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Time–temperature superposition for foaming kinetics of Al-alloy foams
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
journal homepage: www.elsevier.com/locate/jmatprotec
Time–temperature superposition for foaming kinetics of Al-alloy foams Amkee Kim a,∗ , Kazi Tunvir a , Seung-Hoon Nahm b , Seong-Seock Cho c a b c
Division of Mechanical and Automotive Engineering, Kongju National University, Kongju, Chungnam 314-701, Republic of Korea Division of Metrology for Quality Life, Korea Research Institute of Standards and Science, Yuseong, Daejeon 305-340, Republic of Korea School of Materials Engineering, Chungnam National University, Yuseong, Daejeon 305-764, Republic of Korea
a r t i c l e
i n f o
a b s t r a c t
Article history:
Applicability of time–temperature superposition principle to the foaming kinetics of alu-
Received 31 March 2007
minum (Al)-alloy foams produced by powder metallurgical method was investigated.
Received in revised form
Foaming kinetics above melting temperatures of Al–Si–Cu–Mg foams was studied. The
19 August 2007
expansion data at various furnace temperatures were collected. Well-known superposition
Accepted 1 October 2007
parameters such as Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby and Manson-Succop were established based on the linear iso-expansion lines in plots of log(heating time) versus furnace temperature and log(heating time) versus inverse furnace temperature. In order
Keywords:
to study the expansion kinetics of the Al-alloy foams, the expansions were measured in
Time–temperature superposition
terms of pore fraction using an image analyzer. Finally, the linear relationship between the
Aluminum alloy foam
porosity and the superposition parameters was established.
Pore fraction
© 2007 Elsevier B.V. All rights reserved.
Foaming kinetics Powder metallurgy
1.
Introduction
Metal foams have gained a growing interest in automotive and aerospace industries due to their ultra-light weight and other attractive mechanical characteristics such as high specific strength, high impact energy absorption, high damping capacity and high sound absorption capability. Powder metallurgical method for producing closed cell Alfoams from metal powder has been developed. The process consists of mixing the metal and foaming agent powder, and compressing them to a dense precursor which is foamed in the furnace. The porosity of foams has great influence on the mechanical, electrical and thermal properties (Ashby et al., 2000). The density of Al-foams is controlled by many parameters such as the alloy composition, the type of foaming agent, the quality of alloy powder, the morphology and distribution
∗
Corresponding author. Tel.: +82 41 850 8616; fax: +82 41 854 1449. E-mail address: [email protected] (A. Kim). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.10.001
of foaming agent, the compaction pressure, and the time and temperature during foaming process (Degischer and Cottar, 1999). Particularly, time and temperature for foaming of Alalloy are main parameters for the desired porosity. In the case, the simple analysis for evolution of pores in the foam based on phenomenological method will be more practical to avoid the difficulties in determining the individual input parameters although the well-designed theoretical modeling could ¨ provide the details of the evolution (Korner et al., 2002). The superposition principle between heating time and furnace temperature may allow one to control the porosity of foam using the superposition parameters. Time–temperature superposition, often called as the method of reduced variable, is a well-known procedure frequently applied either to determine the temperature dependency of the rheological behavior of material or to expand the time or frequency
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
regime at a given temperature. Time–temperature superposition principle was firstly noticed in the late 1930s in an experimental study of viscoelastic behavior of polymer and polymer fluids (Vinogradov and Malkin, 1980; Tobolsky, 1967). Further studies indicated that the time–temperature superposition could be explained theoretically by some molecular structure models (Phan-Thien, 1979). Afterwards, for decades several time–temperature superposition parameters such as William-Landel-Ferry, Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby, White-Le May, Manson-Succop, MansonHaferd, etc. have appeared in the literature based on different phenomenological hypothesis regarding material behavior (Jeon et al., 2002; Sobrinho and Bueno, 2005; Sen et al., 2002; Alves et al., 2004; Koo and Kim, 2005; Vaidyanathan et al., 2003; Hadid et al., 2004; Strganac and Golden, 1996; Usami et al., 1999; Feng et al., 2005; Pang and Fancey, 2006; Goertzen and Kessler, 2006; Greco et al., 2007). All of these methods have been so far used mostly to predict creep behavior of materials. An interesting application of time–temperature superposition principle which is not associated with the creep can be found in literature (Assink et al., 2005) where the extent of oxygen depletion of low density polyurethane foams aged for various lengths of time was measured. The oxidations of the
451
foam were successfully calculated by using time–temperature superposition parameters such as William-Landel-Ferry. In this paper the time–temperature superposition on the expansion kinetics of Al–Si–Cu–Mg (alloy 322 and alloy 544) foams produced by powder metallurgical method was attempted following the phenomenological analogy to the creep rupture.
