Progress in Materials Science 50 (2005) 341–412 www.elsevier.com/locate/pmatsci
Modelling the semisolid processing of 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 nondendritic microstructure in the semisolid state. The family of innovative manufacturing methods based on this behaviour has been developing over the last 20 years or so and originates from scientiﬁc work at MIT in the early 1970s. Here, a summary is given of: routes to spheroidal microstructures; types of semisolid processing; and advantages and disadvantages of these routes. Background rheology and mathematical theories of thixotropy are then covered as precursors to the main focus of the review on transient behaviour of semisolid alloy slurries and computational modelling. Computational ﬂuid dynamics (CFD) can be used to predict die ﬁlling. However, some of the reported work has been based on rheological data obtained in steady state experiments, where the semisolid material has been maintained at a particular shear rate for some time. In reality, in thixoforming, the slurry undergoes a sudden increase in shear rate from rest to 100 s1 or more as it enters the die. This change takes place in less than a second. Hence, measuring the transient rheological response under rapid changes in shear rate is critical to the development of modelling of die ﬁlling and successful die design for industrial processing. The modelling can be categorised as onephase or twophase and as ﬁnite diﬀerence or ﬁnite element. Recent work by Alexandrou and coworkers and, separately Modigell and coworkers, has led to the production of maps which, respectively summarise regions of stable/unstable ﬂow and regions of laminar/transient/turbulent ﬁll. These maps are of great potential use for the prediction of appropriate process parameters and avoidance of defects. A novel approach to modelling by Rouﬀ and coworkers involves micromodelling of the ‘active zone’ around spheroidal particles. There is little quantitative data on the discrepancies or otherwise between die ﬁll simulations and experimental results (usually obtained through interrupted ﬁlling). There are no direct comparisons of the capabilities of various software packages to model the
*
Tel.: +441162231019; fax: +441162522525. Email address:
[email protected] (H.V. Atkinson).
00796425/$  see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2004.04.003
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ﬁlling of particular geometries accurately. In addition, the modelling depends on rheological data and this is sparse, particularly for the increasingly complex twophase models. Direct ﬂow visualisation can provide useful insight and avoid the eﬀects of inertia in interrupted ﬁlling experiments. 2004 Elsevier Ltd. All rights reserved.
Contents 1.
Introduction to semisolid processing...................................................................................... 1.1. Routes to spheroidal microstructures ....................................................................... 1.2. Types of semisolid processing...................................................................................... 1.3. Advantages and disadvantages....................................................................................
345 346 349 353
2.
Background rheology ................................................................................................................... 355
3.
Origins of thixotropy.................................................................................................................... 358
4.
Mathematical theories of thixotropy ...................................................................................... 4.1. Models based on a structural parameter ............................................................. 4.2. Direct structure theories................................................................................................. 4.3. Simple viscosity theories ................................................................................................
5.
Transient behaviour of semisolid alloys ................................................................................ 364 5.1. Rapid shear rate changes in rheometers .................................................................. 364 5.2. Rapid compression........................................................................................................... 368
6.
Modelling.......................................................................................................................................... 6.1. Model of Brown and coworkers................................................................................. 6.2. Finite diﬀerence modelling............................................................................................ 6.2.1. Onephase ﬁnite diﬀerence based on the model of Brown et al...... 6.2.2. Onephase ﬁnite diﬀerence based on FLOW3D .................................... 6.2.3. Onephase ﬁnite diﬀerence based on MAGMAsoft............................. 6.2.4. Onephase ﬁnite diﬀerence with Adstefan................................................ 6.2.5. Twophase ﬁnite diﬀerence............................................................................. 6.3. Finite element modelling................................................................................................ 6.3.1. Onephase ﬁnite element................................................................................. 6.3.2. Twophase ﬁnite element ................................................................................ 6.3.3. Micromodelling.................................................................................................
7.
Flow visualisation ........................................................................................................................... 404
8.
Concluding remarks....................................................................................................................... 406
362 362 363 364
371 388 388 388 389 392 392 393 393 393 401 402
Acknowledgements........................................................................................................................................ 406 References......................................................................................................................................................... 406
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Nomenclature (in the order in which the symbols appear in the paper) shear stress shear rate viscosity yield stress constant related to the viscosity in Eqs. (1)–(3) shear rate exponent in Eqs. (2) and (3) viscosity as the shear rate tends to inﬁnity viscosity as the shear rate tends to zero fraction solid structural parameter varying between 1 for completely built up and 0 for completely broken down t time a; b; c; d constants in Eq. (5) k1 ; k2 constants for breakdown and buildup in Eq. (6) p; q exponents in Eq. (6) N average number of links per chain in Eq. (7) k0 ; k1 ; k2 rate constants in Eq. (7) P number of single particles per unit volume in Eq. (7) Ne 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 peakstress viscosity in Table 2 gss ‘ﬁrst’ steadystate viscosity in Table 2, i.e. after the ‘fast’ breakdown process as opposed to the ‘slow’ sb ‘ﬁrst’ breakdown time in Table 2 i.e. for the ‘fast’ breakdown process AðkÞ hydrodynamic coeﬃcient as a function of k in Eq. (12), Table 4 and also in Eqs. (25) and (29), Table 5 c eﬀective volume packing fraction solid in Eq. (12), Table 4 and in Eq. (25) and (29), Table 5 cmax maximum eﬀective volume packing fraction solid in Eq. (12), Table 4 and in Eq. (25) and (29), Table 5 gf viscosity of ﬂuid CðT Þ 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 5 G disagglomeration function in Eq. (13), Table 4 and in Eq. (26), Table 5 u velocity vector x inverse of the relaxation time in Eq. (14), Table 4 b1 ; b2 constants in Eq. (15), Table 4 c constant in Eq. (16), Table 4
s c_ g sy k n g1 g0 fs k
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rate constant for thinning in Eq. (18), Table 4 rate constant for thickening in Eq. (18), Table 4 and also in Eq. (24), Table 4 B; k coeﬃcients in Eq. (19), Table 4 j structural parameter in Eq. (19), Table 4, varying between zero (fully broken down) and inﬁnity (fully built up) je equilibrium value of j a; b constants in Eq. (20), Table 4 A; B coeﬃcients in Eq. (23), Table 4 s shear stress tensor in equations in Table 5 Du rate of deformation tensor in equations in Table 5 DII second invariant of the rate of strain tensor in equations in Table 5 m coeﬃcient in Eq. (27), Table 5 u velocity scalar q density K; G coeﬃcients in Eq. (31), Table 5 sij viscous stress tensor in Eq. (33), Table 5 Dij uij þ uji in Eq. (33), Table 5 m coeﬃcient controlling the exponential rise in stress in Eq. (33), Table 5 K coeﬃcient in Eq. (34), Table 5 a; b; c coeﬃcients in Eq. (40), Table 5 ke equilibrium value of the structural parameter in Eq. (41), Table 5 D rate of strain tensor in Eq. (42), Table 5 r ﬂow stress in Eq. (45), Table 5 and subsequent equations e strain in Eq. (45), Table 5 e_ strain rate a; b; m; n coeﬃcients and exponents in Eq. (45), Table 5 T temperature strain rate cutoﬀ in the power law cutoﬀ model, Eqs. (47) and (54), c_ 0 Table 5 K current yield stress in Eq. (48), Table 5 K0 yield stress in Eq. (48), Table 5 e eﬀective strain in Eq. (48), Table 5 true stress at the solidus temperature and a strain rate e_ 0 of 1 s1 in Eq. r0 (50), Table 5 sM Maxwell time g=E where E is the Young’s 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 ðRb =R 0:25Þ Q activation energy for selfdiﬀusion in Eq. (50), Table 5 R ideal gas constant in the exponential term in Eq. (50), Table 5 m material exponent in Eq. (50), Table 5 a b
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c_ g0 c_ c r S K; m; b ecr ; ecr1 b fAs fc D; n Bi Re
345
strain rate tensor viscosity at a characteristic shear strain rate c_ c in Eq. (54), Table 5 characteristic shear strain rate in Eq. (54), Table 5 eﬀective stress in Eq. (56), Table 6 separation coeﬃcient in Eq. (56), Table 6 coeﬃcients in Eq. (56), Table 6 critical strain I and critical strain II in Eq. (56), Table 6 ‘breakage ratio’ in Eq. (58), Table 6 volume solid fraction of the ‘active zone’ in Eq. (60), Table 6 critical fraction solid in Eq. (60), Table 6 material parameters in Eq. (60), Table 6 Bingham number Reynolds number
1. Introduction to semisolid processing Mascara, honey and certain kinds of paint are all thixotropic. When they are sheared they ﬂow, when allowed to stand they thicken up again; their viscosity is shear rate and time dependent. Spencer et al. [1] ﬁrst discovered such behaviour in semisolid metallic alloys in the early 1970s when investigating hot tearing with a rheometer. If the material was stirred continuously during cooling from the fully liquid state to the semisolid state the viscosity was signiﬁcantly lower than if the material was cooled into the semisolid state without stirring. Stirring breaks up the dendrites which would normally be present so that the microstructure in the semisolid state consists of spheroids of solid surrounded by liquid (Fig. 1). It is this microstructure which is a requirement for thixotropic behaviour and for semisolid processing. When such a semisolid microstructure is allowed to stand, the spheroids agglomerate and the viscosity increases with time. If the material is sheared, the agglomerates are broken up and the viscosity falls. In the semisolid state, with between 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 ﬂows with a viscosity similar to that of heavy machine oil. This is the behaviour which is exploited in semisolid processing [2] and which is illustrated in Fig. 2, where the alloy can be cut and spread like butter. Nearly 30 years of work and eﬀort have been invested in the ﬁeld of semisolid processing and the increase in interest in this ﬁeld has been marked by seven international conferences [3–9] with an eighth planned in Cyprus in 2004. Semisolid processing is rivalling other manufacturing routes for military, aerospace and most notably automotive components [10–12]. In Europe, suspension parts, engine brackets and fuel rails for automotives are being produced. In the USA, examples include mechanical parts for mountain bikes and snowmobiles [13], while in Asia there is concentration on the production of electronic components such as computer notebook cases and electrical housing components, particularly in magnesium alloys
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Fig. 1. Micrograph of a typical (a) dendritic microstructure in an ascast sample and (b) globular microstructure in a semisolid alloy sample.
via thixomolding e.g. [12]. Fig. 3 shows some of the components produced by Thixoforming at Stampal for an Alfa Romeo car. 1.1. Routes to spheroidal microstructures There are many diﬀerent routes for obtaining nondendritic microstructures. The main ones are described here. (1) Magnetohydrodynamic (MHD) stirring: This involves stirring electromagnetically (and hence without the contamination, gas entrapment and stirrer erosion involved in mechanical stirring) in the semisolid state to break up the dendrites e.g. [14]. Much of the commercial production of aluminium alloy components to date has been based on MHD material supplied by Pechiney and SAG. There are some problems associated with this route including lack of uniformity and the fact that the spheroids are not completely round with some ‘rosette’ character remaining. (2) Sprayforming: Sprayforming is a relatively expensive route but one which can be used to produce alloys, which cannot be produced in any other way, such as aluminium–silicon alloys with greater than 20 wt.% silicon e.g. [15]. Sprayforming essentially involves the atomisation of a liquid metal stream and collection of the droplets on a former. The resulting microstructure is ﬁne 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. by extrusion. On reheating into the semisolid state, recrystallisation occurs and the liquid penetrates the recrystallised boundaries so resulting in spheroids surrounded by liquid. The SIMA route [17] involves hot working and the RAP route [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 fully rounded than those from the MHD route. The main disadvantages are that there may be variation in the amount of stored work across the section, resulting in
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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) multilink, rear suspension support 8.5 kg, A357, T5; (b) steering knuckle A357, T5 substitution of cast iron part.
inhomogeneity, and extrusion can be diﬃcult and expensive with wider billet diameters. (4) Liquidus/nearliquidus casting: There have been recent developments in producing feedstock by manipulating the solidiﬁcation conditions. The UBE new rheocasting (NRC) process [19,20] is based on this principle with the molten metal at
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nearliquidus temperature poured into a tilted crucible and grain nucleation occurring on the side of the crucible. The grain size is ﬁne because the temperature is near liquidus. An allied technique is the direct thermal method [21]. In the cooling slope method [22], liquid metal is poured down a cooled slope and collects in a mould. Nucleation on the slope ensures the spheroid size is ﬁne. With liquidus casting, a high rate of nucleation can be achieved within the entire volume of undercooled melt [23,24]. (5) ‘New MIT Process’: This is a recently developed hybrid of stirring and near liquidus casting [25] (Fig. 4). A stirrer that also provides the cooling action is inserted into an alloy melt held a few degrees above the liquidus. After some seconds of stirring, the melt temperature decreases to a value which corresponds to a fraction of solid of only a few percent and the stirrer is withdrawn. (6) Grain reﬁnement: Grain reﬁned alloys can give equiaxed microstructures e.g. [26] but it is diﬃcult to ensure the grain structure is uniformly spheroidal and ﬁne and the volume of liquid entrapped in spheroids tends to be relatively high. (7) Semisolid thermal transformation: Spheroidal structure can also be produced by heating a dendritic structure to the semisolid temperature range for a period of time. This is known as semisolid thermal transformation, or SSTT [27]. The structures produced by this route tend to be relatively coarse (around 100 lm diameter particles). Other methods are summarised in [2,12,28].
