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Study on Thixotropic Property of A356 Alloy in Semi-Solid State
Sudip Simlandi1, a, Nilkanta Barman*1,b and Himadri Chattopadhyay1,c 1Department of Mechanical Engineering, Jadavpur University, Kolkata- 700032, India
Keywords: Thixotropic behavior, Modelling, Apparent viscosity
Abstract. In the present work, the thixotropic property of a semisolid aluminium alloy (A356)
under deformation is investigated numerically where the flow between two parallel plates is
considered. The flow field is represented by momentum conservation equations where the non-
Newtonian behavior of the semisolid material is represented by the Herschel-Bulkley model. The
agglomeration and the de-agglomeration phenomena of the suspended particles under shear are
represented using a time dependent structural parameter influenced by the rate of strain and shear
stress. The simulation predicts the flow field, rate of strain and apparent viscosity of the semisolid
materials under transient and steady state conditions. It is found that the apparent viscosity shows a
transient nature during sudden change in the shear rate, and its value decreases with increasing shear
rate and vice-versa. It is also found that the present prediction shows a good agreement with prior
work.
Introduction
The thixoforming is a developing manufacturing technique which produces near-net-shape
components. This technique is much advantageous over the other conventional forming techniques
such as it consumes less energy, and the final products have low porosity and good mechanical
properties. In thixoforming, the alloys are deformed in semisolid state, which exhibits a complex
non-Newtonian flow behavior. The flow behavior is influenced by numerous process factors, and
depends on time and stress history. In literature, it is found that the theoretical models for such
semisolid materials under deformation are less developed and the available constitutive models are
mostly established from experiments. However, for successful implementation of the technique,
proper knowledge on the properties of the semisolid materials under deformation is necessary. In
the present work, therefore, the thixotropic property of a semisolid alloy under deformation is
investigated numerically.
For understanding the modelling of the thixotropic behavior of alloys in semisolid state, related
research works are reviewed. Burgos et al. [1] reported that there exists a shear-dependent finite
yield stress which is modeled using the Herschel-Bulkley fluid model and introducing a structural
parameter to describe the kinetics of the agglomeration and de-agglomeration phenomena. Koke and
Modigell [2] found that the yield stress is strongly depends on the microstructure and the degree of
agglomeration of the solid phase and increases strongly with rest time because of the agglomeration
of the suspended solid particles. They also found that the steady-state rheological behavior is shear
thinning. Gautham and Kapur [3] presented a model for unsteady state shear stress of the semi-solid
metal suspensions by introducing a structural parameter. Dullaert and Mewis [4] presented a general
structural kinetics model to describe the flow behavior of thixotropic systems. A model proposed by
Alexandrou [5] is able to predict the flow behavior of the semi-solid slurries. In that work, the
variation of an apparent viscosity demonstrates the complexity of the flow behavior of slurry.
Alexandrou et al. [5, 6] presented the rate of breakdown and rate of buildup in semi-solid slurry
during shearing. They used the Herschel-Bulkey model as a standard thixotropic model for
modeling of the semi-solid metal suspensions. However, the semisolid alloys show a complex and
distinct flow behavior during semisolid processing. This complex flow behavior during processing
changes the process variables and conditions continuously in a way that is very different than the
convectional processing. Therefore, it is essential to generate concept and ideas of the alloy
Solid State Phenomena Vols. 192-193 (2013) pp 335-340Online available since 2012/Oct/24 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/SSP.192-193.335
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 129.186.1.55, Iowa State University, Ames, United States of America-30/09/13,11:03:41)
behavior in the semisolid state for systematic design of an effective semisolid processing facility.
The present study is motivated toward prediction of the thixotropic behavior of the A356 Al-alloy in
semisolid state. The Al-alloy is considered here because of its light weight, high strength-to-weight
ratio, and high resistance to corrosion properties.
Description of Physical Problem
The present work considers the prediction of the thixotropic behavior of A356 alloy in the
semisolid state. Most of the researchers used mainly the high temperature ‘Searle Rheometer’ for
determination of the semisolid behavior experimentally, where the semisolid metal alloy resides in
an annular space between an outer cylinder and an inner cylinder. In most of the rheometers, the
inner cylinder rotates whereas the outer cylinder is stationary. The temperature of the outer cylinder
is controlled using a resistance heating system with forced air cooling.