2.
Experimental procedure
The commercial grade aluminum alloy powders of chemical composition Al–3 wt.%Si–2 wt.%Cu–2 wt.%Mg (alloy 322) and Al–5 wt.%Si–4 wt.%Cu–4 wt.%Mg (alloy 544) were produced by centrifugal atomization. The particle size of alloy powders ranged from 150 to 900 m, while the shape of the powder particles was ligament type. Ninety-nine percent by weight of Al–Si–Cu–Mg alloy powder and 1% by weight of TiH2 particles were mixed in a rotating V-mixer at a velocity of 300 rpm for 30 min. The decomposition temperature of the foaming agent, TiH2 is 460 ◦ C. In order to facilitate handling slugs and avoid the de-mixing and contamination of the powder, the mixture of alloy powders was cold compacted to a cylindrical billet (with 65% of theoretical density)
Fig. 1 – (a) Microscopic 2D image of alloy 544 foam heated for 8 min at 650 ◦ C and of 80% pore volume fraction, (b) binarised image of the original image shown in (a), (c) microscopic 2D image of alloy 322 foam heated for 6 min at 800 ◦ C and of 73% pore volume fraction and (d) binarised image of the original image shown in (c).
452
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
under the pressure of 4 MPa for 1 min in a metallic mould (152 mm diameter × 304 mm length). Then the compacted billet was hot extruded into rod precursor of diameter 12 mm through eight circular holes at 430 ◦ C with an extrusion ratio of 20:1 by a uni-axial direct extrusion machine having the capacity of 20 MN. Graphite suspension was used as a lubricant. Each of the extruded Al–Si–Cu–Mg alloy rods was foamed by heating them in presence of air, in a preheated furnace at pre-set temperature, above the melting temperatures (630 ◦ C for alloy 322 and 600 ◦ C for alloy 544) of the alloys (Kim et al., 2004). The pre-set furnace temperature for foaming was varied between 650 and 800 ◦ C. Each of the extruded rods (12 mm diameter and 30 mm long) of both the alloys was then placed on the carbon steel plate and foamed in the furnace without any mould. The foaming heat treatment was terminated by simply removing the samples from the furnace after holding for different time periods and then cooled to the room temperature in open air. The detailed foaming process can be also found in literature (Kim et al., 2004). The samples after cooling were bisected along their center lines parallel to the extrusion direction, and the images of areas (12 mm × 15 mm) were taken from the central portions of the cross sections by a microscope as shown in Fig. 1a and c. To evaluate the pore fraction, the binarisation of image (Fig. 1b and d) was performed in such a way that the cell walls are seen to be white while the rest to be black. A ratio of black area to total area of image was then measured as a pore fraction of foam indicating the extent of expansion of foam. Fig. 2a and b represents the evolution behavior of the pore fraction at various pre-set furnace temperatures for alloy 322 and 544 foams respectively.
3.
Superposition parameters
Superposition methodologies such as William-Landel-Ferry (Li, 2000), Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby, White-Le May, Manson-Succop, Manson-Haferd, etc. (Sobrinho and Bueno, 2005; Larson and Miller, 1952) appeared in the literature based on different phenomenological hypothesis of material behavior for trade between the time (tr ) for rupture and the absolute temperature (T). All these methods were so far proposed to analyze mostly creep data of materials. The apparent analogy of a creep rupture strain to a desired porosity of the foam at different temperatures was assumed by similarity between two different experimental curves, i.e. rupture strain versus time in creep (Fig. 3) and pore volume fraction versus heating time in the foaming process (Fig. 2) in this study. Thus the above-mentioned time–temperature superposition methodologies might be applied to obtain a foam with desired density. Creep is a time-based behavior of material since it is not an intrinsic material response and it is highly dependent on environment including temperature and ambient condition. The basic results of a creep test are the strain versus time curves which links creep to an applied stress, and is also defined as a time-based performance test of material under constant stress. This creep process to rupture time under a constant stress is often accelerated by increasing ambient temperature. The typical strain versus time curves for creep data of 2.25Cr–1Mo steel (Levi et al., 2005) until
Fig. 2 – Pore evolution (a) alloy 322 and (b) alloy 544 foams.
