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Modeling of the uniaxial tensile and compression behavior of semi-solid A356 alloys
Computational Materials Science 45 (2009) 633–637
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Modeling of the uniaxial tensile and compression behavior of semi-solid A356 alloys S. Benke a,*, S. Dziallach b, G. Laschet a, U. Prahl b, W. Bleck b a b
Access e. V., RWTH-Aachen, Intzestrasse 5, 52072 Aachen, Germany Institute of Ferrous Metallurgy, RWTH-Aachen, Intzestrasse 1, 52072 Aachen, Germany
a r t i c l e
i n f o
Article history: Received 30 December 2007 Received in revised form 21 July 2008 Accepted 21 August 2008 Available online 14 October 2008 PACS: 46.15.Cc Keywords: Semi-solid state Theory of porous media Finite element method Casting Solidification
a b s t r a c t Uniaxial tensile and compressive tests with the industrial aluminium alloy A356 were carried out at various temperatures in the semi-solid regime to determine the constitutive behavior of the alloy during the solidification in casting processes. The experimental findings show a pronounced strength difference in compression and tension as a function of the solid volume fraction. The semi-solid alloy is modeled on the macroscale as a visco-plastic porous medium saturated with liquid. A generalized creep law is modified to take into account the geometry of the pore space, the coherency of the dendrites and the response of the material to the different loading directions. This provides a continuous description of the constitutive behavior of the semi-solid alloy from the coherency temperature to the creep regime of the fully solidified state. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction During casting processes, most technical alloys solidify with the formation of a two phase region known as the mushy zone in a finite temperature interval. The solidification starts below liquidus temperature by local undercooling and the nucleation of solid germs takes place. At low solid fractions, the solid germs grow independently and are completely free to move. The alloy behaves like a suspension. At some point, the dendrites agglomerate and form a more or less continuous skeleton, which behaves like a fluid saturated porous medium. Below this temperature, defined as the coherency temperature, the semi-solid is able to sustain low tensile strains. Liquid feeding remains possible to accommodate externally applied strains, solidification shrinkage and thermally induced deformations. In general the coherency is not only determined by the temperature and fraction solid but also dependent on the morphology and the size of the solidified particles. In the mushy region below the coherency temperature, the mechanical deformation is influenced by the visco-plastic solid and the flow of the remaining viscous melt in the pore-space. Especially at high solid fractions and high strain rates a considerable liquid pressure will build up which leads together with the inelastic solid to a complex rate dependent constitutive behavior of the mush. * Corresponding author. Tel.: +49 2418098016. E-mail address: [email protected] (S. Benke). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.08.011
The semi-solid alloy shows marked differences in the deformation behavior under shear, compressive and tensile loading. Until now very little is known about the tensile behavior of the mush as most experimental work has been carried out on compressive and shear tests, see [1] for a broad overview. Tensile tests are difficult to carry out, because the measured stresses are low and the amount of strain until the dendrite skeleton looses its load carrying capacity is very small. Additionally, the deformation in the partially melted zone is not homogeneous due to temperature gradients along the specimen. Thus, the strain needs to be measured in a small area of the melted region. In literature one can find several studies which performed tensile tests on aluminium alloys within the general framework of hot tearing research [2–6]. These investigations show a steep drop of the ultimate tensile strength and strain to fracture with increasing liquid fraction entering the semi-solid range. At liquid fractions larger than 0.35 both values gradually drop further. A sophisticated isothermal two phase model was developed by Martin et al. [7– 9] in order to describe the behavior of semi-solid metals using based on the theory of porous media. The model introduces a visco-plastic potential which couples the deviatoric and hydrostatic material properties and is able to describe the material behavior under compression, shear and tension. The material parameters have been identified for a Sn–Pb alloy and an Al–Cu alloy with both a dendritic and a globular structure [9]. For an AA5182 alloy Ludwig et al. [10] identified the parameter for this model by using
634
S. Benke et al. / Computational Materials Science 45 (2009) 633–637
translation shear and tensile tests. For the tensile tests the stress versus strain curves are not identified since only the length change is measured in the experiments. The parameter identification is done by using numerical methods. The aim of this work is the experimental characterization of the thermo-elasto-visco-plastic behavior of the mush at several temperatures between the coherency temperature and the solidus temperature and to develop a numerical model for the material behavior. Therefore, uniaxial tensile and compression tests were carried out at solid fractions between 0.8 and 1.0. A previously developed model [11] is extended to capture the strength difference between tension and compression observed in the experiments. In the current model we focus on the small strain region before break-down due to grain rearrangement and very local deformation at grain boundaries including the loss of coherency of the solid dendrite network.
σ
555°C
6
560°C
4 2
570°C
0
565 °C
−2 −4 −6 −8 − 0.3
560°C − 0.2
− 0. 1
0
0.1
0.2
0.3
0.4
es
Fig. 1. True stress–strain curves at various solid factions f s .
