structural build-up at rest Structural breakdown

under shearCollodial dispersions and

suspensions of solidsFlocculation under inter-particleforces

Break-up of ocs

(1) Paints (deocculation)(2) Coatings(3) Inks(4) Clay slurries(5) Cosmetics(6) Agricultural chemicals

Emulsions Flocculation of droplets Deocculation

Foamed systems Flocculation of bubbles Deocculation(1) Mousses Coalescence

Crystalline systems Interlocking of growing crystals Break up of long needles(1) Waxy crude/fuel oils(2) Waxes(3) Butter/margarine(4) Chocolate

Polymeric systems Agglomeration of macromolecules Deagglomeration(1) Solutions/melts(2) Starch/gums(3) Sauces Entanglement Disentanglement

Fibrous suspensions Agglomeration of brous particles Deagglomeration(1) Tomato ketchup(2) Fruit pulps(3) Fermentation broths Entanglement Disentanglement(4) Sewage sludges

Semisolid metallic systems Agglomeration of particles Deagglomeration



21/72


22/72

Many thixotropic systems show reversibility i.e. the slurries have a steady stateviscosity characteristic of a given shear rate at a given fraction solid regardless of past shearing history. However, in semisolid alloy slurry systems, the evolution of particle shape (and size) with time and stirring (Fig. 17) is irreversible. The measuredviscosity is then expected to depend on the shearing and thermal history. Thesedependencies contribute to the difficulty in modelling.

4. Mathematical theories of thixotropy

Barnes [65] has summarised current mathematical theories of thixotropy. Somedetail on these is given here to enable work on semisolid slurries to be put in thecontext of the wider understanding of thixotropy. The theories fall into three groups:

(1) Those that use a general description of the degree of structural buildup in themicrostructure, denoted by a scalar parameter, typically k, and then use d k=d t as the working parameter.

(2) Those that attempt some direct description of the temporal change of micro-structure as for instance the number of bonds or an attempt at describing the realoc architecture using fractal analysis.

(3) Those that use viscosity time data itself on which to base a theory.

4.1. Models based on a structural parameter k

A completely built structure is represented by k 1 and a completely brokendown structure by k 0. In the simplest case of a typical, inelastic, non-Newtonianuid with upper and lower Newtonian viscosity plateaus (e.g. see Fig. 14), k 1corresponds to g0 and k 0 to g1 . Thixotropy is usually then introduced via thetime derivative of the structure parameter, d k=d t . This is the sum of the breakdownand buildup terms and in the simplest theories these are only controlled by the shearrate and the current level of structure k. The most general description of the rate of breakdown due to shearing is given by the product of the current level of structure

and the shear rate raised to some power and the driving force for buildup as con-trolled by the distance the structure is from its maximum value i.e. 1 k, raised toanother power. Then

dkd t

a1 kb ck _cd 5

where a , b, c, and d are constants for any one system. Overall, if the value of d k=d t isnegative, the system is breaking down towards equilibrium and if it is positive it isbuilding up towards equilibrium. The Moore model [72] is a simplied version of Eq.(5) with b and d set to one. Cheng and Evans [73] set b 1 but allowed d to vary.The next step is to relate the structure k (as calculated using the equations above), tothe stress s or viscosity g in some ow equation. This has been done in a variety of ways which range from a simple Bingham equation (see Eq. (1) in Section 2),



23/72

through the Cross model to a Cross-like model (Eq. (4) in Section 2) containing ayield stress.

4.2. Direct structure theories

Denny and Brodkey [74] applied reaction kinetics to thixotropy via a simplescheme that examined the distribution of broken and unbroken bonds. The numberof these bonds was later linked to viscosity. The rate of structure breakdown wasthen

dunbroken d t

k 1unbroken p k 2broken q 6

where k 1 and k 2 are the rate constants for the breakdown and buildup respectively.This can be solved to give the viscosity by assuming that viscosity is linearly pro-portional to the amount of unbroken structure, with a maximum value when com-pletely structured of g0 and a minimum value when completely destructured of g1 .The rate constant k 2 is assumed to be independent of shear rate, being merely adescription of Brownian collisions leading to restructuring (but note that for semi-solid alloy slurries build up is not due to Brownian collisionsSection 3), while k 1 isrelated to the shear rate by a power law expression.

The Cross model [67] was derived using such considerations. Assuming that astructured liquid was made up of ocs (agglomerates) of randomly linked chains of

particles, Cross obtained a rate equation of the form:d N d t

k 2 P k 0 k 1 _cm N 7

where N was the average number of links per chain, k 2 was a rate constant describingBrownian collision, k 0 and k 1 were rate constants for the Brownian and shear con-tributions to breakdown, P was the number of single particles per unit volume and nwas a constant less than unity. At equilibrium, d N =d t is zero, so

N e k 2 P

k 0 1 k 1k 0 _cn

8

Assuming that the viscosity was given by the constant g1 plus a viscous contributionproportional to the number of bonds N e,

g g1g0 g1

1

1 k 1k 0 _cn

9

which is equivalent to the expression given earlier in Eq. (4) but with k k 1=k 0 .Lapasin et al. [75] used a fractal approach to describe occulated suspensions. In

the relationship they predicted, the viscosity is related to: the number of primaryparticles in a oc when the shear stress becomes innite, a yield stress and the fractaldimension of the oc.