Fig. 3 – Creep data of 2.25 Cr–1 Mo steel at elevated temperature.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
453
Sherby-Dorn are identified by the analysis of the patterns of the iso-pore evolution lines in plots of log(th ) versus 1/T in the following way: (a) Larson-Miller: Assumes the convergence of the iso-pore evolution lines in a point located in the log(th ) axis equal to A. The parameter is given by: PLM = T[A + log(th )]
(1)
where th is the heating time duration in minute and T is the absolute furnace temperature. (b) Orr-Sherby-Dorn: Is based on the parallelism of the isopore evolution lines with slope equal to B. The parameter is given by: POSD = log(th ) −
B T
(2)
(c) Goldhoff-Sherby: Presupposes the convergence of iso-pore evolution lines in a point with coordinates (1/Ta , log(ta )) located below the region of experimental data, being the
Fig. 4 – Variation of log(th ) versus (a) inverse temperature and (b) temperature with iso-pore fraction for alloy 322.
rupture at different temperatures are depicted in Fig. 3. The foaming is also a time-based behavior of material, and the basic result is the expansion versus time curve. The foaming process to a desired expansion of the material can also be accelerated by increasing the furnace temperature providing that the other parameters remain unchanged (Fig. 2a and b). From the figures, a similarity between pore fraction versus heating time curves at various temperatures (Fig. 2) and strain versus loading time curves at various temperatures in creep (Fig. 3) can be observed except for the deceleration of expansion at last portions of curves in Fig. 2. In creep, the Larson-Miller, Orr-Sherby-Dorn, GoldhoffSherby and White-Le May (Sobrinho and Bueno, 2005) are identified by the analysis of the pattern of iso-stress lines in plots of log(tr ) versus 1/T while the Manson-Succop and Manson-Haferd are identified by the analysis of the pattern of iso-stress lines in plots of log(tr ) versus T. The abovementioned methods often show good consistency with the creep deformation process occurring at both low and high temperatures and offer reasonable way to predict the lengthy creep kinetics. The parameters of Larson-Miller and Orr-
Fig. 5 – Variation of log(th ) versus (a) inverse temperature and (b) temperature with iso-pore fraction for alloy 544.
454
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
parameter given by:
PGS =
log(th ) − log(ta ) 1/T − 1/Ta
given by:
(d) White-Le May: Assumes the convergence of iso-pore evolution lines in a point with coordinates (1/Tw , log(tw )), above the region of experimental data, being the parameter given by:
PWL =
1/T − 1/Tw log(th ) − log(tw )
(4)
The method of Manson-Succop and Manson-Haferd, on the other hand, is identified by the analysis of the pattern of the iso-pore evolution lines in plot log(th ) versus T in the following way: (e) Manson-Succop: Is based on the parallelism of the iso-pore evolution lines with slope equal to C. The parameter is given by: PMS = log(th ) + CT
PMH =
(3)
(5)
(f) Manson-Haferd: Presupposes the convergence of iso-pore evolution lines in a point with coordinates (log(tm ), Tm ), above the region of experimental data, being the parameter
4.
T − Tm log(th ) − log(tm )
(6)
Result and discussion
Fig. 2a and b represents the evolution behavior of the pore fraction at various pre-set furnace temperatures for alloy 322 and 544 foams respectively. As one can see, the foaming characteristics depend on the furnace temperature chosen. Apparently, a certain excess temperature above the melting point drives the pore evolution. The reason for this is the decrease in viscosity and the faster decomposition of foaming agent when the temperature is increased. The steeper slopes of the pore fraction versus heating time curves for higher furnace temperatures are the evidence of this fact. Following a nucleation of crack-like pores, the fast expansion of pores is observed in the curves. The deceleration of expansion due to the exhaustion of foaming agent comes after that. In fact, some swollen pores collapse as the trapped gas in the swollen pores is given off the material after all. The lower and upper parts (indicated as dotted lines in Fig. 2a and b) of each curve stand for the nucleation of crack-like pores and collapse stages of pore evolution respectively. As the 2D pore fraction, measured through the image analyzing technique, can be considered representative of the 3D pore volume fraction only in the case of
Fig. 6 – Expansion kinetics of alloy 322 and 544 foams based on time–temperature superposition parameters: (a) Goldhoff-Sherby, (b) Larson-Miller, (c) Orr-Sherby-Dorn and (d) Manson-Succop.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
Table 1 – Constants in equations Foams
Alloy 322 Alloy 544
5.