2. Experimental 2.1. Material The material investigated is an A356 aluminium alloy with the composition as given in Table 1. The samples were taken from die cast billets. No additional heat treatment was applied to the billets and specimens after casting. 2.2. Solid fraction curves The solidus and liquidus temperatures as well as the fraction solid curves f s ðTÞ were determined by differential scanning caliometry (DSC) using a heating cycle as in the mechanical testing and with the Scheil model of ThermoCalc [12]. Both solid fraction curves coincident very well, giving a solidus temperature of T s ¼ 559 C and a liquidus temperature of T l ¼ 611 C. 2.3. Mechanical testing The uniaxial tensile and compressive testing of the A356 alloy in the semi-solid regime was performed with a hot forming simulator described in reference [13]. The apparatus integrates an inductive heating and gas cooling device into a vertical tensile testing machine. The samples are heated by a double wound inductor. Through the inner winding He-gas can be directed on the sample for cooling. A thermocouple was placed on the surface of the specimen to control the thermal history of the remelted zone. The force was measured via a two-range load cell for high and low loads. This way, a very small scatter in the measured stresses is achieved. The data logging of the simulator is supported by a computer based control system. The strains in the remelted zone were measured using a ‘Laser Speckle Extensometer’ (LSE) which enables the determination of the local true strain optically through the surrounding inductor at the hot specimens surface. Using this technique a laser spot is projected onto the surface of the specimen. Video capturing software follows the motion of the reflecting pattern and calculates the strain rate in the illuminated region. With the information about the stress and strain conditions during testing, it is possible to determine true stress-true strain curves in the semi-solid range of specimens with low liquid fractions.
Table 1 Weight percentage of the main alloying elements (wt.%) Si
Fe
Cu
Mn
Mg
Zn
7.31
0.112
0.002
0.007
0.357
0.0034
The tests were conducted starting from the solid regime by first heating the specimen with a rate of 1 K/s to the defined testing temperature between the coherency temperature 585 °C and the solidus temperature of 559 °C [13]. After waiting for 120 s to achieve a thermal equilibrium in the remelted zone, the tension and compression tests were conducted at constant strain rates of e_ ¼ 0:001/s, e_ ¼ 0:003/s and e_ ¼ 0:005/s. The repeatability of data, especially at higher liquid fractions in tension, was a challenge due to the small occurring strength and was successful for the highest strain rate of e_ ¼ 0:005/s. For compression tests, the repeatability of data was easier to realize for all strain rates. 2.4. Results The true stress versus true strain curves obtained for the tensile and compressive tests performed at temperature values of 560 °C (f s ¼ 0:99), 565 °C (f s ¼ 0:93) and 570 °C (f s ¼ 0:8) are presented in Fig. 1 as well as the stress–strain curve for the fully solidified state of the alloy at 555 °C. The stresses and strains are regarded positive in tension and negative in compression. The stress–strain curves show that the maximum stress grows with increasing solid fraction. In addition, two different behaviors in compression and tension are observed: in tension the tensile stress is quite large before fraction but drops very rapidly at the initiation of fracture which occurs at very low strains. Whereas for compression the material behaves softer and the transition from elastic to a viscoplastic behavior is very smooth showing no distinct yield limit as it is observed in tension. After reaching the maximum stress a stress plateau can be observed and the overall load bearing capacity in compression is higher compared to tension.
3. Modeling 3.1. Theoretical framework The adopted model for the macroscopic modeling of the rheological behavior of the mush is summarized hereafter. More details can be found in the reference [11]. The coherent mush is treated as a saturated two phase porous medium consisting of a solid dendrite network us and a viscous melt ul in the pore space. In the framework of the theory of porous media both constituents are assumed to be immiscible and in an ideal state of disarrangement. This leads to a model of superimposed and interacting continua where the local geometrical structure at the current position x at time t is represented by scalar volume fractions:
S. Benke et al. / Computational Materials Science 45 (2009) 633–637
f a ¼ f a ðx; tÞ ¼
dva dv
a ¼ l; s:
ð1Þ
They relate the volume elements of the real phases dva to the volume of the control space dv. Assuming fully saturated conditions, the saturation constraint yields:
f s þ f l ¼ 1:
ð2Þ