24/72

4.3. Simple viscosity theories

Mewis and Schryvers [76] have devised a theory that circumvents the use of anystructural parameter such as k, and instead uses the viscosity as a direct measure of the structure. The rate of change of viscosity is then related to the viscosity differencebetween the steady state ge and the current values of viscosity (not the structuredifference) i.e.:

dgd t

K gs_c gn 10

Thixotropic breakdown has also been described [77,78]

g g1 1 m 1 mkt 1 g0 g1

1 m 11

where g0 and g1 are the asymptotic values of viscosity g (representing the fullystructured and fully destructured states, respectively) measured at time t for anyparticular shear rate, and k and m are material constants.

5. Transient behaviour of semisolid alloys

Computational uid dynamics (CFD) can be used to predict die lling (see Sec-tion 6). However, some of the work reported has been based on rheological data

obtained in steady state experiments, where the semisolid material has been main-tained at a particular shear rate for some time. In reality, in thixoforming the slurryundergoes a sudden increase in shear rate from rest to 100 s 1 or more as it enters thedie. This change takes place in less than a second. Hence, measuring the transientrheological response under rapid changes in shear rate is critical to the developmentof modelling of die lling and successful die design for industrial processing. It canbe investigated with two types of experiment. Firstly, via rapid shear rate changes ina rheometer and secondly, for higher fractions solid (where the torque capability of arheometer is not sufficient), with rapid compression experiments, for example, in thethixoformer itself or in a drop forge viscometer.

5.1. Rapid shear rate changes in rheometers

Studies of transient behaviour have included those by Kumar [79], Quaak [47],Peng and Wang [80], Mada and Ajersch [81,82], Azzi et al. [83] Koke and Modigell[66] Modigell and Koke [84,85] and Liu et al. [71,86]. Two relaxation times werequantied: (1) breakdown time and (2) buildup time. The breakdown time is thecharacteristic time for the slurry to achieve its steady-state condition after a shearrate change from a lower value to a higher value, while the buildup time is for a

change from a higher shear rate to a lower shear rate. These workers found that thetimes for breakdown are faster than those for buildup. This would be expected, asthe breaking up of bonds between spheroidal solid particles in agglomerates is likely



25/72


26/72

move past each other depends on the fraction of liquid medium present, the size of the particles and the degree of agglomeration. The data show that during a change inshear rate, in about 0.15 s the semisolid structure would have broken down from its

initial state. Regardless of the initial shear rate, the breakdown time decreases withincreasing nal shear rate [47,8183,85,86].As far as the existence of isostructure during the jump is concerned, Turng and

Wang [69] and Peng and Wang [80] observed an overshoot in the measured stressduring a rapid increase in shear rate. They found that this overshoot (or undershootin the case of a decrease in shear rate) is proportional to the change in shear rate.Therefore, they argue, for that instant, the material is behaving in a Newtonian way.The viscosity, and hence the structure, is constant, during the change. Peng andWang [80] observed that the overshoot increases with increasing solid fraction.Horsten et al. [87] and Quaak and coworkers [40,47] argued that during this transient

period structure evolution has not had time to occur and the structure correspondsto that of the previous shear rate. Kumar et al. [88] and Koke and Modigell [66]however, nd shear thickening isostructural ow behaviour (e.g. Fig. 21). In [66],

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1 1.2 1.4Time (secs)

S h e a r s t r e s s

( P a

)

0

20

40

60

80

100

120

S h e a r r a

t e ( 1 / s )Shear rate

Shear stress(after 5 hrs rest)

Shear stress(after 0 hr rest)

Shear stress(after 1 hr rest)

Shear stress(after 2 hrs rest)

Fig. 19. Shear rate jumps from 0 to 100 s1

after different rest times for Sn15%Pb alloy at fraction solid0.36 [86].

Table 2Tabulation of parameters obtained from shear rate jump experiments on Sn15%Pb alloys (at F s 0:36)under different rest times [86]

Rest times (h)

0 1 2 5

Shear rate jumps (0100 s 1)

gp (Pa s)a

2.1 5.4 8.0 23.0gss (Pa s) b 0.8 0.8 1.2 2.0sb (s)c 0.18 0.16 0.15 0.12a The errors are within 95% condence limits (0.5).b The errors are within 95% condence limits (0.2).c The errors are within 95% condence limits (0.03).



27/72

after each shear rate jump, the substance is sheared at _c0 to obtain equilibrium beforethe next jump. The plot of shear stress versus shear rate can be tted with a shearthickening HerschelBulkley model with a ow-exponent n 2:07. Koke andModigell [66] argue that this nding is of high importance for simulation of theindustrial process.

Data on the transient behaviour of aluminium alloys is sparse because themajority of the commercially available rheometers do not operate at semisolid alu-minium alloy temperatures.

Fig. 20. Microstructures of Sn15%Pb alloy f s 0:36 at various rest times: (a) 0 h (b) 1 h and (c) 2 h[86].



28/72

5.2. Rapid compression

For high solid fractions, above about 0.5, conventional rheometers do not havesufficient torque capability. Other methods must then be used, introducing com-

plexity because the shear rate is no longer constant throughout the material (as it canbe assumed to be in a concentric cylinder rheometer). Laxmanan and Flemings [68]measured the force and displacement for Sn 15%Pb compressed between parallelplates at low strain rates. The resulting load was not measured directly (but rather,derived from the pressure on the ram) and the rate of compression was much slowerthan in the industrial process. The work of Lou e et al. [89], carried out at highershear rates by backward extrusion on aluminium alloys, resembles industrial thi-xoforming more closely. However, the specimens were heated to temperature over along period of time ( 10 min) and then held isothermally for 30 min before com-pression. Such time periods would be considered long in industrial thixoforming.