Conclusions
Constants A
B
C
1.80 1.13
2570 1690
0.0028 0.0018
well-developed foams with spherical pores, the data for nucleation of crack-like pores and collapse stages were excluded for the analysis. Figs. 4 and 5 depict the linear regression lines of isopore fraction for alloy 322 and 544 foams in the plots of log(th ) versus inverse furnace temperature (1/T) and temperature (T) respectively. As mentioned, Goldhoff-Sherby, White-Le May and Manson-Haferd methods are based on the most convergent points (indicated as black dots in Figs. 4 and 5) of iso-pore evolution lines above or below the experimental data. The convergence points of coordinates (1/Ta = 6.36 × 10−4 , log(ta ) = −0.153 for alloy 322) and (1/Ta = 8.11 × 10−4 , log(ta ) = 0.258 for alloy 544) in Figs. 4a and 5a satisfy the convergence condition for only Goldhoff-Sherby method. Other convergent points for both the foams in the figures fail to satisfy the convergence conditions for WhiteLe May and Manson-Haferd methods because of the absence of convergence of iso-pore evolution lines above the region of experimental data. On the other hand, since the intersections on log(th ) axis and the parallelism of slopes for the constants in Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters were not also determined uniquely from the figures, the average values of constants (as intersections and slopes) for Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters were calculated and utilized in this study. The average values for constants in Eqs. (1), (2) and (5) are contained in Table 1. Fig. 6a shows the relationship between the pore fraction and the Goldhoff-Sherby parameter for both the foams based on the most convergent points of iso-pore fraction lines referring to the log(th ) versus 1/T diagrams (Figs. 4a and 5a) while Fig. 6b–d represents the relationships between the pore fraction and Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters based on intersection or slope respectively. The good correlations between the pore fraction and the parameters are presented by all the aforementioned parameters. Fig. 6 also indicates that the linear kinetics is apparent in terms of Goldhoff-Sherby, Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters for alloy 322 and 544 foams, which can be described by the following equation. Fp = MPTTS + N
455
(7)
Here, FP and PTTS are pore fraction and time–temperature superposition parameters respectively while M and N are constants which may depend on the properties of precursor for the Al-alloy foams such as the alloy composition, the type of foaming agent, the quality of alloy powder, the morphology and distribution of foaming agent, the compaction pressure, etc.
Foaming kinetics above the melting temperature of Al–Si–Cu–Mg alloy foams in the powder metallurgical method was studied using time–temperature superposition principle. Well-known superposition parameters such as Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby and Manson-Succop were established for the of Al-alloy foams based on the linear isoexpansion lines in plots of log(heating time) versus furnace temperature and log(heating time) versus inverse furnace temperature. The parameters turned out to be linearly correlated with the expansion of the Al-alloy foams. Noticing that the furnace temperature and the heating time can be easily controlled by manufacturers in producing the Al foams, the superposition parameters may allow one to obtain the Al-alloy foam with a desired density more efficiently.
Acknowledgement The authors wish to acknowledge the financial support of the Korea Science and Engineering Foundation by grant no. R012002-000-00093-0(2002) from its basic research program, and the BK 21.
references
´ ´ Alves, N.M., Gomez Ribelles, J.F., Gomez Tejedor, J.A., 2004. Departure from the vogel behavior in the glass transition—thermally stimulated recovery. Creep Dyn. Mech. Anal. Stud. Polym. 45, 1007–1017. Ashby, M.F., Evans, A.G., Fleck, N.A., Gibson, L.J., Hutchinson, J.W., Wadley, H.N.G., 2000. Metal Foams: A Design Guide. Butterworth Heinemann, Boston. Assink, R.A., Celina, M., Skutnik, J.M., Harris, D.J., 2005. Use of a respirometer to measure oxidation rates of polymeric materials at ambient temperatures. Polymer 46, 11648–11654. Degischer, H.P., Cottar, A., 1999. On the non-destructive testing of metal foams. In: Banhart, J., Ashby, M.F., Fleck, N.A. (Eds.), Proc. Int. Conf. on Metal Foams and Porous Structure. Verlag MIT Publishing, Bremen, USA, pp. 213–220. Feng, C.W., Keong, C.W., Hsueh, Y.P., Wang, Y.Y., Sue, H.J., 2005. Modeling of long-term creep behavior of structural epoxy adhesives. Int. J. Adhesion Adhesives 25, 427–436. Goertzen, W.W., Kessler, M.R., 2006. Creep behavior of carbon fiber/epoxy matrix composites. Mater. Sci. Eng. A 421, 217–225. Greco, A., Musardo, C., Maffezzoli, A., 2007. Flexural creep behavior of PP matrix woven composite. Comps. Sci. Technol. 67, 1148–1158. Hadid, M., Rechak, S., Tati, A., 2004. Long-term bending creep behavior prediction of injection molded composites using stress–time correspondence principle. Mater. Sci. Eng. A 385, 54–58. Jeon, H.Y., Kim, S.H., Yoo, H.K., 2002. Assessment of long-term performances of polyester geogrids by accelerated creep test. Polym. Test. 21, 489–495. Kim, A., Cho, S.S., Lee, H.J., 2004. Foaming behavior of Al–Si–Cu–Mg alloys. Mater. Sci. Technol. 20, 1615–1620. Koo, H.J., Kim, Y.K., 2005. Lifetime prediction of geogrids for reinforcement of embankments and slopes. Polym. Test. 24, 181–188. ¨ Korner, C., Thies, M., Singer, R.F., 2002. Modeling of metal foaming with lattice Boltzmann automata. Adv. Eng. Mater. 4, 765–769.