The model under consideration incorporates the independent fields, the solid displacement us , the seepage velocity of the melt vls , the fluid pressure p, the chemical composition in the solid cs and the melt cl and the common temperature T ¼ T s ¼ T l of the mixture. We take into account the exchange of mass, momentum, energy and species between the both constituents. The corresponding equations for quasi-static deformation processes can be obtained from the kinematic relationships, the balance equations and the constitutive material models as discussed in detail in [11]. The governing equations are namely the balance of momentum, mass, energy for the mixture and the species balance for solid and liquid as well as the Darcy law for the fluid. In this work we assume both constituents as incompressible on the microscale. Thus, the real densities qaR are only a function of the temperature. Taking the mass exchange between liquid and solid due to the progressing solidification into account, the constitutive relations for the partial Cauchy stresses ra are given by:
ows rs ¼ f s p þ qs s 1 þ rsE ; of ! l l l l ow 1: r ¼ f p þ q of l
ð3Þ ð4Þ
where p denotes the fluid pressure, 1 is the second order identity tensor, rsE is the effective stress in the solid dendrite network, wa are the free Helmholtz energy functions and qa ¼ f a qaR are the effective densities of the constituents. For isothermal deformations without mass exchange expression (3) simplifies to rs ¼ f s p 1 þ rsE . Restricting to isotropic permeability conditions occurring during an equiaxed solidification and assuming a spherical shape of the solid particles with a mean diameter d, the Darcy permeability tensor Kl is given by the Carman–Kozeny relation:
es ¼ es;th þ es;el þ es;vp :
ðf l Þ3 d
180 mlR ð1
f l Þ2
1:
ð5Þ
rsE ¼ f s ðK sR ðTÞ I1 ðes;el Þ 1 þ 2 lsR ðTÞ edev s;el Þ:
ð6Þ
It describes the heat flux on the macroscale induced by an applied temperature gradient using the effective heat conductivity ka of constituent ua . a The solute flux j in the constituent ua is expressed in terms of the gradient of the concentration ca by the first Fick law: a
j ¼ K ac grada ca :
ð9Þ
where the realistic bulk modulus K sR and the shear modulus lsR of the effective solid material depends on the temperature and edev s;el ¼ Idev : es;el characterizes the deviatoric part of the strain tensor. The unit tensor Idev is defined by: Idev ¼ I 13 1 1 using the fourth order unit tensor I. The thermal expansion of a volume element of the solidified skeleton is given by:
_ e_ s;th ¼ asT 1 T:
ð10Þ
In order to simulate the deformation processes during solidification the underlying visco-plastic model has to describe the change of the microstructure form the coherent dendrite network to the solid body continuously with the help of macroscopic structural variables. For the development of a visco-plastic flow potential we follow the concept exploited by Mahnken [14] for rate independent plasticity by the use of a power series for the invariants of the extra stress [15]. The visco-plastic potential X is defined by
XðpÞ ¼
nþ1 I2 ðsdev Þ þ HN ½p D½p : nþ1 g½p
g½p
ð11Þ
The vector p collects the temperature T, the solid fraction f s , the equivalent plastic strain epeq and further possible process variables. The tensor s defines the reduced stress s ¼ rsE S, where S has the representation of a back stress tensor describing the strength difference in tension and compression. The notation sdev characterizes the deviatoric part of the stress tensor sdev ¼ Idev : s. In Eq. (11), g denotes the viscosity parameter and n is the Norton exponent. The inelastic strain rate is given by an associated flow rule:
e_ s;vp ¼
oX : os
ð12Þ
The function HN represents the visco-plastic potential in the hydrostatic plane:
HN ¼
N X
ai ½p Ii1 ½s:
ð13Þ
i¼0
Here, mlR defines the effective melt viscosity. The heat flux in each constituent is determined by the classical Fourier law:
qa ¼ ka grada T:
ð8Þ
The effective elastic stress of the solid is governed by the isotropic Hooke law:
2
Kl ¼
635
ð7Þ
Here, K ac specifies the effective diffusion coefficient of ua . The mass exchange is determined by a solid faction as described in Section 2.2. 3.2. Thermo-visco-plastic material law Since the deformation in casting processes is mainly driven by thermal expansion, we follow the geometrical linear approach. The linearized strain tensor of the solid es is additively decomposed into thermal ðÞs;th , elastic ðÞs;el and visco-plastic ðÞs;vp parts:
Here, HN ¼ H2 is defined as a second order function similar to the yield potential of Green for the rate independent plasticity of porous metals [16]:
H2 ¼ a0 ðf s ; T; epeq Þ þ a2 ðf s Þ I21 ½s:
ð14Þ
The parameter a0 describes the influence of strain hardening. Here, a combination of a saturation term and a linear hardening term will be used:
1 a0 ¼ ðY 0 þ Qðepeq ; TÞÞ2 ðf s Þ2 ; 3
ð15Þ
where
Qðepeq ; TÞ ¼ qð1 expðb epeq ÞÞ þ H epeq :
ð16Þ
The parameters Y 0 , q, b and H are temperature dependent material constants. The parameter a2 is a function of the fraction solid and reflects the internal cohesion of the solid dendrite network due to dendrite interlocking and the presence of liquid films. Here, we assume, that a2 is a linear function of the fraction solid, satisfying a2 ðT s Þ ¼ 0 which leads to the classical von Mises plasticity for the solid body
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S. Benke et al. / Computational Materials Science 45 (2009) 633–637
