Modelling the semisolid processing of metallic alloys

Modelling the semisolid processing of metallic alloys

Progress in Materials Science 50 (2005) 341–412 www.elsevier.com/locate/pmatsci Modelling the semisolid processing of metallic alloys H.V. Atkinson ...

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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 non-dendritic 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 one-phase or two-phase 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 micro-modelling 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.: +44-116-2231019; fax: +44-116-2522525. E-mail address: [email protected] (H.V. Atkinson).

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

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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 two-phase 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. One-phase finite difference based on the model of Brown et al...... 6.2.2. One-phase finite difference based on FLOW3D .................................... 6.2.3. One-phase finite difference based on MAGMAsoft............................. 6.2.4. One-phase finite difference with Adstefan................................................ 6.2.5. Two-phase finite difference............................................................................. 6.3. Finite element modelling................................................................................................ 6.3.1. One-phase finite element................................................................................. 6.3.2. Two-phase finite element ................................................................................ 6.3.3. Micro-modelling.................................................................................................

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 peak-stress viscosity in Table 2 gss ‘first’ steady-state 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 cut-off in the power law cut-off 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 self-diffusion 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 as-cast 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 non-dendritic 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) multi-link, 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/near-liquidus 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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near-liquidus 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) Semi-solid 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 non-dendritic 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 semi-solid 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 non-dendritic 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, anti-lock 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 shear-cooling 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 thin-walled sections) and allows application of the process to higher-strength heat-treatable alloys. (4) Lower processing temperatures reduce the thermal shock on the die, promoting die life and allowing the use of non-traditional 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 non-uniform heating can result in non-uniform 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 non-Newtonian 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 non-linear 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 non-linear 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)

Herschel-Bulkley Bingham Shear Thinning

Newtonian

Shear Thickening

τy

-1

Shear Rate (s ) Viscosity (Pas)

Herschel-Bulkley 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. Shear-stress 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 Ostwald-de-Waele 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 step-change up in shear rate and then a step-change down (after [64]).

fs . The data obey the Cross model, but information on the extremes is sparse. These data are for steady-state 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 3-D 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 build-up 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 inter-particle forces

Break-up 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 cross-sectional 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, non-Newtonian 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 Cross-like 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 1 ðg0  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 steady-state 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 shear-rate drops. Quaak [47] proposes two characteristic times to describe a shear rate jump. He suggests that during a shear-rate 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 peak-stress viscosity, gss is the ‘first’ steady-state 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 solid-particle 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 flow-exponent 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, volume-averaged 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 three-dimensional 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 load-displacement 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 steady-state viscosity determined by Quaak [47] for a 7% Si aluminium alloy, extrapolated to 0.5 fraction solid; this is well below the other results, 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 in-gate, which was narrow as for die-casting, 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, one-phase or two-phase. There is in addition, a paper on micro-modelling [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); two-phase modelling (Section 6.2.5). For the one-phase 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 cut-off (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 semi-solid die filling FEM

One-phase

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. (DEFORM3-D) [110] Jahajeeah et al. (Procast) [111] Rassili et al. (FORGE3) [112] Wahlen (Thixoform) [113] Alexandrou et al. [114] Orgeas et al. (Procast) [115]

Two-Phase

Ilegbusi et al. [102]

Zavaliangos and Lawley (ABAQUS) [103] Zavaliangos [116] Koke et al. [117] Kang and Jung [118] Binet and Pineau [119] Choi et al. [120] Kang and Jung [121] Yoon et al. (CAMPform2D) [122] Kopp and Horst (ABAQUS) [123] Modigell et al. [124]

Micro-modelling

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 One-phase 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, scalar-equation method for free surface, single internal variable ðkÞ constitutive model [127–129], experimentally determined values of agglomeration function H and disagglomeration function G given. Chisel-shaped 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 one-phase and two-phase 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 right-hand 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–de-Waele 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–de-Waele 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 FLOW-3D [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 (FLOW-3D) [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]

Two-phase Ilegbusi et al. [102]

Ostwald–de-Waele compared with Carreau–Yasuda: n1 g  g1 ¼ ½1 þ ð_ckÞa a g0  g1

Compares simulation for die-casting, squeeze-casting 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–de-Waele gives reasonable agreement with partial filling tests

Die-casting 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 One-phase 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 right-hand 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 2-D cavity. Comparison of Newtonian and Bingham filling of 3-D 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



Predicts time evolution of yielded/unyielded regions for sudden 3-D 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_cÞ ðk  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. (DEFORM3-D) [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 cut-off (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