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Larson, F.R., Miller, E.J., 1952. Time–temperature relationship for rupture and creep stresses. Trans. ASME 74, 765–775. Levi, O.B., Vitor, L.S., Luiz, M., 2005. Constant load creep data in air and vacuum on 2.25Cr–1Mo steel from 600 ◦ C to 700 ◦ C. Mater. Res. 8, 401–408. Li, R., 2000. Time–temperature superposition method for glass transition temperature of plastic materials. Mater. Sci. Eng. A 278, 36–45. Pang, J.W.C., Fancey, K.S., 2006. An investigation into long-term viscoelastic recovery of Nylon 6, 6 fibres through accelerated ageing. Mater. Sci. Eng. A 431, 100–105. Phan-Thien, N., 1979. On the time–temperature superposition principle of dilute polymer liquids. J. Rheol. 23 (4), 451–456. Sen, A., Bhattacharya, M., Stelon, K.A., Voller, V.R., 2002. Creep in injection molded starch/synthetic polymer blends. Mater. Sci. Eng. A 338, 60–69. Sobrinho, J.F.D.R., Bueno, L.O., 2005. Correlation between creep and hot tensile behavior for 2.25Cr–1Mo steel from 500 ◦ C to
700 ◦ C. Part 2: An assessment according to different ´ 10 (3), parameterization methodologies. Revista Mater. 463–471. Strganac, T.W., Golden, H.J., 1996. Predictions of nonlinear viscoelastic behavior using a hybrid approach. Int. J. Solids Struct. 33 (30), 4561–4570. Tobolsky, A.V., 1967. Properties and Structure of Polymers. Wiley, New York, pp. 144–159. Usami, S., Suzuki, T., Ejima, H., Asano, K., 1999. Thermo-mechanical properties of epoxy gfrps used in superconducting magnet winding. Cryogenics 39, 905–914. Vaidyanathan, T.K., Vaidyanathan, J., Cherian, Z., 2003. Extended creep behavior of dental composites using time–temperature superposition principle. Dental Mater. 19, 46–53. Vinogradov, G.V., Malkin, A.Y., 1980. Rheology of Polymers. Mir, Moscow, pp. 238–248.
journal homepage: www.elsevier.com/locate/jmatprotec
Time–temperature superposition for foaming kinetics of Al-alloy foams Amkee Kim a,∗ , Kazi Tunvir a , Seung-Hoon Nahm b , Seong-Seock Cho c a b c
Division of Mechanical and Automotive Engineering, Kongju National University, Kongju, Chungnam 314-701, Republic of Korea Division of Metrology for Quality Life, Korea Research Institute of Standards and Science, Yuseong, Daejeon 305-340, Republic of Korea School of Materials Engineering, Chungnam National University, Yuseong, Daejeon 305-764, Republic of Korea
a r t i c l e
i n f o
a b s t r a c t
Article history:
Applicability of time–temperature superposition principle to the foaming kinetics of alu-
Received 31 March 2007
minum (Al)-alloy foams produced by powder metallurgical method was investigated.
Received in revised form
Foaming kinetics above melting temperatures of Al–Si–Cu–Mg foams was studied. The
19 August 2007
expansion data at various furnace temperatures were collected. Well-known superposition
Accepted 1 October 2007
parameters such as Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby and Manson-Succop were established based on the linear iso-expansion lines in plots of log(heating time) versus furnace temperature and log(heating time) versus inverse furnace temperature. In order
Keywords:
to study the expansion kinetics of the Al-alloy foams, the expansions were measured in
Time–temperature superposition
terms of pore fraction using an image analyzer. Finally, the linear relationship between the
Aluminum alloy foam
porosity and the superposition parameters was established.
Pore fraction
© 2007 Elsevier B.V. All rights reserved.
Foaming kinetics Powder metallurgy
1.