and a2 ðT c Þ ¼ 1=3 to obtain zero plastic Poisson ratio at the coherency point. In general the function D controls the deviation from a circle towards the compression and extension radii in the octahedral plane. Due to the lack of experimental data D is currently set to 1. It can be shown that the model includes the visco-plastic potential of [7,9] as a special case [15]. In the experiments we observe that the strength difference in tension and compression depends on the solid fraction. We assume that the evolution law for the back-stress S is an extension of the Prager law:
S_ ¼ c1 e_ s;vp c2 ðf s Þ f_s 1;
ð17Þ
where S0 being the initial value of the back-stress. The parameter c2 measures the response of the microstructure of the dendrite network to the applied load due to buckling of the dendrites in compression and the straightening of the dendrites in tension [15]. 4. Parameter identification Using the implementation of the numerical solution scheme as described in [11] into a finite element software, we simulate the uniaxial tension and compression experiment (see Section 2) in order to identify the model parameters. The used thermo-mechanical data are given in Table 2. The initial yield- and deformation free back-stress due to the initial strength difference as a function of the fraction solid f s is given in Fig. 2. Entering the semi-solid temperature range the yield stress drops from 3.87 to 0.027 MPa at f s ¼ 0:80. Between the measured points the yield strength is interpolated smoothly. The relative strength difference between tension and compression vanishes at solidus temperature and represents a maximum at the coherency
Table 2 Thermo-mechanical data for A356 Parameter
Temperature (°C)
s
f K sR sR
l
a1 q b H n
g
555
560
565
570
1.00 916.7 1375 0.00 2.009 963 18.72 4.74 1.176
0.99 916.7 1375 0.013 1.98 568 108.674 4.81 1.232
0.93 916.7 1375 0.077 0.347 1515 80.435 4.86 1.292
0.80 916.7 1375 0.141 0.009 1892 4.82 4.93 1.354
σ 4
σs
E
560°C
6 4 2
570°C
0
565°C
_2 −4
560°C
−6 − 0.015
−0.01
−0.005
0
0.005
0.01
es
Fig. 3. Calculated stress–strain curves compared to experimental data.
temperature. The absolute value of the back-stress is maximal around a solid fraction of f s ¼ 0:98. Here, the dendritic network is able to sustain some tensile stress but the ligaments of the network still tend to buckle in compression. At higher solid factions the ability to sustain compressive loadings increases and the ligaments get stiffer and more connected [15]. In Fig. 3 the experimental and calculated stress–strain curves are displayed. Here, we consider only small strains before break-down occurs. This is characterized by grain rearrangement and very local deformation at grain boundaries and by the loss of coherency of the dendrite network. 5. Conclusion In this contribution a model was developed for the description of the mechanical behavior of a semi-solid metal, including the effects of isotropic hardening, pressure dependence of yielding, rate dependent visco-plasticity and the strength difference in tension and compression. The development is motivated by the experimental investigation of the semi-solid aluminium alloy A356 in uniaxial tension and compression. The measured true stress-true strain curves for the semi-solid mush show in tension a distinct yield limit and a loss of the load bearing capacity of the solid dendrite network at very small strains. In compression the yield limit is lower as in tension and the transition from elasticity to inelasticity shows no distinct yield limit. For the strength difference an evolution law which depends on the solid fraction has been formulated. The simulation of the uniaxial experiments shows that the model is able to reproduce the experimental findings. For the strain rates observed in casting, the build up of liquid pressure is relatively small. Thus, the deformation is mainly driven by the deformation of the solid dendrite network and the flow of the liquid has only small impact on the overall viscous response of the mush.
3.5
Acknowledgement
3 2.5
The authors thank the Deutsche Forschungsgemeinschaft (DFG) for the financial support within the joint research project PAK222: ‘‘Thermo-mechanical modeling and characterization of the solid– fluid interactions in casting processes”.
2
Y0
1.5 1
S0 References
0.5 0 1
0.98 0.96 0.94
0.92
0.9
0.88 0.86 0.84 0.82
Fig. 2. Yield stress and back-stress as a function of f s .
fs
[1] D.G. Eskin, Suyitno, L. Katgerman, Prog. Mater. Sci. 49 (2004) 629–711. [2] B. Forest, S. Berovici, in: Solidification Technology in the Foundry and Cast House, TMS, London, UK, 1983, pp. 607–612. [3] L. Ohm, S. Engler, Giessereiforschung (1990).
S. Benke et al. / Computational Materials Science 45 (2009) 633–637 [4] G.K. Sigworth, AFS Trans. 106 (1996) 1053–1059. [5] B. Magnin, L. Maenner, L. Katgerman, S. Engler, Mater. Sci. Forum 217,222 (1996) 1209–1214. [6] W.M. van Haaften, W.H. Kool, L. Katgerman, Mater. Sci. Eng. A 336 (1–2) (2002) 1–6. [7] C.L. Martin, D. Favier, M. Suery, Int. J. Plast. 3 (13) (1997) 215–235. [8] C.L. Martin, D. Favier, M. Suery, Int. J. Plast. 3 (13) (1997) 237–259. [9] C.L. Martin, D. Favier, M. Suery, Int. J. Plast. 15 (1999) 981–1008.