In the present work, a flow of semisolid A356 alloy in a Searle Rheometer is considered. Fig.
1(a) shows a schematic part of the Searle Rheometer. It is assumed that the diameter of the
rheometer is sufficiently large compared to the height of the annular space between the cylinders.
Henceforth, the flow of the alloy is represented by flow between two parallel plates (with 98%
correctness) for simplicity of the analysis [see Fig. 1(b)]. In the present system, the lower plate is
moving with a velocity U along X-axis whereas the upper plate is stationary. The alloy (A356)
exists between the plates is deformed isothermally in the semisolid state. The distance between the
plates along Y-axis is H.
Mathematical Model
The present work considers a flow (2-D) of a semisolid alloy (A356) between two parallel plates.
The corresponding flow field (u, v) is represented by the momentum conservation equations as
yt
u
∂∂
=∂∂ τ
(1)
0=v (2)
It is found that the alloys in the semisolid state show viscoplastic behavior. The behavior
primarily depends on the yield strength and the shear rate. The yield strength is the maximum shear
stress at which the shear begins, which mainly depends on the temperature of the semisolid alloy
and decreases with shearing time. Here, this non-Newtonian behavior of the semisolid alloy is
incorporated by representing the shear stress (τ) using the Herschel-Bulkley model [7] as
( )γγ
γλτ
τ ��
�
+= −10 nK (3)
where the rate of strain (γ� ) is given as
y
u
∂∂
=γ� (4)
In Eq.3, K is the consistency index (0.05 Pa-sn) and n is the power law index (0.15). In this work, a
simplified model is considered to represent the time dependent yield stress as considered by
Alexandrou [5] and given as
( ) 00 λτλτ = (5)
where τo is yield stress (110 Pa [8]) and λ is structural parameter. Theλ represents the time
dependent semisolid behavior, which is first introduced by Burgos et al. [1].
The structural parameter (λ ) characterizes the state of the structure of the solid particles in the
semisolid alloys. In a fully structured state, i.e., when all the particles are connected, λ is assumed
to be unity [Fig. 2(a)]. In a fully broken state, when none of the particles are connected, λ is
assumed to be zero [Fig. 2(b)]. The evolution of this structural parameter is defined by a first-order
rate equation, similar to those used to describe the chemical reaction kinetics (Burgos et al. [1]). It is
assumed that the rate of break-down (de-agglomeration) depends on the fraction of links existing at
any instant and on the deformation rate. Similarly, the rate of build-up (agglomeration) is assumed
336 Semi-Solid Processing of Alloys and Composites XII
to be proportional to the fraction of links remained to be formed. The break-down and build-up
mechanisms are depicted in Fig. 3. When shear rate increases, the break-down occurs and vice-
versa. In the present work, the evolution of the structural parameter (λ ) with time (t) is considered
as
( ) ( )γαγλαλαλ
��210 exp1 −−=
Dt
D (6)
The first term in the right hand side of the Eq. 6 represents the recovery term and last term is known
as break-down term. In this work, the value of the break-down ( 1α and 2α ) and recovery ( 0α )
parameters is considered as 0.01 for A356 alloy [5].
Boundary Conditions. For the present flow field, the boundary conditions are
At y = 0, u = U (7)
At y = H , u = 0 (8)
The governing Eqs. (1-8) along with the material properties and the boundary conditions represent
the behavior of the alloy in semisolid state under deformation.
(a)
(b)
(a) λ = 1
(b) λ = 0
Figure 1: (a) A schematic part of a Searle
Rheometer where the inner cylinder is rotating and
the outer cylinder is stationary and (b) Schematic of
the system considered in the present work: a flow
between two parallel plates where lower plate is
moving and upper plate is stationary.