Introduction
Metal foams have gained a growing interest in automotive and aerospace industries due to their ultra-light weight and other attractive mechanical characteristics such as high specific strength, high impact energy absorption, high damping capacity and high sound absorption capability. Powder metallurgical method for producing closed cell Alfoams from metal powder has been developed. The process consists of mixing the metal and foaming agent powder, and compressing them to a dense precursor which is foamed in the furnace. The porosity of foams has great influence on the mechanical, electrical and thermal properties (Ashby et al., 2000). The density of Al-foams is controlled by many parameters such as the alloy composition, the type of foaming agent, the quality of alloy powder, the morphology and distribution
∗
Corresponding author. Tel.: +82 41 850 8616; fax: +82 41 854 1449. E-mail address: [email protected] (A. Kim). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.10.001
of foaming agent, the compaction pressure, and the time and temperature during foaming process (Degischer and Cottar, 1999). Particularly, time and temperature for foaming of Alalloy are main parameters for the desired porosity. In the case, the simple analysis for evolution of pores in the foam based on phenomenological method will be more practical to avoid the difficulties in determining the individual input parameters although the well-designed theoretical modeling could ¨ provide the details of the evolution (Korner et al., 2002). The superposition principle between heating time and furnace temperature may allow one to control the porosity of foam using the superposition parameters. Time–temperature superposition, often called as the method of reduced variable, is a well-known procedure frequently applied either to determine the temperature dependency of the rheological behavior of material or to expand the time or frequency
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
regime at a given temperature. Time–temperature superposition principle was firstly noticed in the late 1930s in an experimental study of viscoelastic behavior of polymer and polymer fluids (Vinogradov and Malkin, 1980; Tobolsky, 1967). Further studies indicated that the time–temperature superposition could be explained theoretically by some molecular structure models (Phan-Thien, 1979). Afterwards, for decades several time–temperature superposition parameters such as William-Landel-Ferry, Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby, White-Le May, Manson-Succop, MansonHaferd, etc. have appeared in the literature based on different phenomenological hypothesis regarding material behavior (Jeon et al., 2002; Sobrinho and Bueno, 2005; Sen et al., 2002; Alves et al., 2004; Koo and Kim, 2005; Vaidyanathan et al., 2003; Hadid et al., 2004; Strganac and Golden, 1996; Usami et al., 1999; Feng et al., 2005; Pang and Fancey, 2006; Goertzen and Kessler, 2006; Greco et al., 2007). All of these methods have been so far used mostly to predict creep behavior of materials. An interesting application of time–temperature superposition principle which is not associated with the creep can be found in literature (Assink et al., 2005) where the extent of oxygen depletion of low density polyurethane foams aged for various lengths of time was measured. The oxidations of the
451
foam were successfully calculated by using time–temperature superposition parameters such as William-Landel-Ferry. In this paper the time–temperature superposition on the expansion kinetics of Al–Si–Cu–Mg (alloy 322 and alloy 544) foams produced by powder metallurgical method was attempted following the phenomenological analogy to the creep rupture.
2.
Experimental procedure
The commercial grade aluminum alloy powders of chemical composition Al–3 wt.%Si–2 wt.%Cu–2 wt.%Mg (alloy 322) and Al–5 wt.%Si–4 wt.%Cu–4 wt.%Mg (alloy 544) were produced by centrifugal atomization. The particle size of alloy powders ranged from 150 to 900 m, while the shape of the powder particles was ligament type. Ninety-nine percent by weight of Al–Si–Cu–Mg alloy powder and 1% by weight of TiH2 particles were mixed in a rotating V-mixer at a velocity of 300 rpm for 30 min. The decomposition temperature of the foaming agent, TiH2 is 460 ◦ C. In order to facilitate handling slugs and avoid the de-mixing and contamination of the powder, the mixture of alloy powders was cold compacted to a cylindrical billet (with 65% of theoretical density)
Fig. 1 – (a) Microscopic 2D image of alloy 544 foam heated for 8 min at 650 ◦ C and of 80% pore volume fraction, (b) binarised image of the original image shown in (a), (c) microscopic 2D image of alloy 322 foam heated for 6 min at 800 ◦ C and of 73% pore volume fraction and (d) binarised image of the original image shown in (c).
452
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
under the pressure of 4 MPa for 1 min in a metallic mould (152 mm diameter × 304 mm length). Then the compacted billet was hot extruded into rod precursor of diameter 12 mm through eight circular holes at 430 ◦ C with an extrusion ratio of 20:1 by a uni-axial direct extrusion machine having the capacity of 20 MN. Graphite suspension was used as a lubricant. Each of the extruded Al–Si–Cu–Mg alloy rods was foamed by heating them in presence of air, in a preheated furnace at pre-set temperature, above the melting temperatures (630 ◦ C for alloy 322 and 600 ◦ C for alloy 544) of the alloys (Kim et al., 2004). The pre-set furnace temperature for foaming was varied between 650 and 800 ◦ C. Each of the extruded rods (12 mm diameter and 30 mm long) of both the alloys was then placed on the carbon steel plate and foamed in the furnace without any mould. The foaming heat treatment was terminated by simply removing the samples from the furnace after holding for different time periods and then cooled to the room temperature in open air. The detailed foaming process can be also found in literature (Kim et al., 2004). The samples after cooling were bisected along their center lines parallel to the extrusion direction, and the images of areas (12 mm × 15 mm) were taken from the central portions of the cross sections by a microscope as shown in Fig. 1a and c. To evaluate the pore fraction, the binarisation of image (Fig. 1b and d) was performed in such a way that the cell walls are seen to be white while the rest to be black. A ratio of black area to total area of image was then measured as a pore fraction of foam indicating the extent of expansion of foam. Fig. 2a and b represents the evolution behavior of the pore fraction at various pre-set furnace temperatures for alloy 322 and 544 foams respectively.