Fig. 4. ‘‘New MIT’’ process (courtesy: Prof. M.C. Flemings, MIT).
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1.2. Types of semisolid processing ‘Semisolid processing’ now covers a whole family of processes. The terminology is as follows. ‘Rheocasting’ refers to the process where the alloy is cooled into the semisolid state and injected into a die without an intermediate solidiﬁcation step. A typical rheocaster is shown in Fig. 5. The nondendritic microstructure can be obtained by a variety of means during cooling (e.g. by mechanical stirring, by stimulated nucleation of solid particles as in the new rheocasting NRC process recently patented by UBE [19,20] (see Fig. 6), or by electromagnetic stirring in the shot sleeve as in the new semisolid metal casting process from Hitachi [30] (see Fig. 7). The NRC process involves pouring molten alloy, at a temperature slightly above the liquidus, into a steel crucible and then controlled cooling to achieve a spheroidal microstructure before transfer to a forming machine. There is no need for specially treated thixoformable feedstock and scrap can be readily recycled within the plant. Hall et al. [20]
Fig. 5. Continuous rheocaster (after [29]).
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Fig. 6. Schematic diagram of the new rheocasting (NRC) process. The inversion procedure causes the oxide skin on the exposed surface to run into the runner and biscuit.
showed that the NRC route has a lower unit cost than Thixoforming, due to the lower starting material cost. ‘Rheomoulding’ is allied to polymer injection moulding, and uses either a single screw [31,32] or a twin screw [33,34] (Fig 8). Liquid metal is fed into a barrel where it is cooled while being mechanically stirred by a rotating screw. The semisolid material is then injected into a die cavity. Such processes are suitable for continuous production of large quantities of components and do not require specially produced feedstock material (at a price premium). ‘Thixo’ usually refers to processes where an intermediate solidiﬁcation step does occur. There are exceptions to this, e.g. ‘Thixomolding’. ‘Thixomolding’ is the process licensed by the ﬁrm called ‘Thixomat’ [35,36]. It is now used by numerous companies, particularly in Japan and the US, to produce magnesium alloy components, e.g. for portable computers and cameras. As for rheomoulding, it is allied to injection moulding of polymers. Magnesium alloy pellets are fed into a continuously rotating screw (Fig. 9) and the energy generated by shearing is suﬃcient to heat the pellets into the semisolid state. The screw action produces the spheroidal microstructure and the material is injected into a die. Although the process is highly eﬀective with magnesium alloys, aluminium alloys in the semisolid state attack the screw and the barrel. Strenuous eﬀorts have been made to overcome these problems but it is not clear that a successful commercial outcome has yet been achieved. ‘Thixoforming’ can cover both ‘thixocasting’, ‘thixoforging’ and an intermediate process called ‘thixoforming’. ‘Thixocasting’ usually means that the alloy is solid initially and has been treated in such a way that when it is heated into the semisolid state it will have a nondendritic microstructure. It is reheated into the semisolid state and ‘casting’ is implying that the liquid content prior to forming is relatively high i.e. above about 50 vol.%. This is the type of process used by Magnetti Marelli in Italy to produce fuel rails [37]. ‘Thixoforging’ describes the process where suitable material is heated into the semisolid state and placed between dies halves e.g. [38]. The parts of the die are then brought together by a ram. The direct insertion of the slurry into the die reduces material use because of the lack of runners, gate and press discard.
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Fig. 7. Electromagnetic stirring in the shot sleeve in the new semisolid metal casting process from Hitachi (after [30]).
‘Thixoforming’ is the process where suitable material is heated into the semisolid state and injected into a die. Usually, the liquid content is between 30 and 50% prior to forming. This is the type of process used by Stampal in Italy to produce the Alfa Romeo 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, antilock brake system valves and automotive steering pumps amongst others. A thixoforming press 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. The specimen is induction heated into the semisolid state. When it has reached the appropriate proportion of liquid it is forced into the die. Usually in commercial
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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 Thixomoldere (see Fig. 9).
Fig. 9. Thixomolding process [35].
thixoforming, the slugs are heated on a carousel. Cycle times are then very comparable with die casting, if not faster because the full solidiﬁcation range does not have to be gone through. The distinctions between rheocasting, thixocasting and thixoforging are illustrated in Fig. 11 [40]. Other processes include the shearcooling roll (SCR) process [41,42] and the cooling slope process [43]. In addition, there is the possibility of using semisolid slurries in solid freeform (SSF) technology [44]. This method deposits a stream of
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Fig. 10. Thixoforming press (courtesy: University of Sheﬃeld).
Fig. 11. Schematic illustration of diﬀerent routes for semisolid metal processing [40].
slurry through a nozzle that moves relative to a substrate. Components are built by building 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 disadvantages in semisolid processing. They are [2,45–47]: Advantages The main advantages of semisolid processing, relative to die casting, are as follows.
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(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 ﬁlling of the die with no air entrapment and low shrinkage porosity gives parts of high integrity (including thinwalled sections) and allows application of the process to higherstrength heattreatable alloys. (4) Lower processing temperatures reduce the thermal shock on the die, promoting die life and allowing the use of nontraditional die materials e.g. [48] and processing of high melting point alloys such as tool steels and stellites [49] that are difﬁcult 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 solidiﬁcation shrinkage gives dimensions closer to near net shape and justiﬁes the removal of machining steps; the near net shape capability (quantiﬁed, 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 facilitate 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 semisolid region require accurate control of the temperature. (6) Liquid segregation due to nonuniform heating can result in nonuniform composition in the component. The economic advantages of thixoforming have been discussed [51,52], including the use of quality function deployment (QFD) to evaluate the interrelationships between thixoforming characteristics (energy usage, near net shape capability, mechanical integrity of product, short cycle time, reduced die wear, raw material cost, process development, skills/wages of work force) and product characteristics (weight, strength, geometry, tolerances, price premium, lead time, ﬂexibility, ﬁnishing operations) and quantiﬁed in software (www.shef.ac.uk/~ibberson/thixo. html). The economics of the NRC process have been analysed [20]. The NRC process does not suﬀer from the disadvantage of (1). Such analysis is important for industries 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 [3–9]. Some of these conferences have been refereed whilst others have
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not. It is therefore the policy in this review to give references as far as possible to refereed 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 published edited by Figueredo [29]. The recent Fan review is comprehensive and the aim here is to complement that by providing a more detailed review of modelling of semisolid processing and the transient rheological experiments required to provide data for that. Millions of components are now made annually by semisolid processing. Aluminium alloy components produced by thixoforming and the NRC process are supplied to the automotive industry. Thixomolding is widely used, particularly in Japan, to produce lightweight magnesium components for mobile phones, laptop computers 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 producing high performance alloys [54–60] and in modelling die ﬁll with its concomitant requirement to obtain the experimental data which can support the modelling. This review focuses on the latter two areas.
2. Background rheology In a Newtonian ﬂuid, the shear stress, s is proportional to the shear rate, c_ , and the constant of proportionality is the viscosity, g. Thixotropic ﬂuids are nonNewtonian i.e. the shear stress is not proportional to the shear rate. The viscosity is then termed the apparent viscosity and is dependent on shear rate, pressure, temperature and time. Some nonlinear ﬂuids also show viscoelasticity i.e. they store some of the mechanical energy as elastic energy. Thixotropic materials do not store energy elastically and show no elastic recovery when the stress is removed. If a ﬂuid exhibits a yield stress and then gives a linear relationship between shear stress and shear rate, it is termed a Bingham material (Fig. 12). Then s ¼ sy þ k c_
ð1Þ
where k is a constant related to the viscosity. The Herschel–Bulkley model is where behaviour is nonlinear after yield i.e.: s ¼ sy þ k c_ n
ð2Þ
There is dispute over whether thixotropic semisolid alloys display yield e.g. [61] and whether they should be modelled as such (e.g. [62]). Barnes et al. [63–65] concluded that the presence of a yield stress as reported by some workers for thixotropic materials (but not semisolid alloys) is probably due to the limitations of their experimental apparatus in not being able to measure shear stresses at very low shear rates. 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 yield stress where the ﬂuid is at rest prior to the application of a shear stress, and a dynamic yield stress where the ﬂuid is being continuously sheared. Their results are
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H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412 Shear Stress (Pa)
HerschelBulkley Bingham Shear Thinning
Newtonian
Shear Thickening
τy
1
Shear Rate (s ) Viscosity (Pas)
HerschelBulkley Shear Thinning Newtonian
Shear Thickening
1
Shear Rate (s )
Fig. 12. Shear stress versus shear rate and associated viscosity versus shear rate curves for a variety of types of rheological behaviour.
Fig. 13. Shearstress ramp experiments after diﬀerent rest times (tr ) for Sn–15%Pb [66]. (a) Shear stress versus deformation angle; (b) yield stress versus rest time. Temperature 195 C, fraction solid 0.5, globular structure prepared by shearing at 100 s1 at a cooling rate of 1 C/min.
shown in Fig. 13. The yield stress increases with rest time prior to deformation because of the increasing degree of agglomeration. In terms of modelling semisolid alloy die ﬁll, the use of a yield stress may be appropriate because a vertical billet does
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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 peak in the load versus displacement curve is detected. Contrary to this though is the fact that at the ‘thixoforming temperature’ the initial peak is so small as to be undetectable. The OstwalddeWaele relationship: s ¼ k c_ n
ð3Þ
is used to describe ﬂuids which do not have a yield point and where there is a power law relationship between the shear stress s and the shear rate c_ . If the exponent n ¼ 1, this reduces to the expression for a Newtonian ﬂuid with the constant k equal to the viscosity g. In Fig. 12, the shear thinning material (whose viscosity decreases as the shear rate increases) would have a value of n of less than 1 and the shear thickening material would have n greater than one. Thixotropic materials are essentially shear thinning but also thicken again when allowed to rest (i.e. all thixotropic materials are shear thinning but not all shear thinning ﬂuids are thixotropic). It is thought that at very high shear rates and at very low shear rates, thixotropic ﬂuids eﬀectively become Newtonian. This is expressed in the Cross model [67]: g g1 g ¼ g1 þ 0 ð4Þ 1 þ k c_ n where as the shear rate c_ ! 0, g ! g0 and as c_ ! 1, g ! g1 . Fig. 14 shows data from a number of studies [67–70] for Sn–15%Pb alloys with various fractions of solid 9
Lax & Flem (Fs=0.5) [68] Fraction solid at 0.5
8
Turng & Wang (Fs=0.5) [69]
Fraction Solid at 0.36
McLelland (Fs=0.5) [70]
7
Liu (Fs=0.5) [71] Cross Model at Fs=0.5 [67]
Log10 viscosity (Pas)
6
Laxmanan (Fs=0.36) [68] 5
Turng & Wang (Fs=0.36) [69]
4 3

McLelland (Fs=0.36) [70]
−
Liu (Fs=0.36) [71] Cross Model (Fs=0.36) [67]
Fraction Solid at 0.3
Lax & Flem (Fs=0.3) [68]
2
Liu (Fs=0.3) [71] Fraction Solid at 0.2
1
Cross Model at Fs=0.3 [67] Lax & Flem (Fs=0.2) [68]
0
Turng & Wang (Fs=0.2) [69] 1
McLelland (Fs=0.2) [70] Liu (Fs=0.2) [71]
2 7
6
5
4
3
2
1
0
1
2
3
4
Cross Model at Fs=0.2 [67]
Log10 Shear rate (s 1 ) Fig. 14. Cross model ﬁtted to apparent viscosities obtained from various works on Sn15%Pb alloys [71].
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Fig. 15. Response of an inelastic thixotropic material to ﬁrstly a stepchange up in shear rate and then a stepchange down (after [64]).
fs . The data obey the Cross model, but information on the extremes is sparse. These data are for steadystate viscosities and, as will be discussed below, it is the transient behaviour which is of importance for the modelling of thixotropic die ﬁll. Viscosity is highly dependent on temperature. For a Newtonian ﬂuid (e.g. the liquid matrix in a semisolid slurry), the viscosity decreases with increase in temperature. Temperature also aﬀects the microstructure. Thus in semisolid slurries, the fraction solid decreases with increase in temperature, with a consequent eﬀect on viscosity (see Fig. 14). In addition, over time, the microstructure will coarsen by diﬀusion and this will be accelerated as the temperature increases. Fig. 14 is for Sn 15%Pb alloy. There is little data on aluminium alloys because there are few commercially 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 equilibrium value 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 with increasing rest time before it recovers back to the equilibrium viscosity of the shear rate speciﬁed.
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 ﬂocculated suspensions
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Fig. 16. Flow curves of a ﬂocculated suspension (after [65]).