Yurko and Flemings [90] designed a drop forge viscometer (Fig. 22) to study uidow behaviour at transient high shear rates. It consists of a lower platen and anupper platen, with an attached platen rod to track platen motion with time. It issimilar to a parallel plate compression viscometer but the upper plate is allowed tofall under the inuence of gravity. A high speed digital camera images the rod as itfalls. The force is calculated from the second derivative of the displacement dataallowing calculation of viscosities at shear rates in excess of 1000 s 1. A typicalexperiment yields instantaneous, volume-averaged viscosity rst under rapidlyincreasing shear rate and then under rapidly decreasing shear rate. Segregation of liquid from solid did not occur at the high shear rates. Liu et al. [91] have carried out

rapid compression in the thixoformer itself using a load cell to record the load versustime signals. The compression rate is then akin to industrial thixoforming and theload is measured directly. A typical signal response is shown in Fig. 23. The peak is

Fig. 21. (a) Series of step changes in shear rate to obtain; (b) isostructural ow curve [66]. The iso-

structural shear stress S iso is measured immediately after each step up in shear rate. In between eachexperiment, the material was sheared at the equilibrium shear rate _c0 of 100 s 1 (Sn15%Pb, 198 C, f s 0:41). The exponent for the t to (b) is 2.07 and hence the isostructural behaviour is termed shearthickening.



29/72

believed to originate from the three-dimensional skeletal structure built up in the

solid phase at rest, which breaks down under load. The width of the peak (or, moreaccurately, the downward part of it) is a measure of the time taken to destroy thisskeletal structure. A rough estimate then gives a breakdown time of about 10 ms, an

Fig. 22. Schematic of the drop forge viscometer [90].

0

0.5

1

1.5

2

2.5

3

3.5

4

01020304050607080

Displacement (mm)

L o a

d ( k N )

InitialContact

PeakLoad ( L )

MinimumLoad (L )

Peak stress can becalculated from thepeak load and the flowstress from theminimum load.

Fig. 23. Typical signal response to rapid compression of a semisolid alloy slug [91].



30/72

order of magnitude less than the relaxation times obtained from shear rate jumps inrheometer experiments (see Section 5.1) and must therefore be related to a differentmechanism. The height of the peak falls with temperature as the skeletal structure isconsumed, and the minimum load beyond the peak also decreases with increasingtemperature, both because a more spheroidal microstructure is developed and thefraction liquid increases (see Fig. 24). In practice, successful thixoforming takes placeat temperatures where there is little or no peak. Viscosity versus shear rate can be

Fig. 24. Load signals and microstructures at different temperatures for Alusuisse A356 aluminium alloy inrapid compression tests (ram speed 500 mm/s, zero soak time) [91].

Fig. 25. Comparison of apparent viscosities obtained by various experimental techniques and conditions[91]. (This work is [91], Loue is [89], Yurko is [90], Quaak is [47].)



31/72

derived from the load-displacement data using a method based on that outlined inLaxmanan and Flemings [68]. This does however assume a Newtonian uid at onestage in the analysis and this may introduce errors.

Data on viscosity versus shear rate for AlSi alloys is summarised in Fig. 25. It isimportant to be aware that small changes in silicon content can affect the resultsquite considerably by changing the solid fraction. The lower values recorded byYurko and Flemings [90] in comparison with those of Liu et al. [91] are derived foran alloy with higher silicon content and also one which has been soaked for longer(giving a larger particle size and consequently lower viscosity). Included in the gureis the steady-state viscosity determined by Quaak [47] for a 7% Si aluminium alloy,extrapolated to 0.5 fraction solid; this is well below the other results, emphasizingthat the steady state is not achieved in those experiments, nor in industrial thixo-forming.

6. Modelling

The recent commercialisation has highlighted the need to model slurry ow intodie cavities. Die design and processing conditions such as ram speed, dwell time andpressure have, to some extent, been a matter of trial and error. In particular, diedesign rules from die casting are not transferable to thixoforming. This is illustratedin Fig. 26, where attempts were made to produce a generic demonstrator consisting

of a round plate with three bolt holes and a central boss. In preliminary trials, there

Fig. 26. Numerical simulation of die lling [93]. (a) Partial lling of die; (b) modelling simulation of (a)where white corresponds to dark on (a); (c) modelling simulation with improved die design showingsmoother lling.



32/72

was difficulty in lling the die. Therefore, partial lling experiments were carriedout which demonstrated that the design of the die, particularly the in-gate, whichwas narrow as for die-casting, led to some jet ow across the cavity (Fig. 26(a))instead of the smooth progressive lling which is the aim in thixoforming. FLOW3D(a computational uid dynamics programme produced by FLOWSCIENCE,Los Alamos) was used to model the ow into the die, trying out different viscositiesin order to nd the range in which the experimental behaviour was mimicked(e.g. Fig. 26(b)). The agreement was promising given that the work did not takeinto account heat transfer in the die nor friction at the die surface. In addition,the model did not, in the version used, incorporate thixotropic behaviour as such(i.e. it assumed the uid had a constant viscosity, independent of the shear rateand time, which in practice is not the case). Changing the design of the die in thelight of the ndings of this simple modelling led to improved lling (e.g. Fig.26(c)). There is, therefore, real potential commercial benet to be obtained frombetter understanding of ow of semisolid material in dies, alongside the academicinterest.