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[10] O. Ludwig, J.-M. Drezet, P. Meneses, C.M. Martin, M. Suery, Mater. Sci. Eng. A (2005) 174–179. [11] S. Benke, G. Laschet, Comp. Mater. Sci. 43 (2008) 92–100. [12] www.thermocalc.se. [13] S. Dziallach, S. Benke, U. Prahl, W. Bleck, Int. J. Cast Met. Res., (2008), in press. [14] R. Mahnken, Int. J. Plast. 18 (2002) 801–831. [15] S. Benke, G. Laschet, Solid State Phenom. 141–143 (2008) 653–658. [16] R.J. Green, Int. J. Mech. Sci. 14 (1972) 215–224.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Modeling of the uniaxial tensile and compression behavior of semi-solid A356 alloys S. Benke a,*, S. Dziallach b, G. Laschet a, U. Prahl b, W. Bleck b a b
Access e. V., RWTH-Aachen, Intzestrasse 5, 52072 Aachen, Germany Institute of Ferrous Metallurgy, RWTH-Aachen, Intzestrasse 1, 52072 Aachen, Germany
a r t i c l e
i n f o
Article history: Received 30 December 2007 Received in revised form 21 July 2008 Accepted 21 August 2008 Available online 14 October 2008 PACS: 46.15.Cc Keywords: Semi-solid state Theory of porous media Finite element method Casting Solidification
a b s t r a c t Uniaxial tensile and compressive tests with the industrial aluminium alloy A356 were carried out at various temperatures in the semi-solid regime to determine the constitutive behavior of the alloy during the solidification in casting processes. The experimental findings show a pronounced strength difference in compression and tension as a function of the solid volume fraction. The semi-solid alloy is modeled on the macroscale as a visco-plastic porous medium saturated with liquid. A generalized creep law is modified to take into account the geometry of the pore space, the coherency of the dendrites and the response of the material to the different loading directions. This provides a continuous description of the constitutive behavior of the semi-solid alloy from the coherency temperature to the creep regime of the fully solidified state. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction During casting processes, most technical alloys solidify with the formation of a two phase region known as the mushy zone in a finite temperature interval. The solidification starts below liquidus temperature by local undercooling and the nucleation of solid germs takes place. At low solid fractions, the solid germs grow independently and are completely free to move. The alloy behaves like a suspension. At some point, the dendrites agglomerate and form a more or less continuous skeleton, which behaves like a fluid saturated porous medium. Below this temperature, defined as the coherency temperature, the semi-solid is able to sustain low tensile strains. Liquid feeding remains possible to accommodate externally applied strains, solidification shrinkage and thermally induced deformations. In general the coherency is not only determined by the temperature and fraction solid but also dependent on the morphology and the size of the solidified particles. In the mushy region below the coherency temperature, the mechanical deformation is influenced by the visco-plastic solid and the flow of the remaining viscous melt in the pore-space. Especially at high solid fractions and high strain rates a considerable liquid pressure will build up which leads together with the inelastic solid to a complex rate dependent constitutive behavior of the mush. * Corresponding author. Tel.: +49 2418098016. E-mail address: [email protected] (S. Benke). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.08.011
The semi-solid alloy shows marked differences in the deformation behavior under shear, compressive and tensile loading. Until now very little is known about the tensile behavior of the mush as most experimental work has been carried out on compressive and shear tests, see [1] for a broad overview. Tensile tests are difficult to carry out, because the measured stresses are low and the amount of strain until the dendrite skeleton looses its load carrying capacity is very small. Additionally, the deformation in the partially melted zone is not homogeneous due to temperature gradients along the specimen. Thus, the strain needs to be measured in a small area of the melted region. In literature one can find several studies which performed tensile tests on aluminium alloys within the general framework of hot tearing research [2–6]. These investigations show a steep drop of the ultimate tensile strength and strain to fracture with increasing liquid fraction entering the semi-solid range. At liquid fractions larger than 0.35 both values gradually drop further. A sophisticated isothermal two phase model was developed by Martin et al. [7– 9] in order to describe the behavior of semi-solid metals using based on the theory of porous media. The model introduces a visco-plastic potential which couples the deviatoric and hydrostatic material properties and is able to describe the material behavior under compression, shear and tension. The material parameters have been identified for a Sn–Pb alloy and an Al–Cu alloy with both a dendritic and a globular structure [9]. For an AA5182 alloy Ludwig et al. [10] identified the parameter for this model by using
634
S. Benke et al. / Computational Materials Science 45 (2009) 633–637
translation shear and tensile tests. For the tensile tests the stress versus strain curves are not identified since only the length change is measured in the experiments. The parameter identification is done by using numerical methods. The aim of this work is the experimental characterization of the thermo-elasto-visco-plastic behavior of the mush at several temperatures between the coherency temperature and the solidus temperature and to develop a numerical model for the material behavior. Therefore, uniaxial tensile and compression tests were carried out at solid fractions between 0.8 and 1.0. A previously developed model [11] is extended to capture the strength difference between tension and compression observed in the experiments. In the current model we focus on the small strain region before break-down due to grain rearrangement and very local deformation at grain boundaries including the loss of coherency of the solid dendrite network.