(Fig. 16). At the shear rate c_ 1 corresponding to point ‘a’, the microstructure consists of a series of large ﬂocs. If the shear rate is increased from c_ 1 to c_ 2 , the ﬂocs break up until the size corresponds to the ﬂow curve which passes through point ‘b’. If the shear rate is then reduced back to c_ 1 , the individual particles begin to collide and agglomerate until an equilibrium size is reached appropriate to the lower shear rate. In semisolid metallic systems, the agglomeration occurs because particles are colliding (either because the shear brings them into contact or, if at rest, because of sintering) and, if favourably oriented, form a boundary. By ‘favourable orientation’ is meant the fact that if the particles are oriented in such a way that a low energy boundary is formed, it will be more energetically favourable for the agglomeration to occur than if a high energy boundary is formed. If a 3D network builds up throughout 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 are broken 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 morphology. The closer the shape to that of a pure sphere, the lower the steady state viscosity [45]. In addition, if liquid is entrapped within particles, it does not contribute to ﬂow. Thus, although the fraction liquid may take a certain value, governed by the temperature (and indeed kinetics as the thermodynamically predicted fraction liquid 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
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Table 1 Examples of thixotropic materials with the mechanisms of recovery (after [64]) Materials
Structural buildup at rest
Structural breakdown under shear
Collodial dispersions and suspensions of solids (1) Paints (2) Coatings (3) Inks (4) Clay slurries (5) Cosmetics (6) Agricultural chemicals
Flocculation under interparticle forces
Breakup of ﬂocs
Emulsions
Flocculation of droplets
Deﬂocculation
Foamed systems (1) Mousses
Flocculation of bubbles
Deﬂocculation Coalescence
Crystalline systems (1) Waxy crude/fuel oils (2) Waxes (3) Butter/margarine (4) Chocolate
Interlocking of growing crystals
Break up of long needles
Polymeric systems (1) Solutions/melts (2) Starch/gums (3) Sauces
Agglomeration of macromolecules
Deagglomeration
Entanglement
Disentanglement
Fibrous suspensions (1) Tomato ketchup (2) Fruit pulps (3) Fermentation broths (4) Sewage sludges
Agglomeration of ﬁbrous particles
Deagglomeration
Entanglement
Disentanglement
Agglomeration of particles
Deagglomeration
Semisolid metallic systems
(deﬂocculation)
of the forces between the particles. Table 1 summarises the phenomena which are occurring during structural buildup and structural breakdown in a variety of systems. In general, the forces between particles include: Van der Waals attraction; steric repulsion due to adsorbed macromolecules; electrostatic repulsion due to the presence of like charges on the particles and a dielectric medium; electrostatic attraction between unlike charges on diﬀerent parts of the particle (e.g. edge/face attraction between clay particles). In semisolid metallic slurries, none of these forces apply. What must actually be occurring in structural buildup is a process akin to adhesion in wear. As shear occurs, particles are forced into contact with each other. If it is energetically favourable for a solid–solid boundary to be formed, the two particles will stay in contact. If not, they will separate again. The process will be inﬂuenced by the rate of shear in two opposing ways. Increasing the rate of shear will increase the possibility of particle–particle contact but it will decrease the time of contact and the formation of a new solid–solid boundary is a time dependent pro
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cess. When the slurry is at rest, gravity will bring the particles into contact. In addition, if the solid fraction is suﬃciently high, the packing density will be such that particles touch each other. Structural breakdown requires the breakdown of particle–particle bonds and this will depend on the crosssectional area of the bond and the radius of the neck which generates a stress concentrating eﬀect. Can this process be represented by a force–distance curve for the particle–particle interaction as has been assumed in other systems? If it can, the forces are very short range (perhaps < 1 nm) and the potential well is deep because many bonds do form. In many thixotropic systems, the Brownian (thermal) randomising force is signiﬁcant. For particles of all shapes, this constant randomisation inﬂuences the radial distribution function (i.e. the spatial arrangement of particles as seen from the centre of any one particle). The Brownian force is strongly size dependent, so that below a particle size of 1 lm it has a big inﬂuence. In semisolid alloy slurries though, the individual particle size tends to be at least 20 lm and so the Brownian force does not play a strong part. The other force which acts on the particles is the viscous force, which is proportional to the local velocity diﬀerence between the particle and the surrounding ﬂuid.
Fig. 17. Schematic illustration of evolution of structure during solidiﬁcation with vigorous agitation: (a) initial dendritic fragment; (b) dendritic growth; (c) rosette; (d) ripened rosette; (e) spheroid [45].
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Many thixotropic systems show ‘reversibility’ i.e. the slurries have a steady state viscosity 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 measured viscosity is then expected to depend on the shearing and thermal history. These dependencies contribute to the diﬃculty in modelling.
4. Mathematical theories of thixotropy Barnes [65] has summarised current mathematical theories of thixotropy. Some detail on these is given here to enable work on semisolid slurries to be put in the context 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 the microstructure, denoted by a scalar parameter, typically k, and then use dk=dt as the working parameter. (2) Those that attempt some direct description of the temporal change of microstructure as for instance the number of bonds or an attempt at describing the real ﬂoc 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 ¼ 1 and a completely broken down structure by k ¼ 0. In the simplest case of a typical, inelastic, nonNewtonian ﬂuid with upper and lower Newtonian viscosity plateaus (e.g. see Fig. 14), k ¼ 1 corresponds to g0 and k ¼ 0 to g1 . Thixotropy is usually then introduced via the time derivative of the structure parameter, dk=dt. This is the sum of the breakdown and buildup terms and in the simplest theories these are only controlled by the shear rate 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 controlled by the distance the structure is from its maximum value i.e. ð1 kÞ, raised to another power. Then dk ¼ að1 kÞb ck_cd dt
ð5Þ
where a, b, c, and d are constants for any one system. Overall, if the value of dk=dt is negative, the system is breaking down towards equilibrium and if it is positive it is building up towards equilibrium. The Moore model [72] is a simpliﬁed version of Eq. (5) with b and d set to one. Cheng and Evans [73] set b ¼ 1 but allowed d to vary. The next step is to relate the structure k (as calculated using the equations above), to the stress s or viscosity g in some ﬂow equation. This has been done in a variety of ways which range from a simple Bingham equation (see Eq. (1) in Section 2),
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through the Cross model to a Crosslike model (Eq. (4) in Section 2) containing a yield stress. 4.2. Direct structure theories Denny and Brodkey [74] applied reaction kinetics to thixotropy via a simple scheme that examined the distribution of broken and unbroken bonds. The number of these bonds was later linked to viscosity. The rate of structure breakdown was then
dðunbrokenÞ p q ¼ k1 ðunbrokenÞ k2 ðbrokenÞ dt
ð6Þ
where k1 and k2 are the rate constants for the breakdown and buildup respectively. This can be solved to give the viscosity by assuming that viscosity is linearly proportional to the amount of unbroken structure, with a maximum value when completely structured of g0 and a minimum value when completely destructured of g1 . The rate constant k2 is assumed to be independent of shear rate, being merely a description of Brownian collisions leading to restructuring (but note that for semisolid alloy slurries build up is not due to Brownian collisions––Section 3), while k1 is related to the shear rate by a power law expression. The Cross model [67] was derived using such considerations. Assuming that a structured liquid was made up of ﬂocs (agglomerates) of randomly linked chains of particles, Cross obtained a rate equation of the form: dN ¼ k2 P ðk0 þ k1 c_ m ÞN dt
ð7Þ
where N was the average number of links per chain, k2 was a rate constant describing Brownian collision, k0 and k1 were rate constants for the Brownian and shear contributions to breakdown, P was the number of single particles per unit volume and n was a constant less than unity. At equilibrium, dN =dt is zero, so Ne ¼
k2 P
k0 1 þ kk10 c_ n
ð8Þ
Assuming that the viscosity was given by the constant g1 plus a viscous contribution proportional to the number of bonds Ne , g g1 1 ¼ g0 g1 1 þ kk1 c_ n 0
ð9Þ
which is equivalent to the expression given earlier in Eq. (4) but with k ¼ k1 =k0 . Lapasin et al. [75] used a fractal approach to describe ﬂocculated suspensions. In the relationship they predicted, the viscosity is related to: the number of primary particles in a ﬂoc when the shear stress becomes inﬁnite, a yield stress and the fractal dimension of the ﬂoc.
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4.3. Simple viscosity theories Mewis and Schryvers [76] have devised a theory that circumvents the use of any structural 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 diﬀerence between the steady state ge and the current values of viscosity (not the structure diﬀerence) i.e.: dg n ¼ K½gs ð_cÞ g dt
ð10Þ
Thixotropic breakdown has also been described [77,78] ðg g1 Þ
1m
¼ ½ð1 mÞkt þ 1ðg0 g1 Þ
1m
ð11Þ
where g0 and g1 are the asymptotic values of viscosity g (representing the fully structured and fully destructured states, respectively) measured at time t for any particular shear rate, and k and m are material constants.
5. Transient behaviour of semisolid alloys Computational ﬂuid dynamics (CFD) can be used to predict die ﬁlling (see Section 6). However, some of the work reported has been based on rheological data obtained in steady state experiments, where the semisolid material has been maintained at a particular shear rate for some time. In reality, in thixoforming the slurry undergoes a sudden increase in shear rate from rest to 100 s1 or more as it enters the die. This change takes place in less than a second. Hence, measuring the transient rheological response under rapid changes in shear rate is critical to the development of modelling of die ﬁlling and successful die design for industrial processing. It can be investigated with two types of experiment. Firstly, via rapid shear rate changes in a rheometer and secondly, for higher fractions solid (where the torque capability of a rheometer is not suﬃcient), with rapid compression experiments, for example, in the thixoformer 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 were quantiﬁed: (1) breakdown time and (2) buildup time. The breakdown time is the characteristic time for the slurry to achieve its steadystate condition after a shear rate 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 the times for breakdown are faster than those for buildup. This would be expected, as the breaking up of ‘bonds’ between spheroidal solid particles in agglomerates is likely
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to be easier than the formation of bonds during shearrate drops. Quaak [47] proposes two characteristic times to describe a shear rate jump. He suggests that during a shearrate change, the slurry undergoes an initial rapid breakdown/buildup followed by a more gradual process dependent on diﬀusion. This can be described by a double exponential expression. Quaak gives Fig. 18 as the microstructural basis. Immediately after a change in shear rate, the structure remains the same (‘isostructure’). This is followed by a very fast process and then a slow process, associated with diﬀusion, giving coarsening and spheroidisation. It is the ‘very fast process’ which is relevant to modelling die ﬁll. In a rheometer, great care must be taken to ensure that inertial eﬀects do not interfere with the results e.g. see. [86]. In addition, instrumental eﬀects must be carefully separated from those of the material itself, particularly when attempting to examine behaviour that occurs in less than a second. For example, electronic switching may occur during the shear rate jump. This can be allowed for by only analysing results after the shear rate has reached 90% of the speciﬁed ﬁnal shear rate (see [86]). The work of Liu et al. [86] involves the fastest data collection rate so far ( 1 kHz capture rate). This is signiﬁcantly faster than that used by other workers (200 Hz in [79], 9 Hz in [47], 200 Hz in [80]) and enables the capture of the very fast process. The results for shear rate jumps from 0 to 100 s1 after diﬀerent rest times are shown in Fig. 19. With longer rest times, the peak stress recorded increases. The breakdown times in Table 2 were obtained by ﬁtting an exponential to the data obtained during the second after 90% of the ﬁnal shear rate was achieved. In Table 2, gp is the peakstress viscosity, gss is the ‘ﬁrst’ steadystate viscosity (given that there are at least two processes going on as mentioned earlier) and sb is the ‘ﬁrst’ breakdown time. Table 2 shows that the longer the rest time prior to the shear rate jump, the lower the breakdown time. This is consistent with microstructural evidence (Fig. 20) showing that increasing the rest time increases the solidparticle sizes and the degree of agglomeration. This increase would impede the movement of the particles upon the imposition of the shear stress. The ease with which particles are able to
Fig. 18. Schematic model describing the fast and slow processes in a semisolid material’s structure after shear rate up and down jumps (taken from Ref. [47]).
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412 2500
120 100
Shear stress (Pa)
2000 Shear rate
80
Shear stress (after 5 hrs rest)
1500
60 Shear stress (after 2 hrs rest) Shear stress (after 1 hr rest) Shear stress (after 0 hr rest)
1000
500
0 0
0.2
0.4
0.6 0.8 Time (secs)
1
1.2
40
Shear rate (1/s)
366
20 0 1.4
Fig. 19. Shear rate jumps from 0 to 100 s1 after diﬀerent rest times for Sn–15%Pb alloy at fraction solid 0.36 [86].
Table 2 Tabulation of parameters obtained from shear rate jump experiments on Sn15%Pb alloys (at Fs ¼ 0:36) under diﬀerent rest times [86] Rest times (h) 0 Shear rate jumps (0–100 s1 ) gp (Pa s)a 2.1 gss (Pa s)b 0.8 0.18 sb (s)c
1
2
5
5.4 0.8 0.16
8.0 1.2 0.15
23.0 2.0 0.12
a
The errors are within 95% conﬁdence limits (±0.5). The errors are within 95% conﬁdence limits (±0.2). c The errors are within 95% conﬁdence limits (±0.03). b
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 in shear 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 with increasing ﬁnal shear rate [47,81–83,85,86]. 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 stress during a rapid increase in shear rate. They found that this overshoot (or undershoot in 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 and Wang [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 corresponds to that of the previous shear rate. Kumar et al. [88] and Koke and Modigell [66] however, ﬁnd shear thickening ‘isostructural’ ﬂow behaviour (e.g. Fig. 21). In [66],
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Fig. 20. Microstructures of Sn–15%Pb alloy ðfs ¼ 0:36Þ at various rest times: (a) 0 h (b) 1 h and (c) 2 h [86].
after each shear rate jump, the substance is sheared at c_ 0 to obtain equilibrium before the next jump. The plot of shear stress versus shear rate can be ﬁtted with a shear thickening Herschel–Bulkley model with a ﬂowexponent n ¼ 2:07. Koke and Modigell [66] argue that this ﬁnding is of high importance for simulation of the industrial process. Data on the transient behaviour of aluminium alloys is sparse because the majority of the commercially available rheometers do not operate at semisolid aluminium alloy temperatures.