In this section modelling is reviewed. Previous reviews include those by Kirkwood[92], Atkinson [93] and Alexandrou [28, Chapter 5]. Table 3 summarises the mainpapers on modelling [94125]. The papers are given in year order and this is carriedover into Tables 46 which give further information. These tables are not exhaustiveand, where authors have published in journals in addition to conferences, it is the journal papers which are cited. The papers are classied as to whether the modelling

is nite difference or nite element, one-phase or two-phase. There is in addition, apaper on micro-modelling [125] which does not strictly t any of these categories. Amajor aim here is to draw out the similarities and the differences between the owand viscosity equations which modellers are using. In Tables 46, this is done byquoting the equations from the papers but converting the symbols as far as possibleto be common. The list of symbols is given in the nomenclature. Where the equationsare given in complex terms they are then reduced to simple shear which allows moredirect comparison. It must be assumed, since this is not made explicit in most papers,that where a derivative, for example with respect to t is given as o=o t or d=d t , thatthis is in fact the substantive derivative D= Dt following the material as it moves.

Tables 46 identify the main features of the models and also observations on sim-ulation results and whether these have been validated. Where commercial code hasbeen used this is identied with the reference. The main threads in the developmentof each of the categories are discussed below, with an initial section on the model of Brown and coworkers since this has been used by a number of researchers. Thus, thenite difference papers are grouped as to whether they are based on: the model of Brown et al. (Section 6.2.1); FLOW3D (Section 6.2.2); MAGMAsoft (Section 6.2.3);Adstefan (Section 6.2.4); two-phase modelling (Section 6.2.5). For the one-phasenite element papers (Section 6.3.1) in some cases it makes sense to group the papersaccording to author. Thus, the headings are: Zavaliangos and Lawley; Backer; Al-

exandrou, Burgos and coworkers; viscoplastic constitutive models; power law cut-off (PLCO) model of Procast; model based on viscoelasticity and thixotropy. The twophase nite element papers are sensibly dealt with as a single section (Section 6.3.2)



33/72

Table 3Classication of models of semi-solid die lling

Finite difference FEM One-phase Ilegbusi and Brown (PHOENICS) [94] Zavaliangos and Lawley (ABAQUS) [103]

Barkhudarov et al. (FLOW3D) [95] Backer (WRAFTS) [104]Barkhudarov and Hirt (FLOW3D) [96] Alexandrou et al. (PAMCASTSIMULOR) [105]Modigell and Koke (FLOW3D) [84] Burgos and Alexandrou [106]Kim and Kang (MAGMAsoft) [97] Alexandrou et al. (PAMCASTSIMULOR) [107Modigell and Koke (FLOW3D) [85] Burgos et al. [108]Ward et al. (FLOW3D) [98] Alexandrou et al. [109]Messmer (FLOW3D) [99] Ding et al. (DEFORM3-D) [110]Seo and Kang (MAGMAsoft) [100] Jahajeeah et al. (Procast) [111]Itamura et al. (Adstefan) [101] Rassili et al. (FORGE3) [112]

Wahlen (Thixoform) [113]Alexandrou et al. [114]Orgeas et al. (Procast) [115]

Two-Phase Ilegbusi et al. [102] Zavaliangos and Lawley (ABAQUS) [103] Zavaliangos [116]Koke et al. [117]Kang and Jung [118]Binet and Pineau [119]Choi et al. [120]Kang and Jung [121]Yoon et al. (CAMPform2D) [122]

Kopp and Horst (ABAQUS) [123]Modigell et al. [124]

Commercial codes employed in work are indicated in brackets. If a code is not given then either the authors have written the cidentied in the text of the paper.


34/72

Table 4Summary of one-phase and two-phase nite difference simulation papers

Comments Flow and viscosity equations Obser

One-phaseIlegbusi and

Brown (PHOEN-ICS) [94]

Slurry incompressible, mass conser-vation, momentum conservation,energy conservation (enthalpy meth-od), solid fraction from [126], inputparameters identied, scalar-equa-tion method for free surface, singleinternal variable k constitutivemodel [127129], experimentallydetermined values of agglomerationfunction H and disagglomerationfunction G given. Chisel-shapedmould

s sy Ak c=cmax

1=3

1 c=cmax 1=3 gf _c

n 1C T k f sgn 1f _cn 12

Bvahtrdk

dt H T ; f s1 k

G T ; f sk _cn 13

Barkhudarovet al. (FLOW3D)[95]

Transport equation for g includesadvection term and relaxation termwhich accounts for thixotropy. Noyield stress, wall slip, or elastic orplastic behaviour at high f s . Inputparameters given

o go t

u:r g x ge g 14 M

terevth

o ko t

u:r k b11 k b2k _c 15

Note: We have changedur k to u:r k for clarityof notation.g g1 ck 16

) g g1 b1c

b1 b2 _c and

x b1 b2 _c 17


35/72

Barkhudarov andHirt (FLOW3D)[96]