σ
555°C
6
560°C
4 2
570°C
0
565 °C
−2 −4 −6 −8 − 0.3
560°C − 0.2
− 0. 1
0
0.1
0.2
0.3
0.4
es
Fig. 1. True stress–strain curves at various solid factions f s .
2. Experimental 2.1. Material The material investigated is an A356 aluminium alloy with the composition as given in Table 1. The samples were taken from die cast billets. No additional heat treatment was applied to the billets and specimens after casting. 2.2. Solid fraction curves The solidus and liquidus temperatures as well as the fraction solid curves f s ðTÞ were determined by differential scanning caliometry (DSC) using a heating cycle as in the mechanical testing and with the Scheil model of ThermoCalc [12]. Both solid fraction curves coincident very well, giving a solidus temperature of T s ¼ 559 C and a liquidus temperature of T l ¼ 611 C. 2.3. Mechanical testing The uniaxial tensile and compressive testing of the A356 alloy in the semi-solid regime was performed with a hot forming simulator described in reference [13]. The apparatus integrates an inductive heating and gas cooling device into a vertical tensile testing machine. The samples are heated by a double wound inductor. Through the inner winding He-gas can be directed on the sample for cooling. A thermocouple was placed on the surface of the specimen to control the thermal history of the remelted zone. The force was measured via a two-range load cell for high and low loads. This way, a very small scatter in the measured stresses is achieved. The data logging of the simulator is supported by a computer based control system. The strains in the remelted zone were measured using a ‘Laser Speckle Extensometer’ (LSE) which enables the determination of the local true strain optically through the surrounding inductor at the hot specimens surface. Using this technique a laser spot is projected onto the surface of the specimen. Video capturing software follows the motion of the reflecting pattern and calculates the strain rate in the illuminated region. With the information about the stress and strain conditions during testing, it is possible to determine true stress-true strain curves in the semi-solid range of specimens with low liquid fractions.
Table 1 Weight percentage of the main alloying elements (wt.%) Si
Fe
Cu
Mn
Mg
Zn
7.31
0.112
0.002
0.007
0.357
0.0034
The tests were conducted starting from the solid regime by first heating the specimen with a rate of 1 K/s to the defined testing temperature between the coherency temperature 585 °C and the solidus temperature of 559 °C [13]. After waiting for 120 s to achieve a thermal equilibrium in the remelted zone, the tension and compression tests were conducted at constant strain rates of e_ ¼ 0:001/s, e_ ¼ 0:003/s and e_ ¼ 0:005/s. The repeatability of data, especially at higher liquid fractions in tension, was a challenge due to the small occurring strength and was successful for the highest strain rate of e_ ¼ 0:005/s. For compression tests, the repeatability of data was easier to realize for all strain rates. 2.4. Results The true stress versus true strain curves obtained for the tensile and compressive tests performed at temperature values of 560 °C (f s ¼ 0:99), 565 °C (f s ¼ 0:93) and 570 °C (f s ¼ 0:8) are presented in Fig. 1 as well as the stress–strain curve for the fully solidified state of the alloy at 555 °C. The stresses and strains are regarded positive in tension and negative in compression. The stress–strain curves show that the maximum stress grows with increasing solid fraction. In addition, two different behaviors in compression and tension are observed: in tension the tensile stress is quite large before fraction but drops very rapidly at the initiation of fracture which occurs at very low strains. Whereas for compression the material behaves softer and the transition from elastic to a viscoplastic behavior is very smooth showing no distinct yield limit as it is observed in tension. After reaching the maximum stress a stress plateau can be observed and the overall load bearing capacity in compression is higher compared to tension.