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Fig. 21. (a) Series of step changes in shear rate to obtain; (b) isostructural ﬂow curve [66]. The isostructural shear stress Siso is measured immediately after each step up in shear rate. In between each experiment, the material was sheared at the equilibrium shear rate c_ 0 of 100 s1 (Sn–15%Pb, 198 C, fs ¼ 0:41). The exponent for the ﬁt to (b) is 2.07 and hence the isostructural behaviour is termed shear thickening.
5.2. Rapid compression For high solid fractions, above about 0.5, conventional rheometers do not have suﬃcient torque capability. Other methods must then be used, introducing complexity because the shear rate is no longer constant throughout the material (as it can be assumed to be in a concentric cylinder rheometer). Laxmanan and Flemings [68] measured the force and displacement for Sn 15%Pb compressed between parallel plates 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 slower than in the industrial process. The work of Loue et al. [89], carried out at higher shear rates by backward extrusion on aluminium alloys, resembles industrial thixoforming more closely. However, the specimens were heated to temperature over a long period of time ( 10 min) and then held isothermally for 30 min before compression. Such time periods would be considered long in industrial thixoforming. Yurko and Flemings [90] designed a drop forge viscometer (Fig. 22) to study ﬂuid ﬂow behaviour at transient high shear rates. It consists of a lower platen and an upper platen, with an attached platen rod to track platen motion with time. It is similar to a parallel plate compression viscometer but the upper plate is allowed to fall under the inﬂuence of gravity. A high speed digital camera images the rod as it falls. The force is calculated from the second derivative of the displacement data allowing calculation of viscosities at shear rates in excess of 1000 s1 . A typical experiment yields instantaneous, volumeaveraged viscosity ﬁrst under rapidly increasing 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 versus time signals. The compression rate is then akin to industrial thixoforming and the load is measured directly. A typical signal response is shown in Fig. 23. The peak is
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Fig. 22. Schematic of the drop forge viscometer [90].
4 3.5
Peak stress can be calculated from the peak load and the flow stress from the minimum load.
Load (kN)
3 2.5 2 1.5 Peak Load ( L )
1
Initial Contact
Minimum Load (L )
0.5 0 80
70
60
50
40
30
20
10
0
Displacement (mm)
Fig. 23. Typical signal response to rapid compression of a semisolid alloy slug [91].
believed to originate from the threedimensional skeletal structure built up in the solid phase at rest, which breaks down under load. The width of the peak (or, more accurately, the downward part of it) is a measure of the time taken to destroy this skeletal structure. A rough estimate then gives a breakdown time of about 10 ms, an
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Fig. 24. Load signals and microstructures at diﬀerent temperatures for Alusuisse A356 aluminium alloy in rapid compression tests (ram speed 500 mm/s, zero soak time) [91].
order of magnitude less than the relaxation times obtained from shear rate jumps in rheometer experiments (see Section 5.1) and must therefore be related to a diﬀerent mechanism. The height of the peak falls with temperature as the skeletal structure is consumed, and the minimum load beyond the peak also decreases with increasing temperature, both because a more spheroidal microstructure is developed and the fraction liquid increases (see Fig. 24). In practice, successful thixoforming takes place at temperatures where there is little or no peak. Viscosity versus shear rate can be
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].)
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derived from the loaddisplacement data using a method based on that outlined in Laxmanan and Flemings [68]. This does however assume a Newtonian ﬂuid at one stage in the analysis and this may introduce errors. Data on viscosity versus shear rate for Al–Si alloys is summarised in Fig. 25. It is important to be aware that small changes in silicon content can aﬀect the results quite considerably by changing the solid fraction. The lower values recorded by Yurko and Flemings [90] in comparison with those of Liu et al. [91] are derived for an 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 ﬁgure is the steadystate viscosity determined by Quaak [47] for a 7% Si aluminium alloy, extrapolated to 0.5 fraction solid; this is well below the other results, emphasizing that the steady state is not achieved in those experiments, nor in industrial thixoforming.
6. Modelling The recent commercialisation has highlighted the need to model slurry ﬂow into die cavities. Die design and processing conditions such as ram speed, dwell time and pressure have, to some extent, been a matter of trial and error. In particular, die design rules from die casting are not transferable to thixoforming. This is illustrated in 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 ﬁlling [93]. (a) Partial ﬁlling of die; (b) modelling simulation of (a) where white corresponds to dark on (a); (c) modelling simulation with improved die design showing smoother ﬁlling.
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was diﬃculty in ﬁlling the die. Therefore, partial ﬁlling experiments were carried out which demonstrated that the design of the die, particularly the ingate, which was narrow as for diecasting, led to some jet ﬂow across the cavity (Fig. 26(a)) instead of the smooth progressive ﬁlling which is the aim in thixoforming. FLOW3D (a computational ﬂuid dynamics programme produced by FLOWSCIENCE, Los Alamos) was used to model the ﬂow into the die, trying out diﬀerent viscosities in order to ﬁnd the range in which the experimental behaviour was mimicked (e.g. Fig. 26(b)). The agreement was promising given that the work did not take into 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 ﬂuid had a constant viscosity, independent of the shear rate and time, which in practice is not the case). Changing the design of the die in the light of the ﬁndings of this ‘simple’ modelling led to improved ﬁlling (e.g. Fig. 26(c)). There is, therefore, real potential commercial beneﬁt to be obtained from better understanding of ﬂow of semisolid material in dies, alongside the academic interest. In this section modelling is reviewed. Previous reviews include those by Kirkwood [92], Atkinson [93] and Alexandrou [28, Chapter 5]. Table 3 summarises the main papers on modelling [94–125]. The papers are given in year order and this is carried over into Tables 4–6 which give further information. These tables are not exhaustive and, where authors have published in journals in addition to conferences, it is the journal papers which are cited. The papers are classiﬁed as to whether the modelling is ﬁnite diﬀerence or ﬁnite element, onephase or twophase. There is in addition, a paper on micromodelling [125] which does not strictly ﬁt any of these categories. A major aim here is to draw out the similarities and the diﬀerences between the ﬂow and viscosity equations which modellers are using. In Tables 4–6, this is done by quoting the equations from the papers but converting the symbols as far as possible to be common. The list of symbols is given in the nomenclature. Where the equations are given in complex terms they are then reduced to simple shear which allows more direct 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=ot or d=dt, that this is in fact the substantive derivative D=Dt following the material as it moves. Tables 4–6 identify the main features of the models and also observations on simulation results and whether these have been validated. Where commercial code has been used this is identiﬁed with the reference. The main threads in the development of 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, the ﬁnite 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); twophase modelling (Section 6.2.5). For the onephase ﬁnite element papers (Section 6.3.1) in some cases it makes sense to group the papers according to author. Thus, the headings are: Zavaliangos and Lawley; Backer; Alexandrou, Burgos and coworkers; viscoplastic constitutive models; power law cutoﬀ (PLCO) model of Procast; model based on viscoelasticity and thixotropy. The two phase ﬁnite element papers are sensibly dealt with as a single section (Section 6.3.2)
Table 3 Classiﬁcation of models of semisolid die ﬁlling FEM
Onephase
Ilegbusi and Brown (PHOENICS) [94] Barkhudarov et al. (FLOW3D) [95] Barkhudarov and Hirt (FLOW3D) [96] Modigell and Koke (FLOW3D) [84] Kim and Kang (MAGMAsoft) [97] Modigell and Koke (FLOW3D) [85] Ward et al. (FLOW3D) [98] Messmer (FLOW3D) [99] Seo and Kang (MAGMAsoft) [100] Itamura et al. (Adstefan) [101]
Zavaliangos and Lawley (ABAQUS) [103] Backer (WRAFTS) [104] Alexandrou et al. (PAMCASTSIMULOR) [105] Burgos and Alexandrou [106] Alexandrou et al. (PAMCASTSIMULOR) [107] Burgos et al. [108] Alexandrou et al. [109] Ding et al. (DEFORM3D) [110] Jahajeeah et al. (Procast) [111] Rassili et al. (FORGE3) [112] Wahlen (Thixoform) [113] Alexandrou et al. [114] Orgeas et al. (Procast) [115]
TwoPhase
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]
Micromodelling
Rouﬀ et al. [125]
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Finite diﬀerence
Commercial codes employed in work are indicated in brackets. If a code is not given then either the authors have written the code themselves or it is not identiﬁed in the text of the paper.
373
374
Comments Onephase Ilegbusi and Brown (PHOENICS) [94]
Barkhudarov et al. (FLOW3D) [95]
Slurry incompressible, mass conservation, momentum conservation, energy conservation (enthalpy method), solid fraction from [126], input parameters identiﬁed, scalarequation method for free surface, single internal variable ðkÞ constitutive model [127–129], experimentally determined values of agglomeration function H and disagglomeration function G given. Chiselshaped mould Transport equation for g includes advection term and relaxation term which accounts for thixotropy. No yield stress, wall slip, or elastic or plastic behaviour at high fs . Input parameters given
Flow and viscosity equations
s ¼ sy þ AðkÞ
ðc=cmax Þ1=3 1 ðc=cmax Þ1=3
Observations
gf c_
þ ðn þ 1ÞCðT Þkfs gnþ1 c_ n f dk ¼ H ðT ; fs Þð1 kÞ dt GðT ; fs Þk_cn
ð12Þ
ð13Þ
og þ u:rg ¼ xðge gÞ ot
ð14Þ
ok þ u:rk ¼ b1 ð1 kÞ þ b2 k_c ot
ð15Þ
Note: We have changed ðurÞk to u:rk for clarity of notation. g ¼ g1 þ ck b1 c ) g ¼ g1 þ b1 þ b2 c_ x ¼ b1 þ b2 c_
Boundary layer at wall (but not clear in velocity vectors diagram). Low temperature region at wall (but using nonheated die) ﬁ solid shell. Jetting at central region
ð16Þ
and ð17Þ
Match to experimental shear stress hysteresis curves (Sn–15%Pb) [130] with reasonable accuracy. Sensitive to exact values of relaxation time. Die swell in thixoextrusion
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Table 4 Summary of onephase and twophase ﬁnite diﬀerence simulation papers
Barkhudarov and Hirt (FLOW3D) [96]
Shear stress is function of yield stress and structural parameter j (which diﬀers from k in that it varies between 0 and 1 rather than 0 and 1). Shear stress is assumed to grow exponentially with increasing solid fraction. Input parameters given
og þ ðurÞg ¼ a minðge g; 0Þ ot þ b maxðge g; 0Þ
ð18Þ
Small droplets of Sn–Pb impacting on ﬂat plate. Droplet shapes inﬂuenced by relaxation times
i.e. if ge g < 0 then the righthand side ¼ aðge gÞ and if ge g > 0 then the right hand side ¼ bðge gÞ
s ¼ sy ðfs Þ þ expðBfs Þk j_cm
ð19Þ
oj ¼ a expðb_cÞðje jÞ ot
ð20Þ
Comparison between Newtonian and thixotropic for ﬂow in a cavity with a round obstacle
In equilibrium: 1 je ¼ ða_cÞmn The equilibrium ﬂow curve is then: s ¼ sy ðfs Þ þ expðBfs Þk c_ n with
ð21Þ
k ¼ k anm Kim and Kang (MAGMAsoft) [97]
Comparison of Newtonian and Ostwald–deWaele with n of )0.48 to +0.45 (depending on T ) under shear rate of 3–2500 s1 . Input parameters given. Predict defects in product from temperature distribution
Ostwald–deWaele for viscosity dependence on shear rate
Good agreement between partial ﬁlling experiment and predicted temperature distribution at 80% ﬁlling
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Modigell and Koke FLOW3D [84]
Transport equation for g with a rate constant for thinning and b rate constant for thickening. g P ge then material is trying to relax towards the lower equilibrium value ge (i.e. thinning). Thixotropic data from [69]. Heat transfer, viscous heating and solidiﬁcation eﬀects included. Heat transfer negligible in time period considered
375
376
Comments
Flow and viscosity equations
Modigell and Koke (FLOW3D) [85]
Die ﬁlling of steering axle assumed isothermal with wall adhesion
s ¼ ðsy ðfs Þ þ k ðfs Þ_cmðfs Þ Þj
Observations
Ward, Atkinson, Kirkwood and Chin (FLOW3D) [98]
As for Barkhudarov and Hirt [95]
As for Barkhudarov and Hirt [95]
Messmer (FLOW3D) [99]
Thixoforging using approach of Barkhudarov et al. [95]
ge ¼ A expðBfs Þ_cm
ð23Þ
dg ¼ bðge gÞ dt
ð24Þ
ð22Þ