Visco-plastic 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 load-stroke curve and actual. No examination of liquid segregation in the samples Brake calliper divided into different regions each with different cut-off 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]

2-D 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 cut-off 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 Poiseuille-type 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’) well-simulated

383

384

Table 6 Two-phase finite element simulation papers and micro-modelling 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 near-undrained conditions and minimal phase segregation

Koke et al. [117]

Liquid phase assumed Newtonian. Solid phase is pseudo-fluid 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, Carman-Kozeny 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 visco-plastic 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 cross-sections. Higher strain rates gave less segregation

Yoon et al. (CAMPform2D) [122]

Von Mises yield criterion. Semi-solid 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 non-isothermal Al2024 alloy given

Equations are summarised in Fig. 1 in the paper

Isothermal predictions of liquid segregation in good agreement with experiment. Non-isothermal 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. Pseudo-fluid approach for the solid phase. All non-Newtonian 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

Micro-modelling 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 micro-modelling 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 built-up 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 non-linearly 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 shear-induced 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. One-phase 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

H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412

389

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. One-phase 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 built-up) 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 step-change 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

390

H.V. Atkinson / Progress in Materials Science 50 (2005) 341–412

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 one-dimensional 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 FLOW-3D [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. One-phase 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–de-Waele 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–de-Waele power law are closer to the experimental findings. Seo and Kang [100] also used MAGMAsoft, comparing the Ostwald–de-Waele 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–de-Waele 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. One-phase 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. Two-phase 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 one-phase (Table 5) and two-phase (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. One-phase 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 three-dimensional 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 Non-Newtonian 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 three-dimensional 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 power-law 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 two-dimensional 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 right-hand 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 shear-rate step-up experiments is faster than that for the step-down 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 DEFORM-3D software with a six-fingered die heated to the initial temperature of the billet, i.e. the conditions are quasi-isothermal. The die fingers are of different cross-sectional 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 cut-off (PLCO) model of Procast Jahajeeah et al. [111] and Orgeas et al. [115] have both used the power law cut-off (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 cut-off 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 cut-off 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 cut-off c_ 0 and this is determined by geometry. The shear rate cut-off 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 ‘ratchet-type’ 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 eutectic-rich 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 ‘sponge-like’ 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) Cross-section through the semisolid material solidified in a tube in a Poiseuille-type experiment. The eutectic-rich 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 connected-particle network and the viscous flow of a suspension of solid particles. 6.3.2. Two-phase finite element Some of the two-phase 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 two-phase approaches. In the two-phase models, the semisolid material is considered as a saturated two-phase 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 Darcy-type 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]. Two-phase 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 pseudo-fluid 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. non-symmetric in tension and compression). • Modigell et al. [124] use the pseudo-fluid approach for the solid phase [117]. All the non-Newtonian properties of the material are shifted to the solid phase and the liquid is treated as Newtonian. Two-dimensional 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 three-dimensional process window for A356 aluminium alloy, based on laminar filling, is also identified (Fig. 36). These results are highly significant. 6.3.3. Micro-modelling 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. Three-dimensional 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 three-dimensional 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 T-shaped 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 T-shaped 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 one-phase and some are two-phase. 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.

Acknowledgements I would like to acknowledge helpful discussions with Profs. A.C.F. Cocks and A.R.S. Ponter at the University of Leicester, and with my coworkers Drs. P.J. Ward, T.Y. Liu, S.B. Chin and D.H. Kirkwood at the University of Sheffield. I would also like to thank Dr. D Liu for assistance in preparing this manuscript. References [1] Spencer DB, Mehrabian R, Flemings MC. Metall Trans 1972;3:1925–32. [2] Kirkwood DH. Int Mat Rev 1994;39:173–89. [3] Proc. 1st Int. Conf. on Semi-Solid Processing of Alloys and Composites, Ecole Nationale Superieure des Mines de Paris, Societe Francaise de Metallurgie, Cemef, Sophia-Antipolis, France, 1990. [4] Brown SB, Flemings MC, editors. In: Proc. 2nd Int. Conf. on Semi-Solid Processing of Alloys and Composites, Cambridge, MA, USA, June 1992. Warrendale: TMS; 1992. [5] Kiuchi M, editor. In: Proc. 3rd Int. Conf. on Semi-Solid Processing of Alloys and Composites, Tokyo, Japan, June 1994. Tokyo Institute of Industrial Science, 1994.

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