Transport equation for g with a rateconstant for thinning and b rateconstant for thickening. g P ge thenmaterial is trying to relax towards thelower equilibrium value ge (i.e. thin-ning). Thixotropic data from [69].Heat transfer, viscous heating andsolidication effects included. Heattransfer negligible in time periodconsidered

o go t

ur g a min ge g; 0

b max ge g; 0 18

Sr

i.e. if ge g < 0 then the right-handside age g and if ge g > 0

then the right hand side bge g

Modigell andKoke FLOW-3D[84]

Shear stress is function of yield stress and structuralparameter j (which differs fromk in that it varies between 0 and1 rather than 0 and 1). Shearstress is assumed to growexponentially with increasing

solid fraction. Input parametersgiven

s sy f s exp Bf sk j _cm 19 tr

o jo t

a expb _cj e j 20

In equilibrium:

j e 1a _cm n

The equilibrium ow curveis then:

s sy f s exp Bf sk _cn with 21

k k a n m

Kim and Kang(MAGMAsoft)[97]

Comparison of Newtonian andOstwaldde-Waele with n of ) 0.48to +0.45 (depending on T ) under

shear rate of 32500 s 1 . Inputparameters given. Predict defectsin product from temperaturedistribution

Ostwaldde-Waele for viscositydependence on shear rate

Gpp

a


36/72


37/72

Seo and Kang(MAGMAsoft)[100]

Simple upsettingexperiments to obtainrheological data with A356.Input parameters given

Ostwaldde-Waele comparedwith CarreauYasuda:

Npste

g g1g0 g1

1 _ck an 1

a

Itamura et al.(Adstefan) [101]

Compares simulation for die-casting,squeeze-casting and rheocasting forboth metal ow and solidication

No details given DisL

Two-phaseIlegbusi et al.[102]

Single phase equations solved forwhole lling phase. Trajectories of given number of particles computed,assuming they disappear when theyhit a wall or are trapped in recircu-lation zone. Measure of segregation

obtained by comparing number of particles at given distance from inletto total number of injected particles


38/72


39/72

(In comparison with Ilegbusi andBrown [94], the liquid viscosity gf appears to have been included inthe function A, a velocity u ispresent in the rst term andn 1gn 1f is included in the func-tion C in the second term)The transport equation for k isdqkdt

r :quk K 1 k

G k2 ffiffiffiffi DII =2p 31which, if q is treated as a constantand ou xo x ;

o u y o y ;

o u z o z 0, reduces to

q o k

o t qu:r k K 1 k G k

2 _c 32(which is very similar to Barkhu-

darov et al. [95] but with a k2 in thesecond term on the right-hand siderather than k)

Alexandrou et al.(PAMCASTSIMULOR)[105]

HerschelBulkleysy 9615 f

3s

0:6 f sFluid assumed incompressible.n 1 then BinghamT dependence introduced throughmaking sy function of T

s ij sy1 exp m ffiffiffi DII =2p ffiffiffiffi DII =2p ( )

Dij

33

g K DII =2n1=2 34

For simple shear the equationsreduce tos sy1 exp m_c 35

g K _cn 1 36

Boaippsoe


40/72

Table 5 ( continued )

Comments Flow and viscosity equations Obse

Burgos and Alex-androu [106]

HerschelBulkley (as for [105]) Preie

Alexandrou et al.(PAMCASTSIMULOR)[107]

Bingham uid. Continuous modeldue to Papanastasiou [131] to avoiddiscontinuity at yield surface

_c 0 s 6 sy FPa

s g sy_c

_c s > sy 37

s g" sy 1 exp mj_cjj_cj #_c 13138

Simplifying:

s g _c sy1 exp m_c 39

Burgos et al. [108] HerschelBulkley expanded toinclude effect of evolution of microstructure. sy , K , n areassumed functions of f s and k.Single internal variable k. Assumetransient behaviour at constantstructure is shear thickening.Material parameters from [133]for Sn15%Pb with f s 0:45

o ko t

u:r k a1 k bk DII

2 1=2

exp c DII

2 1=2 ! 40

Fltdctwhich, for simple shear and where

kis not changing spatially, reduces to

o ko t

a1 k bk _c exp c _c

a b _c exp c _c k ke 41

(which for c 0 is equivalent to theMoore equation [72])

s K f s;k DII

2 n f s ;k 1=2

sy f s;k1 exp mj ffiffiffi DII =2p j ffiffiffiffi DII =2p

D

42


41/72

which, in simple shear, reduces to

s K f s; k_cn f s ;k sy f s; k

1 exp m_c 43

(Note that Eq. (8) in [108] is notcorrect)

Alexandrou et al.[109]

Bingham but using [131] to avoidsingularity. Simple compressiontest. No account taken of evolutionof samples internal structure

s g" sy 1 exp m ffiffi DII =2p ffiffiffi DII =2p #_c

44

Aitva

Ding et al. (DE-FORM3-D) [110]

Rigid viscoplastic constitutivemodel. Flow stress for AlSi7Mgobtained from compression onGleeble machine, ignoring initial

transient. LevyMises ow rule

r expa bT em _en 45 DtaM

Soes

When written in shear stress termsthis is equivalent to

s expa bT cm _cn 46

Jahajeeah et al.(Procast) [111]

Power law cut-off (PLCO) modelof Procast [134] i.e. isotropic,purely viscoplastic, independentof pressure, deformationhomogeneous

g_c; T g0T _cnT 0 for _c 6 _c0

g_c; T g0T _cnT for _c > _c047

Br_crw

Rassili et al.(FORGE3) [112]