Models step change of shear rate experiments quite well. Die ﬁlling of steering axle. Above critical inlet velocity ﬁlling no longer laminar Modelling of shear rate jumps for Sn– 15%Pb. All variable values to ﬁt shear rate jumps consistent with Cross equation and with rate data, except the initial viscosity, which was 2–5 times lower than experimental values. This suggests the initial breakdown of the slurry is very rapid, possibly beyond the detection limits of the data collection system. Modelling of rapid compression in a thixoformer suggests aluminium slurries undergo an initial very rapid breakdown and that the subsequent breakdown rate is not strongly shear rate dependent Forming force measured at end of stroke corresponds well with simulated force. Early part of stroke not well simulated. Attributed to use of only one thinning rate
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Table 4 (continued)
Seo and Kang (MAGMAsoft) [100]
Twophase Ilegbusi et al. [102]
Ostwald–deWaele compared with Carreau–Yasuda: n1 g g1 ¼ ½1 þ ð_ckÞa a g0 g1
Compares simulation for diecasting, squeezecasting and rheocasting for both metal ﬂow and solidiﬁcation
Single phase equations solved for whole ﬁlling phase. Trajectories of given number of particles computed, assuming they ‘disappear’ when they hit a wall or are trapped in recirculation zone. Measure of segregation obtained by comparing number of particles at given distance from inlet to total number of injected particles
No details given
No ﬁlling results for Carreau–Yasuda presented. Ostwald–deWaele gives reasonable agreement with partial ﬁlling tests
Diecasting gives air entrapment, cf. squeeze casting and rheocasting. Less shrinkage defects in rheocasting
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Itamura et al. (Adstefan) [101]
Simple upsetting experiments to obtain rheological data with A356. Input parameters given
377
Comments Medium volume fraction solid fs 6 0:6–0:7
Single internal variable ðkÞ constitutive model [127–129]. Same equations as for Ilegbusi and Brown [94] but without yield stress. Isothermal
Flow and viscosity equations
Observations
1=3
s ¼ AðkÞ
ðc=cmax Þ
1=3
1 ðc=cmax Þ
Sn–15%Pb. Free standing billet collapse for fs 6 0:5. Thixoforming of a simple shape. No validation available
gf c_
þ ðn þ 1ÞCðT Þkfs gnþ1 c_ n f ð25Þ dk ¼ H ðT ; fs Þð1 kÞ GðT ; fs Þk_cn dt ð26Þ
Backer (WRAFTS) [104]
(1) Herschel–Bulkley
ð1Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s ¼ sy mj DII =2jn1 Du ð27Þ
Note that this equation is given here exactly as in the Backer paper but it is not clear whether sy is being treated as a tensor. In its present form the equation is dimensionally incorrect For simple shear, the Backer equation reduces to ð28Þ s ¼ sy þ k c_ n (2) Bingham combined with power law dependence. Single internal variable k [127–129]. . . apparently the same equations as for Ilegbusi and Brown [94] but without a yield stress. Value of n ¼ 4
ð2Þ
s ¼ ½A
ðc=cmax Þ1=3
u 1 ðc=cmax Þ1=3 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n1 Ckfs j DII =2j Du ð29Þ
which, for simple shear, reduces to s¼A
ðc=cmax Þ1=3 1 ðc=cmax Þ1=3
u_c þ Ckfs c_ n ð30Þ
Comparison of Newtonian, Herschel– Bulkley and internal variable results for complex die. Latter tends to ﬁll from side runner rather than bottom because material has ﬂowed further in the runner and disagglomerated in the process. No validation with experiment
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Zavaliangos and Lawley (ABAQUS) [103]
378
Table 5 Onephase ﬁnite element simulation papers
which, if q is treated as a constant ou
and ouoxx ; oyy ; ouozz ¼ 0, reduces to ok qðu:rkÞ q ot
Alexandrou et al. (PAMCASTSIMULOR) [105]
Herschel–Bulkley fs3 sy ¼ 9615 0:6f s Fluid assumed incompressible. n ¼ 1 then Bingham T dependence introduced through making sy function of T
¼ Kð1 kÞ Gk2 c_ ð32Þ (which is very similar to Barkhudarov et al. [95] but with a k2 in the second term on the righthand side rather than k) ( pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ) sy ð1 expðm DII =2ÞÞ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sij ¼ Dij DII =2 ð33Þ ðn1Þ=2
ð34Þ g ¼ KðDII =2Þ For simple shear the equations reduce to ð35Þ s ¼ sy ð1 expðm_cÞÞ g ¼ K c_ n1
ð36Þ
Bingham set constant and independent of processing conditions. . .fairly good agreement between modelling and ﬁlling for complex part. Local ﬂow not predicted as well as bulk ﬁlling. Comparison of Newtonian and Bingham ﬁlling of simple 2D cavity. Comparison of Newtonian and Bingham ﬁlling of 3D cavity with core. Results show eﬃcacy of ‘overﬂows’ on dies beyond ‘rewelding’ areas
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
(In comparison with Ilegbusi and Brown [94], the liquid viscosity gf appears to have been included in the function A, a velocity u is present in the ﬁrst term and ðn þ 1Þgnþ1 is included in the funcf tion C in the second term) The transport equation for k is dqk ¼ r:quk þ Kð1 kÞ dt pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Gk2 DII =2 ð31Þ
379
380
Table 5 (continued) Comments Herschel–Bulkley (as for [105])
Alexandrou et al. (PAMCASTSIMULOR) [107]
Bingham ﬂuid. Continuous model due to Papanastasiou [131] to avoid discontinuity at yield surface
c_ ¼ 0
Observations
s 6 sy
sy c_ s > sy gþ ð37Þ c_ " # 1 expðmj_cjÞ c_ ½131 s ¼ g þ sy j_cj
s¼
Predicts time evolution of yielded/unyielded regions for sudden 3D square expansion Five diﬀerent ﬂow patterns ‘mound’, ‘disk’, ‘shell’, ‘bubble’ and ‘transition ﬂow’ agreeing with observations by Paradies and Rappaz [132]. Map of ﬂow patterns as a function of Reynolds and Bingham numbers
ð38Þ Simplifying: s ¼ g_c þ sy ð1 expðm_cÞÞ Burgos et al. [108]
Herschel–Bulkley expanded to include eﬀect of evolution of microstructure. sy , K, n are assumed functions of fs and k. Single internal variable k. Assume transient behaviour at constant structure is shear thickening. Material parameters from [133] for Sn–15%Pb with fs ¼ 0:45
ð39Þ
1=2 Flow in simple straight channel. Power ok DII þ u:rk ¼ að1 kÞ bk law index decreases with k, but consisot 2 1=2 ! tency index and sy decrease. BreakDII exp c ð40Þ down is less in the core and in the 2
which, for simple shear and where k is not changing spatially, reduces to ok ¼ að1 kÞ bk_c expðc_cÞ ot ¼ ½a þ b_c expðc_cÞðk ke Þ ð41Þ (which for c ¼ 0 is equivalent to the Moore equation [72]) ðnðfs ;kÞ1Þ=2 DII s ¼ Kðfs ; kÞ 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sy ðfs ; kÞ½1 expðmj DII =2jÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D þ DII =2
ð42Þ
corners of the square channel than in the higher shear regions
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Burgos and Alexandrou [106]
Flow and viscosity equations
which, in simple shear, reduces to s ¼ Kðfs ; kÞ_cnðfs ;kÞ þ sy ðfs ; kÞ ð1 expðm_cÞÞ
Ding et al. (DEFORM3D) [110]
Jahajeeah et al. (Procast) [111]
Rassili et al. (FORGE3) [112]
Bingham but using [131] to avoid singularity. Simple compression test. No account taken of evolution of sample’s internal structure
(Note that Eq. (8) in [108] is not correct) " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ # 1 expðm DII =2Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s ¼ g þ sy c_ DII =2 ð44Þ
Rigid viscoplastic constitutive model. Flow stress for AlSi7Mg obtained from compression on Gleeble machine, ignoring initial transient. Levy–Mises ﬂow rule
r ¼ expða bT Þem e_ n
Power law cutoﬀ (PLCO) model of Procast [134] i.e. isotropic, purely viscoplastic, independent of pressure, deformation homogeneous
gð_c; T Þ ¼ g0 ðT Þ_c0
for c_ 6 c_ 0
gð_c; T Þ ¼ g0 ðT Þ_cnðT Þ
for c_ > c_ 0
Viscoplastic constitutive model from force recordings of extrusion tests. Friction assumed very low. No time dependence
Kðe; T Þ ¼ K0 expðb=T Þen
ð45Þ
When written in shear stress terms this is equivalent to s ¼ ðexpða bT ÞÞcm c_ n ð46Þ
nðT Þ
ð47Þ
ð48Þ
K is equivalent to the current yield stress, K0 to a yield stress and e to an eﬀective strain. In shear terms, this is analogous to s ¼ sy expðb=T Þcn
ð49Þ
A356 simple compression. Shape during compression reproduced in simulation using g and sy from ﬁtting load versus time curve. Unyielded material at top and bottom in stagnant layers Die with six rectangular oriﬁces heated to 580–586 C (i.e. isothermal). Good agreement with interrupted ﬂow tests. Metal in biggest oriﬁce ﬂows fastest. Some discrepancy between prediction of loadstroke curve and actual. No examination of liquid segregation in the samples Brake calliper divided into diﬀerent regions each with diﬀerent cutoﬀ values c_ 0 . Reasonable agreement with interrupted ﬁlling tests. Defect prediction with less than optimum runner design
381
Several combinations for tool displacement. Ejector goes up, punch starts to go down when ejector stops. . .buckling leading to lap. Punch and ejector move simultaneously or punch goes down ﬁrst then ejector goes up both avoid buckling but lap is formed on each ‘ear’ of part. Estimation of forging force
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Alexandrou et al. [109]
ð43Þ
382
Table 5 (continued) Comments Model based on viscoelasticity and thixotropy
Flow and viscosity equations 3 e fs k Rb r ¼ r0 1 exp e_ sM 2 R " #m Q e_ exp ð50Þ RT e_ 0 This could be written in shear stress terms as:
3 c fs k Rb s ¼ sy 1 exp c_ sM 2 R " #m Q c_ exp ð51Þ RT c_ 0
Observations Good agreement between model and curve of ﬂow stress versus true strain. Cylindrical specimens produced in backward extrusion. Results allow prediction of temperature of transition from plastic deformation of interconnected particles to viscous ﬂow of a suspension of solid particles. Discrepancies between prediction and experiment though
Note: R means both the particle size in the Rb =R term and the gas constant in the exponential term Alexandrou et al. [114]
2D jets of Bingham and Herschel– Bulkley ﬂuids impacting on vertical surface at distance from die exit in order to account for ﬂow instabilities (e.g. ‘toothpaste’ eﬀect) in semisolid processing
Papanastasiou model [131]: 1 expðm_cÞ s ¼ g þ sy c_ c_
ð52Þ
Generalized to Herschel–Bulkley ﬂuid by specifying g ¼ j_cn1
ð53Þ
‘Bubble’ pattern gives unstable jet, ‘shell’, ‘disk’ and ‘mound’ stable along with most ‘transition’ cases. Instabilities are result of ﬁnite yield stress and the way yielded and unyielded regions interact. Plots of Bingham number versus Reynolds number identify stable and unstable regions
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Wahlen (Thixoform) [113]
Orgeas et al. (Procast) [115]
PLCO model as for Jahajeeah et al. [111] (see above) but with only one value of cutoﬀ c_ 0 (determined by geometry). n and g0 dependent on fs
g ¼ g0
c_ 0 c_ c
g ¼ g0
c_ c_ c
!n1 for c_ < c_ 0 ð54Þ
!n1 for c_ P c_ 0
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Bifurcation of Poiseuilletype ﬂow near a shaft inserted in a tube. Without shaft, experimentally, thin segregated layer of liquid at wall led to ‘plug ﬂow’. Reliable results for ﬂow around the shaft but some instabilities/discrepancies when ﬂow ﬁrst encounters shaft. Filling of reservoir (giving ‘disk + shell’) wellsimulated
383
384
Table 6 Twophase ﬁnite element simulation papers and micromodelling Flow and viscosity equations
Observations
High volume fraction solid fs P 0:7. Porous viscoplasticity model; ﬂuid ﬂow in porous medium; continuity equations. Isothermal. Behaviour symmetric under tension and compression
See paper for details
Compression of semisolid billet indicating liquid segregation. At higher strain rates less liquid is lost. No experimental validation
Zavaliangos [116]
Deformable porous medium saturated with liquid. Stress partitioned into stress carried by solid phase and purely hydrostatic component for pressure in liquid phase. Solid phase has two limits: fully cohesive porous solid and cohesionless granular material. Degree of cohesion represented by internal variable which does evolve with deformation (cf. single internal variable in [127–129]). Permeability equation implies that solidliquid segregation decreases as the grain size decreases. Behaviour not symmetric under tension and compression
See paper
Converging (conical) channel. High strain rates result in nearundrained conditions and minimal phase segregation
Koke et al. [117]
Liquid phase assumed Newtonian. Solid phase is pseudoﬂuid with Herschel– Bulkley viscosity. Darcy Law, Carmen– Kozeny capillary approach. fs P 0:5
gs ¼
sy þ k c_ m1 k þ gf c_
ð55Þ
At equilibrium, k ¼ k, the coeﬃcient in the Herschel–Bulkley power law term (see Backer [104]), and m ¼ n. sy and k are assumed to increase exponentially with fs . This equation gives a diﬀerent expression for shear stress from that due to Brown and coworkers [127– 129]. Note: It is not clear where k has gone to in Eq. (16)
Vertical compression of cylindrical billet. Phase segregation. Qualitative agreement with experiment [135]
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Comments Zavaliangos and Lawley (ABAQUS) [103]
Kang and Jung [118]
¼ K expðSÞ_em exp r
Q RT
½1 bfl 2=3
ð56Þ
for e < ecr ; e > ecr1
r ¼ K expð1 SÞ_em exp
Q RT
The higher the strain rate the more homogeneous the distribution of the solid fraction. In compression forming, macroscopic phase segregation occurred with densiﬁcation of the remaining solid in the central region
½1 bfl 2=3 ð57Þ
for ecr < e < ecr1 These equations could be written in shear with s, e with c, e_ stress terms by replacing r with c_ etc. Binet and Pineau [119]