Figure 2: The structural parameter
(a) when fully structured state and (b) when
fully broken state
Solution Method. The thixotropic behavior of A356 alloy is predicted in transient and steady
state conditions. Hence, to incorporate the sudden change in velocity of the moving plate and for the
simplicity in solution, an apparent viscosity ( aη ) of the semisolid alloy is considered so that the
shear stress in Eq. 1 may represented as
γητ �a= (9)
The sudden increase or decrease in the velocity ( LPU ) of the lower plate is considered as
increamentsteadyLP UUU += (see Fig. 4) where steadyU is the steady velocity and increamentU is the
increment in velocity. The corresponding time dependent velocity distribution (Kundu and Cohen
[9]) is given as
−−
−+
−= ∑∞
=12
22 sinexp12
11k
aincreamentsteady
H
yk
H
tk
kH
yU
H
yUu
πρη
ππ
(10)
where ρ (2685.0 kg/m3) is the density of A356 alloy. The shear rate is calculated as
−+−−= ∑
∞
=12
22 cosexp21
k
aincreament
steady
H
yk
H
tk
HHU
H
U πρη
ππ
πγ� (11)
Solid State Phenomena Vols. 192-193 337
In the Eq. 10-11, an approximate value of the apparent viscosity ( aη ) is assumed. The apparent
viscosity is updated by using the Herschel-Bulkley model (Eq. 3) and the evolution of the structural
parameter (Eq. 6) as
+= −10 n
a K γγλτ
η �
� (12)
Finally, a FORTRAN based program is developed to solve the governing equations.
Figure 3: Break-down & recovery phenomena Figure 4: Variation of the moving plate velocity
Results and Discussion
In this work, the semisolid alloy is sheared between two parallel plates where the upper plate is
assumed stationary and the lower plate is moving at a velocity U. The transient behavior of the
semisolid alloy is incorporated by suddenly increasing or decreasing the velocity of the lower plate.
Initially, the evolution of velocity distribution is performed under different flow conditions. Finally,
the work involves evolution of the apparent viscosity of the semisolid A356 alloy under different
shear rates at transient and steady state conditions, and then the predicted results are validated with
an existing work.
Velocity Distribution along Y-direction (Plate Height) with Time. The thixotropic behavior of
the alloys in the semisolid state depends on the flow field developed under deformation. In the
present section, the distribution of the velocity is presented with time under different flow
conditions: (i) when plate velocity increases suddenly and (ii) when plate velocity decreases
suddenly. Fig.5(a) shows the distribution of velocity along the Y-direction between the plates when
the plate velocity increases suddenly where Vm = 0 m/s and Vin = 0.20 m/s. With the increase in
velocity, the material adjacent to the moving plate yields first then it proceeds towards the upper
plate. The velocity increases throughout the entire gap until the yielding reaches to the fixed plate
and after a certain time, the velocity distribution attains a steady state condition. In the present
condition, the shear rate (the absolute value) also increases to a high value initially and then reaches
to a low steady state value [Fig. 6(a)]. Fig. 5(b) shows the distribution of velocity along Y-direction
between the plates when the plate velocity decreases suddenly where Vm = 0.40 m/s and Vin = - 0.20
m/s. In the fig. 5(b), the velocity of the yielded material, near to the moving plate, suddenly drops to
the moving plate velocity. Thereafter, the velocity of the yielded material decreases with time to a
steady state condition. Under such condition, the shear rate (the absolute value) shows an undulation
and then reaches to a higher steady state value [Fig. 6(a)].
Prediction of Apparent Viscosity during Isothermal Deformation. The alloys in the semisolid
state behave as a non-Newtonian fluid. It depends on its temperature and shear rate applied during
deformation. In addition, it also exhibits a time dependent characteristic. The time and shear rate
dependent property of the alloys in the semisolid state is known as the thixotropic property. In this
work, the thixotropic property of the A356 alloy is predicted, which represented by the variation of
338 Semi-Solid Processing of Alloys and Composites XII
the apparent viscosity ( aη ) of the semisolid alloy with time at different shear rates. The evolution of
the apparent viscosity ( aη ) is performed at a constant temperature (T∼590°C). The imposed shear
rate is varied with changing the velocity of the lower plate suddenly. Fig. 6(a) shows the variation of
the shear rate developed during deformation at position Y = 0.001m, measured from the lower plate,
with time. Fig. 6(b) shows the evolution of the apparent viscosity at Y = 0.001m with time.
The work considers sudden increase and decrease in the lower plate velocity. Initially, the shear
rate (the magnitude of the shear rate) at the position (Y = 0.001m) is high which leads to a low
apparent viscosity of the semisolid slurry. Thereafter, the shear rate decreases to a steady state value
and accordingly, the apparent viscosity also increases gradually to a steady state value. It is noticed
that, with increasing shear rate, the apparent viscosity decreases. When plate velocity decreases
suddenly, an undulation in the shear rate is found. Correspondingly, the apparent viscosity shows an
undulation and then gradually increases to a higher steady value. Finally, the present numerical
prediction is validated against an existing work by Zhang et al. [10] as shown in Fig. 7 where a
good agreement is found.