Visco-plastic constitutive modelfrom force recordings of extrusiontests. Friction assumed very low.No time dependence

K e; T K 0 expb=T en 48

K is equivalent to the current yieldstress, K 0 to a yield stress and e toan effective strain. In shear terms,this is analogous to

s sy expb=T cn 49

Smglsbo


42/72


Comments Flow and viscosity equations O

Wahlen (Thixo-form) [113]

Model based on viscoelasticityand thixotropy r r 0 1 exp

e_esM

f sk2

Rb R

3

_e_e0

exp Q RT " #

m

50

This could be written in shear stressterms as:

s sy 1 exp c_cs M

f sk

2

Rb

R 3

_c_c0

exp Q RT " #

m

51

Note: R means both the particle sizein the Rb = R term and the gasconstant in the exponential term

Alexandrou et al.[114]

2-D jets of Bingham and Herschel Bulkley uids impacting on verticalsurface at distance from die exit in

order to account for ow instabil-ities (e.g. toothpaste effect) insemisolid processing

Papanastasiou model [131]:

s g sy1 exp m_c

_c _c 52

Generalized to HerschelBulkleyuid by specifying

g j _cn 1 53


43/72

Orgeas et al. (Pro-cast) [115]

PLCO model as for Jahajeeahet al. [111] (see above) but withonly one value of cut-off _c0(determined by geometry).n and g0 dependent on f s

g g0_c0_cc !

n 1

for _c < _c0

g g0_c

_cc !n 1

for _c P _c0

54


44/72

Table 6Two-phase nite element simulation papers and micro-modelling

Comments Flow and viscosity equations Observ

Zavaliangos andLawley (ABA-

QUS) [103]

High volume fraction solid f s P 0:7.Porous viscoplasticity model; uid ow

in porous medium; continuity equations.Isothermal. Behaviour symmetric undertension and compression

See paper for details Coming

ratva

Zavaliangos [116] Deformable porous medium saturatedwith liquid. Stress partitioned into stresscarried by solid phase and purelyhydrostatic component for pressure inliquid phase. Solid phase has two limits:fully cohesive porous solid and cohe-sionless granular material. Degree of cohesion represented by internal variable

which does evolve with deformation (cf.single internal variable in [127129]).Permeability equation implies that solid-liquid segregation decreases as the grainsize decreases. Behaviour not symmetricunder tension and compression

See paper Constrcotio

Koke et al. [117] Liquid phase assumed Newtonian. Solidphase is pseudo-uid with Herschel Bulkley viscosity. Darcy Law, Carmen Kozeny capillary approach. f s P 0:5

gs sy

_c k _cm 1 k gf 55

VePhwi

At equilibrium, k k , the coefficient in the

HerschelBulkley power law term (see Backer[104]), and m n. sy and k are assumed toincrease exponentially with f s . This equationgives a different expression for shear stressfrom that due to Brown and coworkers [127 129]. Note: It is not clear where k has gone toin Eq. (16)


45/72

Kang and Jung[118]

Compressible viscoplastic model for thesolid phase and Darcys law for the owof liquid through a porous medium.Separation coefficient introducedS S 0 1 S 0 eecr , where S 0 is the ratioof the actual separation to the initialseparation, e is the equivalent strain andecr a critical strain

r K expS _em exp Q RT 1 b f l 2=3 56 Thho

somawiin

for e < ecr ; e > ecr1

r K exp1 S _em exp Q RT 1 b f l 2=3

57

for ecr < e < ecr1These equations could be written in shearstress terms by replacing r with s , e with c, _ewith _c etc.

Binet and Pineau

[119]

Mixture approach. Hydrodynamic part

same as for most incompressible CFDcodes but velocity eld represents veloc-ities of the mixture and a source term isadded to the momentum equations totake account of the diffusion velocities of the individual phases. Relative velocitiescalculated from interaction force betweenphases. Darcys Law, Carman-Kozenyrelation. Rheological data from [136]

See paper Pre

enin

Choi et al. [120] Compressible visco-plastic solid, liquidphase following Darcys law. Kuhns

yield criterion [137] for deformation of solid phase. Friction equation at die/material surface from [138]

See paper Heafo

taktemucme


46/72


Comments Flow and viscosity equations Observ

Kang and Jung[121]

As for Kang and Jung [118] As for Kang and Jung [118] Predictcona

ratYoon et al.

(CAMPform2D)[122]

Von Mises yield criterion. Semi-solidtreated as single phase with incompress-ibility. Flow stress as function of strain(with f s and breakage ratio). Mixturetheory and DArcys Law to update f s .Material properties for Sn15%Pb from[139]. Input parameters for non-isother-mal Al2024 alloy given

Equations are summarised in Fig. 1in the paper

IsotioNogiv

r f K e

ecr !n

exp b_em for e < ecr 58

r f K exp be est

ecr est ! !_em for e P ecr59

These equations could be written in shearstress terms in an analogous way to Kang andJung above

Kopp and Horst(ABAQUS)[123]

DruckerPrager yield criterion (yieldstrength different in tensile and in com-pressive strain). Finite element meshattached to solid phase

Modigell et al.[124]