Mixture approach. Hydrodynamic part same as for most incompressible CFD codes but velocity ﬁeld represents velocities of the mixture and a source term is added to the momentum equations to take account of the diﬀusion velocities of the individual phases. Relative velocities calculated from interaction force between phases. Darcy’s Law, CarmanKozeny relation. Rheological data from [136]
See paper
Predictions of segregation at corners of entrance and outlet of diverging channel in a simple die
Choi et al. [120]
Compressible viscoplastic solid, liquid phase following Darcy’s law. Kuhn’s yield criterion [137] for deformation of solid phase. Friction equation at die/ material surface from [138]
See paper
Head of a trench mortar shell in which forward and backward extrusion are taking place simultaneously. Higher die temperature (400 C) gives better product. Qualitative agreement with experiment for segregation of liquid
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Compressible viscoplastic model for the solid phase and Darcy’s law for the ﬂow of liquid through a porous medium. Separation coeﬃcient introduced S ¼ S0 þ ð1 S0 Þ eecr , where S0 is the ratio of the actual separation to the initial separation, e is the equivalent strain and ecr a critical strain
385
386
Table 6 (continued) Flow and viscosity equations
Observations
Kang and Jung [121]
As for Kang and Jung [118]
As for Kang and Jung [118]
Prediction of overﬂow positions in scroll component. Liquid segregation in the narrow crosssections. Higher strain rates gave less segregation
Yoon et al. (CAMPform2D) [122]
Von Mises yield criterion. Semisolid treated as single phase with incompressibility. Flow stress as function of strain (with fs and ‘breakage ratio’). Mixture theory and D’Arcy’s Law to update fs . Material properties for Sn–15%Pb from [139]. Input parameters for nonisothermal Al2024 alloy given
Equations are summarised in Fig. 1 in the paper
Isothermal predictions of liquid segregation in good agreement with experiment. Nonisothermal simulation for ball joint gives qualitatively useful information
!n
f ¼ K r
e ecr
f ¼ K r
e est exp b ecr est
ðexpðbÞÞe_m
for e < ecr
!! e_m
ð58Þ
for e P ecr ð59Þ
These equations could be written in shear stress terms in an analogous way to Kang and Jung above Kopp and Horst (ABAQUS) [123]
Drucker–Prager yield criterion (yield strength diﬀerent in tensile and in compressive strain). Finite element mesh attached to solid phase
Modigell et al. [124]
Equilibrium ﬂow behaviour modelled with Herschel–Bulkley. Thixotropy modelled with structural parameter following ﬁrst order diﬀerential equation. Pseudoﬂuid approach for the solid phase. All nonNewtonian properties shifted to the solid. Liquid Newtonian. Continuity and momentum equations solved for each phase. Interaction between phases modelled with Darcy law
Simulation and experiment for Sn– 15%Pb match well for fs > 0:55. Maps of laminar, transient and full turbulent ﬁlling produced
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Comments
Micromodelling Rouﬀ et al. [125]
Good agreement with experimental data on viscosity versus shear rate for fAs > f c
ð60Þ
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
Volume solid fraction of the fAs ¼ 0 for fs 6 f c ‘active zone’, fAs , is the internal variable. The ‘active zone’ consists of the fs s solid bonds between spheroids and the fA ¼ f þ Dð1 f Þ_cn s s liquid between spheroids which is not internally entrapped. During deformation the bonds are broken and liquid is released. Spheroids and ‘active zone’ treated as isotropic and incompressible
387
388
H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412
but highlighting the distinctive features of the papers. The work of Rouﬀ et al. [125] on micromodelling is covered in Section 6.3.3. 6.1. Model of Brown and coworkers Brown and coworkers [127–129] presented a constitutive model based on the ‘single internal variable’ concept (see Section 4.1), where the structural parameter k varies between 0 and 1, depending on whether the structure is fully broken down or fully builtup respectively. Their model assumes that ﬂow resistance is due to hydrodynamic ﬂow of agglomerates and deformation of solid particles within the agglomerates. It has been used by a number of workers both for ﬁnite diﬀerence and for ﬁnite element modelling (Ilegbusi and Brown [94]], Ilegbusi et al. [102], Zavaliangos and Lawley [103], Backer [104]). Ilegbusi and Brown [94] used it in their ﬁnite diﬀerence work but also introduced a yield stress sy . The second term on the right hand side in Eq. (12) (at the top of Table 4) represents the hydrodynamic interaction among agglomerates, with AðkÞ a hydrodynamic coeﬃcient depending on the size, distribution and morphology of the particle agglomerates. The term is linear in shear rate and increases nonlinearly with the solid fraction, fs (the eﬀective volume fraction of solid c ¼ fs ð1 þ 0:1kÞ). It depends weakly on fs for fs < 0:5 but then increases at an increasing rate towards an inﬁnite asymptote at a given solid fraction and state of agglomeration. The third term on the right hand side represents the deformation resistance due to energy dissipated in the plastically deforming particle– particle bonds. Under isothermal conditions and at constant structure this term indicates a shear rate thickening response with n > 1––during rapid shear rate transients, the deformation resistance increases with increasing shear rate (consistent with experimental work by Kumar et al. [129]). There is debate as to whether shear rate thickening is the constant structure response (e.g. see [84]). The term exhibits a strong inverse dependence on temperature through CðT Þ ¼ C0 expðnQ=RT Þ, which brings in the temperature dependence of diﬀusional processes and the temperature dependence of the solid deformation. Eq. (13) (the second equation in Table 4) represents evolution of the structure parameter as a function of ﬂow conditions and state variables. H is an agglomeration function and G a disagglomeration function representing the shearinduced rupture of the particle–particle bonds. Overall the model of Brown et al. predicts an increase in deformation resistance with the solid fraction and this becomes rapid between 0.5 and 0.6 fs . Brown et al. state that it is not valid beyond this sharp increase in deformation resistance and is applicable only for fs < 0:5–0:6. 6.2. Finite diﬀerence modelling 6.2.1. Onephase ﬁnite 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 ﬂow into a chisel shaped cavity. This showed the importance
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of heating the die and the heat transfer coeﬃcient, as a solid shell formed at the mould wall leading to ‘jetting’ in the centre of the cavity. 6.2.2. Onephase ﬁnite diﬀerence based on FLOW3D The approach to thixotropic modelling in the FLOW3D code is outlined in Barkhudarov et al. [95] and Barkhudarov and Hirt [96]. Barkhudarov et al. [95] use a transport equation (Eq. (14) in Table 4) for the viscosity g rather than a transport equation for k because this is more convenient for CFD, which requires a value for g. This is therefore reminiscent of the simple viscosity theories in Section 4.3. The transport equation includes an advection term and a relaxation term which accounts for the thixotropy of the material. The relaxation term is based on two variables, the steady state viscosity ge and the relaxation time 1=x, both of which may be functions of shear rate and solid fraction. No yield stress, wall slip or elastic or plastic behaviour at high solid fractions are included. This simple model therefore applies for fs < 0:6–0:7. It can be related to the Moore model (see Section 4.1 and [72]) and ok=ot then includes agglomeration and disagglomeration terms. Here disagglomeration is dependent on c_ and not c_ n as in the Model of Brown et al. [127–129]. Barkhudarov et al. [95] used their model to predict hysteresis curves for Sn 15%Pb with reasonable accuracy and to predict die swell when the relaxation time is similar to the time it takes for the metal to ﬂow through the nozzle. The equation in Barkhudarov and Hirt [96] (Eq. (18) in Table 4) is an extension of that in Barkhudarov et al. [95]. If the local viscosity is greater than the equilibrium viscosity ge then the local viscosity is driven towards ge at the thinning rate a, if the local viscosity is less than the equilibrium viscosity then it is driven towards ge at the thickening rate b. In this work it is assumed that a and b are constants, but practically it is likely that they are dependent on shear rate. The test problem is one of Sn–15%Pb droplets impacting on a ﬂat plate. The results show that droplet shape is inﬂuenced by relaxation time. The approach outlined here is essentially the basis for the thixotropic module in the FLOW3D code, which is the basis of thixotropic modelling by a number of workers including Modigell and Koke [84], Modigell and Koke [85], Ward et al. [98], Messmer [99]. Modigell and Koke [84] ﬁtted the steady state ﬂow curve for Sn–15%Pb to a Herschel–Bulkley model (see Section 2 Eq. (2)) with a yield stress sy dependent on the fraction solid. The second term on the right hand side in the expression for shear stress (Eq. (19) in Table 4) includes a structural parameter j which describes the current state of agglomeration but diﬀers from k in that it varies from zero (fully broken down) to inﬁnity (fully builtup) rather than zero and one. The time evolution of j is described with ﬁrst order reaction kinetics (Eq. (20) in Table 4) with a expðb_cÞ the rate constant for the approach of j to the equilibrium value je . After parameter evaluation, the model ﬁts stepchange of shear rate experiments with Sn– 15%Pb quite well. Simulation of die ﬁll involved a cavity with a cylindrical obstacle, highlighting the diﬀerent behaviour of Newtonian and thixotropic ﬂuids (see Fig. 27). The equation for shear stress s in Modigell and Koke [85] (Eq. (22) in Table 4) appears to be slightly diﬀerent from that in Modigell and Koke [84] in that the yield
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Fig. 27. Comparison between simulation of ﬂow into a cavity with a round obstacle assuming Newtonian behaviour and assuming thixotropic behaviour [84].
stress sy is now multiplied by the structural parameter j and the exponential in the second term in Eq. (19) in Table 4 is no longer present. This may have been rolled into the consistency index k as the text states ‘k . . .increased exponentially with the solid fraction’. The experimental rheological data was for Sn–15%Pb but the material used for die ﬁlling simulation was A356 aluminium alloy. The parameters used for the model were adjusted empirically during the simulation study (but are not given in the paper). Modigell and Koke found that above a critical inlet velocity
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the ﬁlling was not laminar any more. The transition between laminar and turbulent ﬁlling (in the sense of a smooth ﬂow front and one that is starting to break up) could be represented reasonably well. Ward et al. [98,140,141] found that, in modelling force versus time for a rapid compression test (see Section 5.2), the implicit solver tended to give an initial peak regardless of whether the viscosity was Newtonian or thixotropic, especially if the software was allowed to choose its own time step and if the model started with a gap between the slug and the platen. Provided the time step was small enough not to be a factor in determining the results, the implicit solver could reproduce the downward slope of the initial peak and the subsequent force proﬁle. The explicit solver could accurately model shear rate jumps in a rotational viscometer (see Section 5.1) but was inordinately slow. FLOWSCIENCE have therefore produced a new alternating direction implicit (ADI) solver to cope with the large ranges and changes of pressure associated with thixotropic slurries. Fig. 28 shows the ADI result for a shear rate jump in Sn–15%Pb compared with experimental results (three repeats of the same experiment) and the results from a onedimensional spread sheet calculation. The contrast between implicit and explicit solvers is not mentioned elsewhere and it is not clear whether workers have tested their modelling against the artefacts found with the implicit solver. Ward et al. [98] found that to model the shear rate jumps, an initial viscosity was required which was 2–5 times lower than the experimental values. This suggests that the initial breakdown of the slurry is very rapid, possibly beyond 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 existing system (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 then brought together by a ram. In simulating the thixoforging process, moving dies must therefore be modelled. The apparent viscosity depends on fraction solid fs , shear rate c_ and time t. The fraction solid is calculated using the Scheil equation (i.e. assuming a simple binary). The equilibrium viscosity is given by Eq. (23) in Table 4 and the time
Fig. 28. Shear rate jump from 1 to 100 s1 in SnPb alloy ðfs ¼ 0:36Þ, showing repeats of the same experiment and modelled ﬁts using a spreadsheet and FLOW3D [98].