0 0.05 0.1 0.15 0.20
1
2
3
4
5
6x 10
-3
Velocity (m/s)
Y(m
)
t
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
1
2
3
4
5
6x 10
-3
Velocity (m/s)
Y(m
)
t
(a) (b)
Figure 5: (a) Velocity distribution along the height (Y) between the plates when the plate velocity
increases suddenly (Vm = 0 m/s and Vin = 0.20 m/s) and (b) Velocity distribution along the height
(Y) between the plates when the plate velocity decreases suddenly (Vm = 0.40 m/s and Vin = - 0.20
m/s)
Conclusion
The present work predicts the thixotropic property of a semisolid aluminium alloy (A356)
numerically. The semisolid alloy is sheared between two parallel plates. The flow behavior is
modeled considering the transient momentum equations where the non-Newtonian behavior of the
semisolid alloy is incorporated with the Herschel-Bulkley model. A non-dimensional structural
parameter (λ) is used to represent the agglomeration and de-agglomeration phenomena under
varying shear rate. To represent the time dependent yield stress, a simplified linear model is
considered. In this work, the apparent viscosity represents the thixotropic property of the semisolid
alloy. It is found that the apparent viscosity shows a transient value during sudden change in the
shear rate, and its value decreases with increasing shear rate and vice-versa. It is also found that the
present prediction shows a good agreement with an existing work.
Solid State Phenomena Vols. 192-193 339
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
Time (s)
Norm
aliz
ed A
ppare
nt V
iscosity a
Shear rate : 100 (1/s)
Zhang et el. (2006)
Shear rate: 50 (1/s)
Figure 7: Variation of
normalized apparent
viscosity [(η-ηe)/(η0-ηe)]
with time
References
[1] G. R. Burgos, N. Andreas, A. N. Alexandrou, V. Entov, Thixotropic rheology of semisolid metal
suspensions, Journal of Materials Processing Technology, 110 (2001) 164-176.
[2] J. Koke, M. Modigell, Flow behaviour of semi-solid metal alloys, J. Non-Newtonian Fluid
Mech., 112 (2003) 141–160.
[3] B. P. Gautham, P. C. Kapur, Rheological model for short duration response of semi-solid
metals, Materials Science and Engineering A, 393 (2005) 223–228.
[4] K. Dullaert, J. Mewis, A structural kinetics model for thixotropy, J. Non-Newtonian Fluid
Mech., 139 (2006) 21–30.
[5] A. N. Alexandrou, On the Modeling of semisolid suspentions, Solid State Phenomena, 141-143
(2008) 17-23.
[6] A. N. Alexandrou, G. Georgiou, On the early breakdown of semisolid suspensions, Non-
Newtonian Fluid Mech., 142 (2007) 199–206
[7] H. V. Atkinson, Modelling the semisolid processing of metallic alloys, Progress in Materials
Science, 50 (2005) 341–412
[8] W. C. Keung, Y.F. Lee, W. Shan, S. Luo, Thixotropic Strength and Thixotropic Criteria in
Semisolid Processing, Solid State Phenomena, 141-143 (2008)319-323
[9] P. K. Kundu, I. M. Cohen, Fluid Mechanics, 2nd
Edition, Academic Press, New York, 2002
[10] Y. Zhang, W. Mao, Z.. Zhao, Z. Liu, Rheological Behavior of Semisolid A356 alloy at steady
state ,Acta Metallurgica Sinica, 42(2) (2006) 163-166
(a)0 200 400 600 800 1000 1200 1400 1600 1800
-300
-250
-200
-150
-100
-50
0
50
TIME(s)
SH
EA
R R
AT
E(1
/s)
(b)0 200 400 600 800 1000 1200 1400 1600 1800
0
1
2
3
4
5
TIME(s)
VIS
CO
SIT
Y(P
a-s
)
Figure 6: (a) variation of the shear rate with time at Y= 0.001m and (b) evolution of apparent
viscosity with time at Y= 0.001m
340 Semi-Solid Processing of Alloys and Composites XII
Semi-Solid Processing of Alloys and Composites XII 10.4028/www.scientific.net/SSP.192-193 Study on Thixotropic Property of A356 Alloy in Semi-Solid State 10.4028/www.scientific.net/SSP.192-193.335