3.
Superposition parameters
Superposition methodologies such as William-Landel-Ferry (Li, 2000), Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby, White-Le May, Manson-Succop, Manson-Haferd, etc. (Sobrinho and Bueno, 2005; Larson and Miller, 1952) appeared in the literature based on different phenomenological hypothesis of material behavior for trade between the time (tr ) for rupture and the absolute temperature (T). All these methods were so far proposed to analyze mostly creep data of materials. The apparent analogy of a creep rupture strain to a desired porosity of the foam at different temperatures was assumed by similarity between two different experimental curves, i.e. rupture strain versus time in creep (Fig. 3) and pore volume fraction versus heating time in the foaming process (Fig. 2) in this study. Thus the above-mentioned time–temperature superposition methodologies might be applied to obtain a foam with desired density. Creep is a time-based behavior of material since it is not an intrinsic material response and it is highly dependent on environment including temperature and ambient condition. The basic results of a creep test are the strain versus time curves which links creep to an applied stress, and is also defined as a time-based performance test of material under constant stress. This creep process to rupture time under a constant stress is often accelerated by increasing ambient temperature. The typical strain versus time curves for creep data of 2.25Cr–1Mo steel (Levi et al., 2005) until
Fig. 2 – Pore evolution (a) alloy 322 and (b) alloy 544 foams.
Fig. 3 – Creep data of 2.25 Cr–1 Mo steel at elevated temperature.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
453
Sherby-Dorn are identified by the analysis of the patterns of the iso-pore evolution lines in plots of log(th ) versus 1/T in the following way: (a) Larson-Miller: Assumes the convergence of the iso-pore evolution lines in a point located in the log(th ) axis equal to A. The parameter is given by: PLM = T[A + log(th )]
(1)
where th is the heating time duration in minute and T is the absolute furnace temperature. (b) Orr-Sherby-Dorn: Is based on the parallelism of the isopore evolution lines with slope equal to B. The parameter is given by: POSD = log(th ) −
B T
(2)
(c) Goldhoff-Sherby: Presupposes the convergence of iso-pore evolution lines in a point with coordinates (1/Ta , log(ta )) located below the region of experimental data, being the
Fig. 4 – Variation of log(th ) versus (a) inverse temperature and (b) temperature with iso-pore fraction for alloy 322.
rupture at different temperatures are depicted in Fig. 3. The foaming is also a time-based behavior of material, and the basic result is the expansion versus time curve. The foaming process to a desired expansion of the material can also be accelerated by increasing the furnace temperature providing that the other parameters remain unchanged (Fig. 2a and b). From the figures, a similarity between pore fraction versus heating time curves at various temperatures (Fig. 2) and strain versus loading time curves at various temperatures in creep (Fig. 3) can be observed except for the deceleration of expansion at last portions of curves in Fig. 2. In creep, the Larson-Miller, Orr-Sherby-Dorn, GoldhoffSherby and White-Le May (Sobrinho and Bueno, 2005) are identified by the analysis of the pattern of iso-stress lines in plots of log(tr ) versus 1/T while the Manson-Succop and Manson-Haferd are identified by the analysis of the pattern of iso-stress lines in plots of log(tr ) versus T. The abovementioned methods often show good consistency with the creep deformation process occurring at both low and high temperatures and offer reasonable way to predict the lengthy creep kinetics. The parameters of Larson-Miller and Orr-
Fig. 5 – Variation of log(th ) versus (a) inverse temperature and (b) temperature with iso-pore fraction for alloy 544.
454
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
parameter given by:
PGS =
log(th ) − log(ta ) 1/T − 1/Ta
given by:
(d) White-Le May: Assumes the convergence of iso-pore evolution lines in a point with coordinates (1/Tw , log(tw )), above the region of experimental data, being the parameter given by:
PWL =
1/T − 1/Tw log(th ) − log(tw )
(4)
The method of Manson-Succop and Manson-Haferd, on the other hand, is identified by the analysis of the pattern of the iso-pore evolution lines in plot log(th ) versus T in the following way: (e) Manson-Succop: Is based on the parallelism of the iso-pore evolution lines with slope equal to C. The parameter is given by: PMS = log(th ) + CT
PMH =
(3)
(5)
(f) Manson-Haferd: Presupposes the convergence of iso-pore evolution lines in a point with coordinates (log(tm ), Tm ), above the region of experimental data, being the parameter
4.