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dependent thixotropic eﬀects in Eq. (24). The viscosity parameters were obtained by ﬁtting the simulation to experimental results with A356 aluminium alloy. Comparison between forming forces, measured on the lower die at the end of the punch stroke, and simulated forces suggests that the initial thinning rate is much higher than that in the ﬁnal stages. This is consistent with the results of Ward et al. [98] and with the proposal by Quaak [47] that at least two diﬀerent relaxation processes are operating, with diﬀerent characteristic relaxation times. The die ﬁlling for a suspension part was modelled where the material has to ﬂow around a core and weld on the opposite side. The die was modiﬁed to ensure this welding occurred in the area of the overﬂow as required.
6.2.3. Onephase ﬁnite diﬀerence based on MAGMAsoft MAGMAsoft and FLOW3D are commercial competitors in the simulation of ﬂow processes. It is not the intention here to discuss the relative merits of various commercial codes but rather to identify how those codes have been used. Kim and Kang [97] compared the output from MAGMAsoft with a Newtonian ﬂuid and assuming the viscosity of the ﬂuid obeyed the Ostwald–deWaele power law with n varying between )0.48 and +0.45 depending on the temperature. The data for this is from MAGMAsoft. It is not clear in the text what relationship between n and fs is being used to obtain this temperature relationship or how this relates to the experimental ﬁndings for A356 by Quaak et al. [142], who found n values of )0.2 and )0.3 for solidiﬁed fractions between 0.2 and 0.4 (i.e. temperatures of 605 and 589 C respectively) and by Loue et al. [143] who found n of )1.0 almost independent of temperature for temperatures between 603 and 590 C. A value of n of )0.2 does ﬁt the curve in Fig. 15 in Kim and Kang [97] at 605 C but the other values are a long way oﬀ. It is also of note that values of n of less than zero imply that the shear stress decreases with increasing shear rate. This is not easily explained (see discussion in [142] and also [61]). Notwithstanding these comments about the basis for the curve of n versus temperature, the Newtonian analysis does not agree well with the experimental results of partial ﬁlling experiments, whereas the results using the Ostwald–deWaele power law are closer to the experimental ﬁndings. Seo and Kang [100] also used MAGMAsoft, comparing the Ostwald–deWaele power law model, which has a limited range of shear rates over which it is applicable, with the Carreau–Yasuda equation, which allows viscosity at very low and very high shear rates to be considered. In the paper, only results of simulation with the Ostwald–deWaele model are presented and show reasonable agreement with partial ﬁlling results for an automotive component (but one that does not involve parting and rewelding of ﬂow fronts or very big changes in section thickness).
6.2.4. Onephase ﬁnite diﬀerence with Adstefan Itamura et al. [101] compares simulation for die casting, squeeze casting and rheocasting for both metal ﬂow and solidiﬁcation. Few details are given. The results indicate that air entrapment would occur for die casting, in contrast with squeeze
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casting and rheocasting. There would be fewer shrinkage defects in rheocasting than with the other processes. 6.2.5. Twophase ﬁnite diﬀerence There appears to be only one paper using a ﬁnite diﬀerence model for a two phase analysis, that by Ilegbusi et al. [102]. The single phase equations are solved for the whole ﬁlling phase. Trajectories of a given number of particles are computed, assuming they will ‘disappear’ when they hit a wall or are trapped in a recirculation zone. A measure of segregation is obtained by comparing the number of particles at a given distance from the inlet to the total number of injected particles. 6.3. Finite element modelling The consideration of ﬁnite element modelling is divided into onephase (Table 5) and twophase (Table 6) treatments. A number of diﬀerent commercial and other codes are used. Note that in the discussion below the papers are grouped and therefore are not discussed in the order in which they appear in the tables (which is by year). 6.3.1. Onephase ﬁnite element Zavaliangos and Lawley Zavaliangos 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 analysis is for Sn–15%Pb and, for fractions solid less than about 0.5, it is predicted that a free standing billet will collapse. The thixoforming of a simple shape is simulated. No experimental validation of the results is given. Zavaliangos and Lawley deal with a two phase analysis in the same paper (see Section 6.3.2). Backer Various rheological models were programmed into the WRAFTS software by Backer [104] including: a Newtonian; a Herschel–Bulkley model (i.e. combining a yield stress with a power law––see Eqs. (27) and (28) in Table 5); and a Bingham ﬂuid (see Section 2) combined with a power law dependence (see Eqs. (29) and (30) in Table 5). Note that it is not clear that the description of ‘‘a Bingham ﬂuid combined with 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 c_ n . An internal variable model is also used (viz. Brown et al. in Section 6.1). In this the structural parameter k (which varies between 0 and 100% rather than 0 and 1) is perceived as a chemical concentration and a convective transport equation is written for its spatial and temporal variation (Eq. (31) in Table 5). When this equation is simpliﬁed by assuming the ﬂuid density is constant 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 the second term on the right hand side rather than k as in Eq. (15) in Table 4. This reduces the contribution of disagglomeration.
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With the Newtonian rheological model, due to the relatively low viscosity of the liquid metal, there are a number of locations behind cores in a complex die that remain unﬁlled while the liquid ﬂows past them. Such locations would increase gas entrapment and would show ﬂow lines in a casting. For the power law rheological model, the ﬂuid ﬁlls behind the cores. With the internal variable model, a larger percentage of the ﬂow into the cavity arises from the runner at the side of the die cavity, in contrast with the Newtonian and Bingham/power law models which predict a larger ﬂow rate from the bottom runner. The reason for the larger ﬂow rate from the side runner is that the structural parameter (‘agglomeration variable’) k is reduced as the mixture ﬂows 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 at the bottom runner is 40%; thus, the material in the side runner is less viscous and can ﬂow into the cavity more readily. Experimental validation is not presented in the paper. Alexandrou, Burgos and coworkers There are several papers by Alexandrou, Burgos and coworkers [105–109,114] sometimes with Alexandrou and Burgos working together and sometimes in cooperation with others. The papers have similar threads running through and therefore are treated together here. In [105], Alexandrou et al. use the commercial code PAMCASTSIMULOR to compare Newtonian and Bingham ﬁlling of a threedimensional cavity with a core. The yield stress sy is a function of the fraction solid fs . Temperature dependence is introduced through this relationship. Eqs. (33) and (34) in Table 5 give the viscous stress tensor and the viscosity expression. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃDII is the second invariant of the deformation rate tensor. In simple shear, DII =2 ¼ c_ . When the local stress is larger than sy , the slurry behaves as a NonNewtonian ﬂuid. m controls the exponential rise in the stress at small rates of strain and, depending on the value of the power law coeﬃcient n, the behaviour is either shear thinning ðn < 1Þ or shear thickening ðn > 1Þ. The continuous Bingham law in Eq. (33) is based on that introduced by Papanastasiou [131] to avoid the discontinuity at the yield stress. In [105], the value of n is taken to be 1. In the simulation, ﬁrstly pipe ﬂow was studied, demonstrating that the ﬂow at the outlet was identical with the analytical solution. Due to the ﬁnite yield stress, the Bingham case shows a large unyielded area where the material in the centre ﬂows like a solid. For a three dimensional cavity with a cylindrical core, there is a strong contrast between Newtonian and Bingham behaviour (Fig. 29). In the Newtonian case, the velocity vectors at the rewelding front (i.e. where ﬂow fronts must remerge beyond a core) point towards the core, whereas in the Bingham case, they point away from the core, allowing oxide skins to be transported into overﬂows. Burgos and Alexandrou [106] examined the ﬂow development of Herschel– Bulkley ﬂuids in a sudden threedimensional square expansion, using again the Papanastasiou model [131]. The results show that during the evolution of ﬂow, two core regions and dead zones at the corners are formed. The extent of the core regions decreases with the pressure gradient and the Reynolds number and increases with the powerlaw index.
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Fig. 29. Comparison of Newtonian (on the left) and Bingham (on the right) ﬁlling behaviour for a threedimensional cavity with a cylindrical obstacle [105].
Fig. 30. Flow patterns found by modelling [107].
The relative importance of the inertial, viscous and yield stress eﬀects on the ﬁlling proﬁle in a twodimensional cavity with a Bingham ﬂuid is examined in Alexandrou et al. [107]. The analysis is as for the previous two papers. The results identify ﬁve diﬀerent ﬂow patterns (see Fig. 30): ‘shell’ (large Reynolds numbers but small Bingham numbers), ‘mound’ (low Reynolds and Bingham numbers), ‘bubble’ (larger Bingham numbers), ‘disk’ (occurs between shell and bubble ﬁlling), and ‘transition’.
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Fig. 31. Map showing ﬂow patterns in Fig. 30 as a function of Reynolds number Re and the ratio of the Bingham number to the Reynolds number (Bi=Re) [107]. Curves (a) and (b) represent simpliﬁed analyses.
These can be plotted using the Saint–Venant number, (which is deﬁned as the ratio of the Bingham number sy H =gV to the Reynolds number qVH =g, where H and V are characteristic length and velocity scales), which indicates the importance of the yield stress relative to the inertia forces, and the Reynolds number (see Fig. 31). This is a very helpful approach to schematising the diﬀerent types of behaviour, particularly in identifying the vulnerability to defects. Transition ﬂow occupies a narrow region between the disk and bubble patterns. Since the ﬂow initially starts as disk and then switches to bubble ﬁlling, this region may be prone to instabilities. In Burgos et al. [108], the Herschel–Bulkley model in Eq. (33) in Table 5 and the approach of the previous papers [105–107] is expanded to include the eﬀect of the evolution of microstructure via an equation for ok=ot very similar to Eq. (15) in Table 4 [95] but including an exponential factor in the second term on the righthand side (i.e. the disagglomeration term: see Eq. (41) in Table 5). This exponential dependence is included to account for the fact that experimentally e.g. [80–82,86] the shear stress evolution for shearrate stepup experiments is faster than that for the stepdown case. (It is not clear why this is not taken into account by the constants a and b or whether this is, in fact, a way of introducing two relaxation processes as in Quaak [47]). In addition, the yield stress, consistency index K and power law index n are now all assumed to be functions of the volume fraction solid fs and the structural parameter k. There are then six material parameters in the model: a, b, c, Kðfs ; kÞ, nðfs ; kÞ, sy ðfs ; kÞ. Burgos et al. [108] obtain these data on Sn–15%Pb from Modigell et al. [133]. The behaviour of the material is shear thickening for isostructure during a shear rate jump. The power law index decreases with the structural parameter while the consistency index and the yield stress increase. For ﬂow through a straight square channel, disagglomeration is small in the corners and in the core region of the channel.
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Alexandrou et al. [109] analysed a simple compression test assuming a Bingham ﬂuid and not taking account of the evolution of the internal structure. The viscosity and yield stress were obtained from ﬁtting a load versus time curve. There is unyielded material at the top and at the bottom of the compressed cylinder, in stagnant layers. The identiﬁcation of ﬂow regimes prone to instabilities in [107] has led to a more recent paper (Alexandrou et al. [114]) analysing two dimensional jets of Bingham and Herschel–Bulkley ﬂuids impacting on a vertical surface at a distance from the die entrance. A bubble pattern gives an unstable jet, whereas shell, disk and mound are all stable, along with most transition cases. Instabilities are the result of ﬁnite yield stress and the way yielded and unyielded regions interact. Plots of Bingham number versus Reynolds number identify stable and unstable regions. This identiﬁcation of instability provides an important explanation of the common defect in semisolid processing sometimes termed the ‘toothpaste eﬀect’ (Fig. 32). Viscoplastic constitutive models Ding et al. [110] established a rigid viscoplastic constitutive model for AlSi7Mg alloy through compression tests. They neglected the ﬂow stress during the initial breakdown stage and only ﬁtted the ﬂow stress in the steady state. In the simulation, they assumed that the deformation of semisolid materials is governed by the Levy– Mises ﬂow rule. They used DEFORM3D software with a sixﬁngered die heated to the initial temperature of the billet, i.e. the conditions are quasiisothermal. The die ﬁngers are of diﬀerent crosssectional areas. It appears from the diagrams that the material is initially in position in the die and is then compressed by the punch. Metal in the larger oriﬁce ﬁngers ﬂows faster, contrary to what would be expected with thixotropic breakdown. The simulation and the experimental results agree well but there is no analysis of whether liquid phase has segregated out from the solid, and whether it is this which is giving the results which are contrary to expectations. Rassili et al. [112] also obtained a viscoplastic constitutive model from force recordings of extrusion tests. Their simulation is aimed at thixoforging steels. There is no time dependence in the constitutive equation ((49) in Table 5) in contrast with the Ding et al. equation (46) in Table 5 which does include a c_ term. This is therefore essentially a forging simulation. The friction is assumed to be very low. There are
Fig. 32. Flow instability of the ‘toothpaste’ type in semisolid processing. The metal is ﬁlling from the right to the left [114].