Equilibrium ow behaviour modelledwith HerschelBulkley. Thixotropymodelled with structural parameter fol-lowing rst order differential equation.Pseudo-uid approach for the solidphase. All non-Newtonian propertiesshifted to the solid. Liquid Newtonian.Continuity and momentum equationssolved for each phase. Interaction be-tween phases modelled with Darcy law

Si15of l


47/72

Micro-modellingRouff et al. [125]

Volume solid fraction of theactive zone, f sA , is the internal vari-able. The active zone consists of thesolid bonds between spheroids and theliquid between spheroids which is notinternally entrapped. During deforma-tion the bonds are broken and liquid isreleased. Spheroids and active zonetreated as isotropic and incompressible

f sA 0 for f s 6 f c

Goon

f sA f s

f s D1 f s_cn for f sA > f

c 60


48/72

but highlighting the distinctive features of the papers. The work of Rouff et al. [125]on micro-modelling is covered in Section 6.3.3.

6.1. Model of Brown and coworkers

Brown and coworkers [127129] presented a constitutive model based on thesingle internal variable concept (see Section 4.1), where the structural parameter kvaries between 0 and 1, depending on whether the structure is fully broken down orfully built-up respectively. Their model assumes that ow resistance is due tohydrodynamic ow of agglomerates and deformation of solid particles within theagglomerates. It has been used by a number of workers both for nite difference andfor nite element modelling (Ilegbusi and Brown [94]], Ilegbusi et al. [102], Zaval-

iangos and Lawley [103], Backer [104]). Ilegbusi and Brown [94] used it in their nitedifference work but also introduced a yield stress sy. The second term on the righthand side in Eq. (12) (at the top of Table 4) represents the hydrodynamic interactionamong agglomerates, with Ak a hydrodynamic coefficient depending on the size,distribution and morphology of the particle agglomerates. The term is linear in shearrate and increases non-linearly with the solid fraction, f s (the effective volumefraction of solid c f s1 0:1k). It depends weakly on f s for f s < 0:5 but thenincreases at an increasing rate towards an innite asymptote at a given solid fractionand state of agglomeration. The third term on the right hand side represents thedeformation resistance due to energy dissipated in the plastically deforming particle particle bonds. Under isothermal conditions and at constant structure this termindicates a shear rate thickening response with n > 1during rapid shear ratetransients, the deformation resistance increases with increasing shear rate (consistentwith experimental work by Kumar et al. [129]). There is debate as to whether shearrate thickening is the constant structure response (e.g. see [84]). The term exhibits astrong inverse dependence on temperature through C T C 0 expnQ= RT , whichbrings in the temperature dependence of diffusional processes and the temperaturedependence of the solid deformation.

Eq. (13) (the second equation in Table 4) represents evolution of the structureparameter as a function of ow conditions and state variables. H is an agglomerationfunction and G a disagglomeration function representing the shear-induced ruptureof the particleparticle bonds.

Overall the model of Brown et al. predicts an increase in deformation resistancewith the solid fraction and this becomes rapid between 0.5 and 0.6 f s. Brown et al.state that it is not valid beyond this sharp increase in deformation resistance and isapplicable only for f s < 0:5 0:6.

6.2. Finite difference modelling

6.2.1. One-phase nite difference based on the model of Brown et al.Ilegbusi and Brown [94] use the Brown et al. model (see Section 6.1), but with a

yield stress, to examine ow into a chisel shaped cavity. This showed the importance



49/72


50/72

stress sy is now multiplied by the structural parameter j and the exponential in thesecond term in Eq. (19) in Table 4 is no longer present. This may have been rolledinto the consistency index k as the text states k . . . increased exponentially with thesolid fraction. The experimental rheological data was for Sn15%Pb but the

material used for die lling simulation was A356 aluminium alloy. The parametersused for the model were adjusted empirically during the simulation study (but arenot given in the paper). Modigell and Koke found that above a critical inlet velocity

Fig. 27. Comparison between simulation of ow into a cavity with a round obstacle assuming Newtonianbehaviour and assuming thixotropic behaviour [84].



51/72

the lling was not laminar any more. The transition between laminar and turbulentlling (in the sense of a smooth ow front and one that is starting to break up) couldbe represented reasonably well.

Ward et al. [98,140,141] found that, in modelling force versus time for a rapidcompression test (see Section 5.2), the implicit solver tended to give an initial peakregardless of whether the viscosity was Newtonian or thixotropic, especially if thesoftware was allowed to choose its own time step and if the model started with a gapbetween the slug and the platen. Provided the time step was small enough not to be afactor in determining the results, the implicit solver could reproduce the downwardslope of the initial peak and the subsequent force prole. The explicit solver couldaccurately model shear rate jumps in a rotational viscometer (see Section 5.1) butwas inordinately slow. FLOWSCIENCE have therefore produced a new alternatingdirection implicit (ADI) solver to cope with the large ranges and changes of pressureassociated with thixotropic slurries. Fig. 28 shows the ADI result for a shear rate jump in Sn15%Pb compared with experimental results (three repeats of the sameexperiment) and the results from a one-dimensional spread sheet calculation. Thecontrast between implicit and explicit solvers is not mentioned elsewhere and it is notclear whether workers have tested their modelling against the artefacts found withthe implicit solver. Ward et al. [98] found that to model the shear rate jumps, aninitial viscosity was required which was 25 times lower than the experimental val-ues. This suggests that the initial breakdown of the slurry is very rapid, possiblybeyond the limits of the viscometer data collection system, even though the system

used in this experimental work has the fastest data collection rate of any existingsystem (see Section 5.1 and [86]).Messmer [99] used FLOW3D to simulate thixoforging rather than thixoforming

i.e. the slurry is inserted directly into open dies and the parts of the die are thenbrought together by a ram. In simulating the thixoforging process, moving dies musttherefore be modelled. The apparent viscosity depends on fraction solid f s, shear rate_c and time t . The fraction solid is calculated using the Scheil equation (i.e. assuming asimple binary). The equilibrium viscosity is given by Eq. (23) in Table 4 and the time

Fig. 28. Shear rate jump from 1 to 100 s 1 in SnPb alloy f s 0:36, showing repeats of the sameexperiment and modelled ts using a spreadsheet and FLOW-3D [98].