T − Tm log(th ) − log(tm )
(6)
Result and discussion
Fig. 2a and b represents the evolution behavior of the pore fraction at various pre-set furnace temperatures for alloy 322 and 544 foams respectively. As one can see, the foaming characteristics depend on the furnace temperature chosen. Apparently, a certain excess temperature above the melting point drives the pore evolution. The reason for this is the decrease in viscosity and the faster decomposition of foaming agent when the temperature is increased. The steeper slopes of the pore fraction versus heating time curves for higher furnace temperatures are the evidence of this fact. Following a nucleation of crack-like pores, the fast expansion of pores is observed in the curves. The deceleration of expansion due to the exhaustion of foaming agent comes after that. In fact, some swollen pores collapse as the trapped gas in the swollen pores is given off the material after all. The lower and upper parts (indicated as dotted lines in Fig. 2a and b) of each curve stand for the nucleation of crack-like pores and collapse stages of pore evolution respectively. As the 2D pore fraction, measured through the image analyzing technique, can be considered representative of the 3D pore volume fraction only in the case of
Fig. 6 – Expansion kinetics of alloy 322 and 544 foams based on time–temperature superposition parameters: (a) Goldhoff-Sherby, (b) Larson-Miller, (c) Orr-Sherby-Dorn and (d) Manson-Succop.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 450–456
Table 1 – Constants in equations Foams
Alloy 322 Alloy 544
5.
Conclusions
Constants A
B
C
1.80 1.13
2570 1690
0.0028 0.0018
well-developed foams with spherical pores, the data for nucleation of crack-like pores and collapse stages were excluded for the analysis. Figs. 4 and 5 depict the linear regression lines of isopore fraction for alloy 322 and 544 foams in the plots of log(th ) versus inverse furnace temperature (1/T) and temperature (T) respectively. As mentioned, Goldhoff-Sherby, White-Le May and Manson-Haferd methods are based on the most convergent points (indicated as black dots in Figs. 4 and 5) of iso-pore evolution lines above or below the experimental data. The convergence points of coordinates (1/Ta = 6.36 × 10−4 , log(ta ) = −0.153 for alloy 322) and (1/Ta = 8.11 × 10−4 , log(ta ) = 0.258 for alloy 544) in Figs. 4a and 5a satisfy the convergence condition for only Goldhoff-Sherby method. Other convergent points for both the foams in the figures fail to satisfy the convergence conditions for WhiteLe May and Manson-Haferd methods because of the absence of convergence of iso-pore evolution lines above the region of experimental data. On the other hand, since the intersections on log(th ) axis and the parallelism of slopes for the constants in Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters were not also determined uniquely from the figures, the average values of constants (as intersections and slopes) for Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters were calculated and utilized in this study. The average values for constants in Eqs. (1), (2) and (5) are contained in Table 1. Fig. 6a shows the relationship between the pore fraction and the Goldhoff-Sherby parameter for both the foams based on the most convergent points of iso-pore fraction lines referring to the log(th ) versus 1/T diagrams (Figs. 4a and 5a) while Fig. 6b–d represents the relationships between the pore fraction and Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters based on intersection or slope respectively. The good correlations between the pore fraction and the parameters are presented by all the aforementioned parameters. Fig. 6 also indicates that the linear kinetics is apparent in terms of Goldhoff-Sherby, Larson-Miller, Orr-Sherby-Dorn and Manson-Succop parameters for alloy 322 and 544 foams, which can be described by the following equation. Fp = MPTTS + N
455
(7)
Here, FP and PTTS are pore fraction and time–temperature superposition parameters respectively while M and N are constants which may depend on the properties of precursor for the Al-alloy foams such as the alloy composition, the type of foaming agent, the quality of alloy powder, the morphology and distribution of foaming agent, the compaction pressure, etc.
Foaming kinetics above the melting temperature of Al–Si–Cu–Mg alloy foams in the powder metallurgical method was studied using time–temperature superposition principle. Well-known superposition parameters such as Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby and Manson-Succop were established for the of Al-alloy foams based on the linear isoexpansion lines in plots of log(heating time) versus furnace temperature and log(heating time) versus inverse furnace temperature. The parameters turned out to be linearly correlated with the expansion of the Al-alloy foams. Noticing that the furnace temperature and the heating time can be easily controlled by manufacturers in producing the Al foams, the superposition parameters may allow one to obtain the Al-alloy foam with a desired density more efficiently.
Acknowledgement The authors wish to acknowledge the financial support of the Korea Science and Engineering Foundation by grant no. R012002-000-00093-0(2002) from its basic research program, and the BK 21.
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