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several combinations possible for the tool displacement. If the ejector goes up and the punch starts to go down when the ejector stops, buckling occurs and leads to a lap. If the punch and ejector move simultaneously, or the punch goes down ﬁrst and then the ejector goes up, the buckling disappears but a lap is formed on each ‘ear’ of the part. The simulation agrees reasonably with a lead prototype but thixoforging results with steel are not presented. Power law cutoﬀ (PLCO) model of Procast Jahajeeah et al. [111] and Orgeas et al. [115] have both used the power law cutoﬀ (PLCO) model in Procast commercial software [134]. The major assumption of the PLCO model is that the material is a purely viscous isotropic, incompressible ﬂuid. The versions used are slightly diﬀerent. In the Jahajeeah et al. work, the demonstrator component is divided into diﬀerent regions each with diﬀerent cutoﬀ values c_ 0 . It is not clear how these values are determined. If c_ 6 c_ 0 then g ¼ g0 c_ n0 , whereas if c_ > c_ 0 then g ¼ g0 c_ n i.e. shear thinning is only occurring if the cutoﬀ value is exceeded. If it is not exceeded, the viscosity is not aﬀected by local shearing and is calculated using c_ 0 . There is reasonable agreement between the simulation and the results of interrupted ﬁlling tests with A356 aluminium alloy. With less than optimum runner design, defects are predicted and these were found in the identiﬁed areas in practice. In the work by Orgeas et al. [115], there is only one value of the cutoﬀ c_ 0 and this is determined by geometry. The shear rate cutoﬀ c_ 0 was initially used in ﬁnite element codes to improve numerical convergence for shear thinning materials ðn < 1Þ when the shear rate decreased towards zero. Orgeas et al. adopt a diﬀerent point of view. Firstly, they assume that agglomeration and coalescence of grains probably does not take place over the very short injection times characteristic of thixoforming. Therefore, a decrease of c_ will not lead to an increase of the viscosity g. Secondly, a sudden increase of the shear rate c_ will lead to a decrease in viscosity. In eﬀect, an increase of c_ beyond the largest shear rate c_ 0 experienced so far will lead to a decrease in viscosity (and modify the maximum shear rate c_ 0 ). A decrease of c_ below c_ 0 will not modify the viscosity (and leaves c_ 0 unchanged). This ‘ratchettype’ behaviour could be fully modelled, but in the work by Orgeas et al. they have assumed only one value of c_ 0 because their experiments involve a transition between a shot sleeve and a small injection aperture and most of the change in viscosity is occurring at that point. The calculation of the value of c_ 0 is given in the paper. Orgeas et al. obtain the parameters for their model from experiments measuring the pressures and temperatures in a tube with a shaft in it. They then use the model to simulate the ﬁlling of a reservoir. Comparison between the simulations and interrupted ﬁlling experiments for A356 aluminium alloy are shown in Fig. 33. It should be noted that Orgeas et al. found eutecticrich concentric rings in the tube in a Poiseuille type experiment (Fig. 34(a)). These were due to veins of liquid formed in the shot sleeve as a result of mechanical instabilities generated in a solid skeleton which is not uniformly sheared (see Fig. 34). The vein of eutectic liquid at 45 was due to the ‘dead’ zone at the bottom right corner of the shot sleeve i.e. a zone which is almost not sheared and only compressed. The compression of the dead zone in
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Fig. 33. Comparison between experimental and simulated ﬁlling of a cavity (initial ram velocity 0.8 m s1 , diameter of tube 25 mm) [115]: (a) fraction solid 0.52; (b) fraction solid 0.58; (c) fraction solid 0.73. In each ﬁgure, the upper part is the simulation and the lower part the experimental result obtained with interrupted ﬁlling.
duces a ‘spongelike’ eﬀect. Such complex behaviour cannot be predicted with a onephase model. It should be noted that this highlights the need for such dead zones to be avoided in die design for semisolid processing.
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Fig. 34. (a) Crosssection through the semisolid material solidiﬁed in a tube in a Poiseuilletype experiment. The eutecticrich concentric rings are due to veins of liquid formed in the shot sleeve; (b) shows such a vein formed at the limit of the dead zone in the shot sleeve (see inset) [115].
Model based on viscoelasticity and thixotropy Wahlen [113] presents a model based on viscoelasticity and thixotropy. Thixotropic materials do not normally display viscoelasticity. It would seem that this could only really occur if the fraction solid is relatively high. In Eq. (50) in Table 5, the ﬁrst term in brackets is the viscoelastic term and the last term in brackets is essentially a creep term. It is not clear why these terms have been multiplied rather than being treated as additive. There is good agreement between the model and the curve of ﬂow
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stress versus true strain but this is for a temperature of 570 C where the A356 would be expected to be almost fully solid (see diﬀerential scanning calorimetry data in [59]). The simulation results are compared with backward extrusion samples for various temperatures. There is a transition at around 570 C, which is interpreted as the transition between plastic deformation of a connectedparticle network and the viscous ﬂow of a suspension of solid particles. 6.3.2. Twophase ﬁnite element Some of the twophase ﬁnite element modelling papers are presented in Table 6 but that approach is less useful when the equations are so complex. (Gebelin et al. [144] have presented a useful mathematical comparison of one phase and two phase approaches.) Others are discussed here in a more general background section. Orgeas et al. [115] have reviewed twophase approaches. In the twophase models, the semisolid material is considered as a saturated twophase medium i.e. made of the liquid and solid phases. Each phase has its own behaviour, which can be inﬂuenced by the presence of the other phase via interfacial contributions. The conservation equations can be written within a mixture theory background [145] and the solid phase (solid skeleton) can be modelled as a purely viscous and compressive medium [146,147]. Momentum exchanges between the solid and the Newtonian liquid are handled by a Darcytype term appearing in the momentum equations [148]. These models are able to predict phase separation e.g. [118,149]. However, the determination of the rheological parameters which are required is not straightforward e.g. [146,147]. Twophase models also usually require the simultaneous calculation of a solid fraction ﬁeld, a pressure ﬁeld, two velocity ﬁelds (for the liquid and the solid) and a temperature ﬁeld (although in most cases the simulation is isothermal). Such simulations therefore require very high computation time. The distinctive features of the papers identiﬁed in Table 6 are as follows: • Zavaliangos [116]: The degree of cohesion is represented by an internal variable which evolves with deformation (cf. the internal variable in the Brown et al. model [127–129] in Section 6.1). The permeability equation implies that solid–liquid segregation decreases as the grain size decreases. Behaviour is not symmetric under tension and compression. • Koke et al. [117]: The solid phase is assumed to be a pseudoﬂuid with a Herschel– Bulkley viscosity. • Kang and Jung [118] treated the solid phase as compressible and introduced a separation coeﬃcient expressing the actual separation of the particles in relation to their initial separation. The higher the strain rate the more homogeneous the distribution of the solid fraction. In compression forming, macroscopic phase segregation occurred with densiﬁcation of the remaining solid in the central region. • Binet and Pineau [119] adopt a mixture approach where the hydrodynamic part is the same as for most incompressible CFD codes but the velocity ﬁeld represents the velocities of the mixture. A source term is added to the momentum equations to take account of the diﬀusion velocities of the individual phases.
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• Choi et al. [120]: The solid is assumed to be viscoplastic. Kuhn’s yield criterion is used for the solid phase. (i.e. behaviour is symmetric for tension and compression and the hydrostatic component of stress is included). • Yoon et al. [122] used Von Mises yield criterion (i.e. symmetric in tension and compression). The semisolid is treated as a single incompressible phase. • Kopp and Horst [123] adopt the Drucker–Prager yield criterion (i.e. nonsymmetric in tension and compression). • Modigell et al. [124] use the pseudoﬂuid approach for the solid phase [117]. All the nonNewtonian properties of the material are shifted to the solid phase and the liquid is treated as Newtonian. Twodimensional contour maps showing the transitions between laminar, transient and full turbulent ﬁlling are plotted (Fig. 35). The dimensionless groups used for this mapping are not given in detail in this short paper. The threedimensional process window for A356 aluminium alloy, based on laminar ﬁlling, is also identiﬁed (Fig. 36). These results are highly significant. 6.3.3. Micromodelling Rouﬀ et al. [125] present a novel and interesting approach. Spherical inclusions (i.e. particles) containing entrapped liquid are assumed to deform very little and can slip relative to each other if the restriction between them is released. They are surrounded by solid bonds and the ‘not entrapped’ liquid where deformation generally takes place. This ‘active zone’, associated with the strain localization, is gathered in a layer surrounding the inclusions (Fig. 37). The volume solid fraction of the active zone, fAs , is the internal variable. During deformation, the bonds are broken and liquid is released. Thus, the bimodal liquid–solid distribution changes with the strain
Fig. 35. Map of types of ﬂow [124]: (a) laminar, (b) transient, (c) turbulent. Bi is the Bingham number, Kc a rheological number, C1 , C2 geometric constant and Re the Reynolds number. Kc , C1 and C2 are not speciﬁed in the paper.
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Fig. 36. Threedimensional process window for aluminium alloy A356 [124]. Mechanical properties are given in the two top boxes. The smaller boxes summarise the process parameter windows to obtain those mechanical properties (DT is the temperature window, Dfs the solid fraction window, Dl the wall thickness and Dvm ). The higher the required mechanical properties, the smaller the threedimensional process window (compare the right hand diagram with the left).
Fig. 37. Schematic representation of the semisolid microstructure with inclusions of solid and entrapped liquid surrounded by liquid and solid bonds [125].
rate. Both the liquid and the solid are assumed isotropic and incompressible. The liquid and solid are then embedded in a homogeneous equivalent medium having the eﬀective properties of the inclusion or the active zone. The viscosity of each inclusion and the viscosity of the active zone can then be determined and the eﬀective viscosity of the semisolid is a mixture of these. This approach enables very accurate prediction of the viscosity of Sn–15%Pb as a function of shear rate and shows great promise for further development.
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7. Flow visualisation Virtually all the experimental validation of die ﬁlling patterns reported in Section 6 involves interrupted ﬁlling. The diﬃculty with this is that the eﬀects of inertia
Fig. 38. Filling by Sn–12%Pb of a Tshaped cavity with a glass side showing the eﬀect of piston velocity on the shape of the ﬂow front [150]. Piston velocity: (a) 10 mm/s; (b) 50 mm/s; (c) 100 mm/s.
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Fig. 39. Eﬀect of obstacle size and shape on meeting of split ﬂow fronts, Sn–15%Pb, 189 C, 0.25 m s1 , splayed die entrance [98]. Left to right: obstacle 30 mm diameter; obstacle 20 mm diameter; experimental and standard ‘spiders’ for extruding PVC pipes. Note how broader obstacle leads to ﬂow fronts meeting with the die more full. Far right: experimental spider at 1 m s1 .
Fig. 40. Shots from ﬁlmed die ﬁlling with Al A357 [98]. Left to right: 576 C, 0.25 m s1 , parallel entrance; 576 C 1 m s1 , parallel entrance; 1 m s1 splayed entrance; 0.25 m s1 splayed entrance with the experimental ‘spider’ and the same at 1 m s1 , all at 577 C.
compromise the results, with the material continuing to travel even when the ram has stopped. The most appropriate way of checking the position of the ﬂow front during die ﬁll is with in situ observation. The main recent work with transparent sided dies is that by Petera et al. [150] and Ward et al. [98]. Petera et al. [150] use a Tshaped die, covered with a glass plate on one side. The die is integrated into an oven to ensure that conditions are isothermal. Experiments have been carried out with Sn–12%Pb. The eﬀect of piston velocity on the ﬂow front is shown in Fig. 38. At low piston velocity (Fig. 38(a)), no detachment of material from the walls of the die could be observed. At much higher piston velocity (Fig. 38(c)), there is signiﬁcant detachment, with the potential to form cavities in the ﬁnal product. Ward et al. [98] established an arrangement which could be used with both SnPb and with aluminium alloys. Various obstacle shapes were placed in the path of the ﬂowing material to observe ﬂow fronts remerging. Fig. 39 illustrates the results for Sn–15%Pb and Fig. 40 for A357 aluminium alloy. The die entrance was either parallel sided or splayed. Obstacles included cylinders of various diameters and ‘spiders’ used in the manufacture of PVC pipe. Remerging was sensitive to ram velocity and obstacle shape.
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8. Concluding remarks The main focus for this review has been the modelling of semisolid processing. As background for that, routes to spheroidal microstructures, types of semisolid processing and the advantages and disadvantages have been summarised. Groundwork on rheology and the origins of thixotropy have been laid and mathematical theories of thixotropy introduced. Experimental data for input into modelling is crucial and depends on measuring behaviour during rapid transients, either through rapid shear rate jumps in rheometers or through rapid compression testing. The review of modelling has then been divided into those models based on ﬁnite diﬀerence methods and those based on ﬁnite element. In addition, some models are onephase and some are twophase. Papers on modelling are summarised in Tables 4–6 and in the text are dealt with in sections, grouped together where there is common ground. There are a multiplicity of approaches to the modelling of semisolid forming. What emerges clearly here is the lack of quantitative measures of the accuracy of the results and a lack of direct means of comparison. There is also a serious need for more rheological data, both for Sn–15%Pb (the classic ‘model’ alloy for semisolid thixotropic studies), for aluminium alloys used in commercial forming, and for other materials such as steels where there is signiﬁcant interest in commercial use. This rheological data is diﬃcult to obtain and great care has to be taken to avoid artefacts and to ensure the data is appropriate for the application. For example, thixoforming is essentially a rapid transient rather than a steady state process. Despite these difﬁculties, accurate modelling can be a great aid in die design, predicting appropriate processing conditions and minimising defects. The recent development of ‘maps’ by Alexandrou and coworkers [114] and by Modigell et al. [124] is highly signiﬁcant in this respect.
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