52/72


53/72

casting and rheocasting. There would be fewer shrinkage defects in rheocasting thanwith the other processes.

6.2.5. Two-phase nite differenceThere appears to be only one paper using a nite difference model for a two phase

analysis, that by Ilegbusi et al. [102]. The single phase equations are solved for thewhole lling phase. Trajectories of a given number of particles are computed,assuming they will disappear when they hit a wall or are trapped in a recirculationzone. A measure of segregation is obtained by comparing the number of particles ata given distance from the inlet to the total number of injected particles.

6.3. Finite element modelling

The consideration of nite element modelling is divided into one-phase (Table 5)and two-phase (Table 6) treatments. A number of different commercial and othercodes are used. Note that in the discussion below the papers are grouped andtherefore are not discussed in the order in which they appear in the tables (which isby year).

6.3.1. One-phase nite elementZavaliangos and LawleyZavaliangos and Lawley [103] use identical equations to Ilegbusi and Brown [94]

from the Brown et al. model (see Section 6.1) but without a yield stress. The analysisis for Sn15%Pb and, for fractions solid less than about 0.5, it is predicted that a freestanding billet will collapse. The thixoforming of a simple shape is simulated. Noexperimental validation of the results is given. Zavaliangos and Lawley deal with atwo phase analysis in the same paper (see Section 6.3.2).

BackerVarious rheological models were programmed into the WRAFTS software by

Backer [104] including: a Newtonian; a HerschelBulkley model (i.e. combining a

yield stress with a power lawsee Eqs. (27) and (28) in Table 5); and a Bingham uid(see Section 2) combined with a power law dependence (see Eqs. (29) and (30) inTable 5). Note that it is not clear that the description of a Bingham uid combinedwith a power law dependence is correct here as there is no yield stress in Eq. (30)but rather one term dependent on _c and one on _cn . An internal variable model is alsoused (viz. Brown et al. in Section 6.1). In this the structural parameter k (which variesbetween 0 and 100% rather than 0 and 1) is perceived as a chemical concentrationand a convective transport equation is written for its spatial and temporal variation(Eq. (31) in Table 5). When this equation is simplied by assuming the uid density isconstant and the velocity does not vary spatially, it can be compared to that due to

Barkhudarov et al. [95]. There is a strong similarity but with a factor of k2 in thesecond term on the right hand side rather than k as in Eq. (15) in Table 4. Thisreduces the contribution of disagglomeration.



54/72

With the Newtonian rheological model, due to the relatively low viscosity of theliquid metal, there are a number of locations behind cores in a complex die thatremain unlled while the liquid ows past them. Such locations would increase gasentrapment and would show ow lines in a casting. For the power law rheologicalmodel, the uid lls behind the cores. With the internal variable model, a largerpercentage of the ow into the cavity arises from the runner at the side of the diecavity, in contrast with the Newtonian and Bingham/power law models which pre-dict a larger ow rate from the bottom runner. The reason for the larger ow ratefrom the side runner is that the structural parameter (agglomeration variable) k isreduced as the mixture ows through the runner system from a maximum value of 100% in the shot sleeve to less than 20% at the end of the side runner. The value atthe bottom runner is 40%; thus, the material in the side runner is less viscous and canow into the cavity more readily. Experimental validation is not presented in thepaper.

Alexandrou, Burgos and coworkersThere are several papers by Alexandrou, Burgos and coworkers [105109,114]

sometimes with Alexandrou and Burgos working together and sometimes in coop-eration with others. The papers have similar threads running through and thereforeare treated together here.

In [105], Alexandrou et al. use the commercial code PAMCASTSIMULOR tocompare Newtonian and Bingham lling of a three-dimensional cavity with a core.The yield stress sy is a function of the fraction solid f s. Temperature dependence isintroduced through this relationship. Eqs. (33) and (34) in Table 5 give the viscousstress tensor and the viscosity expression. DII is the second invariant of the defor-mation rate tensor. In simple shear, ffiffiffiffiffiffiffiffiffiffi DII =2p _c. When the local stress is larger thansy, the slurry behaves as a Non-Newtonian uid. m controls the exponential rise inthe stress at small rates of strain and, depending on the value of the power lawcoefficient n, the behaviour is either shear thinning n < 1 or shear thickeningn > 1. The continuous Bingham law in Eq. (33) is based on that introduced byPapanastasiou [131] to avoid the discontinuity at the yield stress. In [105], the valueof n is taken to be 1. In the simulation, rstly pipe ow was studied, demonstratingthat the ow at the outlet was identical with the analytical solu

tixoformabilidade geral de atkinson

Documents