3. Modeling 3.1. Theoretical framework The adopted model for the macroscopic modeling of the rheological behavior of the mush is summarized hereafter. More details can be found in the reference [11]. The coherent mush is treated as a saturated two phase porous medium consisting of a solid dendrite network us and a viscous melt ul in the pore space. In the framework of the theory of porous media both constituents are assumed to be immiscible and in an ideal state of disarrangement. This leads to a model of superimposed and interacting continua where the local geometrical structure at the current position x at time t is represented by scalar volume fractions:
S. Benke et al. / Computational Materials Science 45 (2009) 633–637
f a ¼ f a ðx; tÞ ¼
dva dv
a ¼ l; s:
ð1Þ
They relate the volume elements of the real phases dva to the volume of the control space dv. Assuming fully saturated conditions, the saturation constraint yields:
f s þ f l ¼ 1:
ð2Þ
The model under consideration incorporates the independent fields, the solid displacement us , the seepage velocity of the melt vls , the fluid pressure p, the chemical composition in the solid cs and the melt cl and the common temperature T ¼ T s ¼ T l of the mixture. We take into account the exchange of mass, momentum, energy and species between the both constituents. The corresponding equations for quasi-static deformation processes can be obtained from the kinematic relationships, the balance equations and the constitutive material models as discussed in detail in [11]. The governing equations are namely the balance of momentum, mass, energy for the mixture and the species balance for solid and liquid as well as the Darcy law for the fluid. In this work we assume both constituents as incompressible on the microscale. Thus, the real densities qaR are only a function of the temperature. Taking the mass exchange between liquid and solid due to the progressing solidification into account, the constitutive relations for the partial Cauchy stresses ra are given by:
ows rs ¼ f s p þ qs s 1 þ rsE ; of ! l l l l ow 1: r ¼ f p þ q of l
ð3Þ ð4Þ
where p denotes the fluid pressure, 1 is the second order identity tensor, rsE is the effective stress in the solid dendrite network, wa are the free Helmholtz energy functions and qa ¼ f a qaR are the effective densities of the constituents. For isothermal deformations without mass exchange expression (3) simplifies to rs ¼ f s p 1 þ rsE . Restricting to isotropic permeability conditions occurring during an equiaxed solidification and assuming a spherical shape of the solid particles with a mean diameter d, the Darcy permeability tensor Kl is given by the Carman–Kozeny relation:
es ¼ es;th þ es;el þ es;vp :
ðf l Þ3 d
180 mlR ð1
f l Þ2
1:
ð5Þ
rsE ¼ f s ðK sR ðTÞ I1 ðes;el Þ 1 þ 2 lsR ðTÞ edev s;el Þ:
ð6Þ
It describes the heat flux on the macroscale induced by an applied temperature gradient using the effective heat conductivity ka of constituent ua . a The solute flux j in the constituent ua is expressed in terms of the gradient of the concentration ca by the first Fick law: a
j ¼ K ac grada ca :
ð9Þ
where the realistic bulk modulus K sR and the shear modulus lsR of the effective solid material depends on the temperature and edev s;el ¼ Idev : es;el characterizes the deviatoric part of the strain tensor. The unit tensor Idev is defined by: Idev ¼ I 13 1 1 using the fourth order unit tensor I. The thermal expansion of a volume element of the solidified skeleton is given by:
_ e_ s;th ¼ asT 1 T:
ð10Þ
In order to simulate the deformation processes during solidification the underlying visco-plastic model has to describe the change of the microstructure form the coherent dendrite network to the solid body continuously with the help of macroscopic structural variables. For the development of a visco-plastic flow potential we follow the concept exploited by Mahnken [14] for rate independent plasticity by the use of a power series for the invariants of the extra stress [15]. The visco-plastic potential X is defined by
XðpÞ ¼
nþ1 I2 ðsdev Þ þ HN ½p D½p : nþ1 g½p
g½p
ð11Þ
The vector p collects the temperature T, the solid fraction f s , the equivalent plastic strain epeq and further possible process variables. The tensor s defines the reduced stress s ¼ rsE S, where S has the representation of a back stress tensor describing the strength difference in tension and compression. The notation sdev characterizes the deviatoric part of the stress tensor sdev ¼ Idev : s. In Eq. (11), g denotes the viscosity parameter and n is the Norton exponent. The inelastic strain rate is given by an associated flow rule:
e_ s;vp ¼
oX : os
ð12Þ
The function HN represents the visco-plastic potential in the hydrostatic plane:
HN ¼
N X
ai ½p Ii1 ½s:
ð13Þ
i¼0
Here, mlR defines the effective melt viscosity. The heat flux in each constituent is determined by the classical Fourier law:
qa ¼ ka grada T:
ð8Þ
The effective elastic stress of the solid is governed by the isotropic Hooke law:
2
Kl ¼
635
ð7Þ
Here, K ac specifies the effective diffusion coefficient of ua . The mass exchange is determined by a solid faction as described in Section 2.2. 3.2. Thermo-visco-plastic material law Since the deformation in casting processes is mainly driven by thermal expansion, we follow the geometrical linear approach. The linearized strain tensor of the solid es is additively decomposed into thermal ðÞs;th , elastic ðÞs;el and visco-plastic ðÞs;vp parts:
Here, HN ¼ H2 is defined as a second order function similar to the yield potential of Green for the rate independent plasticity of porous metals [16]:
H2 ¼ a0 ðf s ; T; epeq Þ þ a2 ðf s Þ I21 ½s:
ð14Þ
The parameter a0 describes the influence of strain hardening. Here, a combination of a saturation term and a linear hardening term will be used:
1 a0 ¼ ðY 0 þ Qðepeq ; TÞÞ2 ðf s Þ2 ; 3
ð15Þ
where
Qðepeq ; TÞ ¼ qð1 expðb epeq ÞÞ þ H epeq :
ð16Þ
The parameters Y 0 , q, b and H are temperature dependent material constants. The parameter a2 is a function of the fraction solid and reflects the internal cohesion of the solid dendrite network due to dendrite interlocking and the presence of liquid films. Here, we assume, that a2 is a linear function of the fraction solid, satisfying a2 ðT s Þ ¼ 0 which leads to the classical von Mises plasticity for the solid body
636
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and a2 ðT c Þ ¼ 1=3 to obtain zero plastic Poisson ratio at the coherency point. In general the function D controls the deviation from a circle towards the compression and extension radii in the octahedral plane. Due to the lack of experimental data D is currently set to 1. It can be shown that the model includes the visco-plastic potential of [7,9] as a special case [15]. In the experiments we observe that the strength difference in tension and compression depends on the solid fraction. We assume that the evolution law for the back-stress S is an extension of the Prager law:
S_ ¼ c1 e_ s;vp c2 ðf s Þ f_s 1;
ð17Þ
where S0 being the initial value of the back-stress. The parameter c2 measures the response of the microstructure of the dendrite network to the applied load due to buckling of the dendrites in compression and the straightening of the dendrites in tension [15]. 4. Parameter identification Using the implementation of the numerical solution scheme as described in [11] into a finite element software, we simulate the uniaxial tension and compression experiment (see Section 2) in order to identify the model parameters. The used thermo-mechanical data are given in Table 2. The initial yield- and deformation free back-stress due to the initial strength difference as a function of the fraction solid f s is given in Fig. 2. Entering the semi-solid temperature range the yield stress drops from 3.87 to 0.027 MPa at f s ¼ 0:80. Between the measured points the yield strength is interpolated smoothly. The relative strength difference between tension and compression vanishes at solidus temperature and represents a maximum at the coherency
Table 2 Thermo-mechanical data for A356 Parameter
Temperature (°C)
s
f K sR sR
l
a1 q b H n
g
555
560
565
570
1.00 916.7 1375 0.00 2.009 963 18.72 4.74 1.176
0.99 916.7 1375 0.013 1.98 568 108.674 4.81 1.232
0.93 916.7 1375 0.077 0.347 1515 80.435 4.86 1.292
0.80 916.7 1375 0.141 0.009 1892 4.82 4.93 1.354
σ 4
σs
E
560°C
6 4 2
570°C
0
565°C
_2 −4
560°C
−6 − 0.015
−0.01
−0.005
0
0.005
0.01
es
Fig. 3. Calculated stress–strain curves compared to experimental data.
temperature. The absolute value of the back-stress is maximal around a solid fraction of f s ¼ 0:98. Here, the dendritic network is able to sustain some tensile stress but the ligaments of the network still tend to buckle in compression. At higher solid factions the ability to sustain compressive loadings increases and the ligaments get stiffer and more connected [15]. In Fig. 3 the experimental and calculated stress–strain curves are displayed. Here, we consider only small strains before break-down occurs. This is characterized by grain rearrangement and very local deformation at grain boundaries and by the loss of coherency of the dendrite network. 5. Conclusion In this contribution a model was developed for the description of the mechanical behavior of a semi-solid metal, including the effects of isotropic hardening, pressure dependence of yielding, rate dependent visco-plasticity and the strength difference in tension and compression. The development is motivated by the experimental investigation of the semi-solid aluminium alloy A356 in uniaxial tension and compression. The measured true stress-true strain curves for the semi-solid mush show in tension a distinct yield limit and a loss of the load bearing capacity of the solid dendrite network at very small strains. In compression the yield limit is lower as in tension and the transition from elasticity to inelasticity shows no distinct yield limit. For the strength difference an evolution law which depends on the solid fraction has been formulated. The simulation of the uniaxial experiments shows that the model is able to reproduce the experimental findings. For the strain rates observed in casting, the build up of liquid pressure is relatively small. Thus, the deformation is mainly driven by the deformation of the solid dendrite network and the flow of the liquid has only small impact on the overall viscous response of the mush.
3.5
Acknowledgement
3 2.5
The authors thank the Deutsche Forschungsgemeinschaft (DFG) for the financial support within the joint research project PAK222: ‘‘Thermo-mechanical modeling and characterization of the solid– fluid interactions in casting processes”.
2
Y0
1.5 1
S0 References
0.5 0 1
0.98 0.96 0.94
0.92
0.9
0.88 0.86 0.84 0.82
Fig. 2. Yield stress and back-stress as a function of f s .
fs
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