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Relation between local necking and cavitation during high-temperature tensile deformation of polycrystalline pure aluminum
Materials Science and Engineering A 526 (2009) 118–122
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Relation between local necking and cavitation during high-temperature tensile deformation of polycrystalline pure aluminum T. Morita a,∗ , Y. Sakurai b , Y. Tanaka c , T. Hasegawa b a b c
Department of Chemistry, Faculty of Education, Aichi University of Education, Kariya, Aichi 448-8542, Japan Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan Division of Diversity Science, Graduate School of Science and Technology, Chiba University, Yayoi, Inage-ku, Chiba 263-8522, Japan
a r t i c l e
i n f o
Article history: Received 1 April 2009 Received in revised form 1 July 2009 Accepted 6 July 2009
Keywords: Cavitation Local necking Pure aluminum High temperature tensile deformation Grain boundary Small-angle X-ray scattering
a b s t r a c t For polycrystalline aluminum with grain size of 170 and 346 m, tensile deformation was carried out at a true strain rate of 1 × 10−2 /s and a temperature of 553 K (=0.59Tm , Tm : melting temperature in K). Onset and progress of local necking during the deformation were investigated by measurements of smallangle X-ray scattering (SAXS) from a viewpoint of cavity formation on grain boundaries. The cavitation was observed in the change of the SAXS intensities as a function of local strain along the gauge part of specimen. The intensities increase rather slowly during the beginning uniform deformation. However, it increases rapidly in the gauge part where the local necking has occurred. The degree of cavitation is found to be more remarkable for a specimen with smaller grain size. Cavity size and size distribution are evaluated from the SAXS profiles. The local necking phenomenon is discussed from a viewpoint of cavitation. © 2009 Elsevier B.V. All rights reserved.
1. Introduction There have been proposed the following two aspects of localized neck formation during tensile deformation of metallic materials, though correlations between them have not been discussed. One aspect is related to the deformation parameters such as strain ˙ and strain hardening parameter rate sensitivity m (=∂ ln /∂ ln ε) (=∂ ln /∂ε). Hart proposed the relation for uniform deformation as follows [1]: m + ≥ 1.
(1)
In fact, in a previous paper [2], in MM Al–1.1 at.% Mg–1.2 at.% Cu alloy, m and have been descriptively represented by several deformation parameters that have been obtained from computer simulation of stress–strain curves on basis of a dislocation dynamics viewpoint [3]. Further, the shape of the gauge part of the specimen has been continuously recorded by use of a high speed camera. It was observed that local necking starts when the Hart’s criterion given by Eq. (1) breaks down [2]. Another aspect of necking is the formation of cavities or voids during tensile deformation. It has been widely observed that, in practical metallic materials, voids are formed at the boundaries of inclusions or the second phase particles, and this leads to the local necking in the specimen
∗ Corresponding author. Tel.: +81 566 26 2352; fax: +81 566 26 2352. E-mail address: [email protected] (T. Morita). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.07.011
gauge part [4,5]. However, even in polycrystalline pure aluminum (99.99 mass%) in which few inclusions or few second phase particles were observed, local necking certainly occurs during tensile deformation at high temperatures. Further, it is concluded that the thermal recovery due to the annihilation of crystal lattice dislocations into grain boundaries strongly affects the stress–strain behavior of pure aluminum [6]. Such absorption of crystal lattice dislocations into grain boundaries may cause the grain boundary sliding to reduce local strains along grain boundaries. Since grain boundary becomes wavy during hightemperature deformation due to the interaction of grain boundaries and sub-boundaries formed near grain boundaries [7], the sliding of such wavy boundaries should cause the cavitation at grain boundaries. From works mentioned above, we believe that, even in pure aluminum, cavitation occurs at grain boundaries during high-temperature deformation, and such cavity formation should be related to the onset of local necking in tensile deformation. The present paper is concerned with measurements of cavitation and population of cavities as a function of strain by small-angle X-ray scattering (SAXS) method. The method is a powerful tool for deriving quantitative information on cavities such as size, size distribution, number and shape [8–11]. The method endures for structural investigations of voids or vacancies in addition to particles, based on the Babinet’s theorem [8]. Further, the information is comprehensively obtained from the interior of a material, as well as from the surface. A crystal defect and dislocation itself are negligible for change of SAXS signals, because the signal corresponds
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to scattering from inhomogeneous region in electron density of the order from 10 to 103 Å. Pure aluminum used for the present sample is the only metallic material of which absorption coefficient for the used X-rays is adequate for accurate measurement, and in which other unconformities except for the grain boundary almost do not exist originally. Here, we report the relation between the cavitation and the onset and growth of local necking during high-temperature tensile deformation of polycrystalline pure aluminum, studied by the SAXS method. 2. Experimental procedure 2.1. Material preparation Pure aluminum of 99.99 mass% was melted in air and cast into a mold after degassing. It was then forged and finally swaged at ambient temperature to a bar in 12 mm diameter and the swaged bar was then machined on lathe to tensile specimens with a gauge length and diameter of 20 and 8 mm, respectively. The machined specimens were then annealed at temperatures higher than the deformation temperature 553 K, for preventing the grain growth during deformation. The resultant grain sizes were 170 and 346 m. The grain was equi-axial, and the grain size was measured on an optical microscope by using a linear intercept method. The main impurities were Si, Fe and Cu of the order of 0.001 mass%. 2.2. Tensile test Tensile tests were conducted at 553 K (the ratio to the melting temperature: 0.59) on a hydraulic testing machine in which the true strain rate of 1 × 10−2 /s was maintained constant throughout straining by continuously increasing the crosshead speed for compensating the instantaneous gauge length change. During the test, the load and extension were recorded on a chart recorder and/or a storage oscilloscope, and the nominal stress n and nominal strain εn were converted into the true stress t and true strain εt by using the well-known relations, t = n (1 + εn ) and εt = ln(1 + εn ). Further, during each test, the shape change of the gauge part was continuously recorded by a high speed camera, and hence the onset of local necking could be detected from a series of photographs taken during deformation. 2.3. Evaluation of cavitation 2.3.1. Sample preparation for SAXS experiment The deformed specimen was sectioned to disks of 0.6 mm in thickness by using a wire-electrical discharge machine. The sectioned disks were mechanically polished to about 0.16 mm in thickness. Finally, the sample thickness was reduced to almost uniform thickness of 0.10 mm by the use of electrochemical polishing in a solution of 90 vol.% methanol and 10 vol.% perchloric acid. This thickness was deliberately optimized to meet the requirements for balance between sufficient intensity of scattering from cavities formed on grain boundaries and consideration of the large X-ray absorption coefficient of aluminum for used X-rays. The samples were prepared from several gauge part, such as parts of uniform deformation, onset of local necking, growth of the necking and grip of the specimen for zero strain. Fig. 1 shows an example of specimen deformed beyond the onset of local necking. Strain in each part was determined from the diameter of that part, and is defined here as local nominal strain, (εn )local . 2.3.2. Measurements of SAXS SAXS measurements were carried out using the apparatus situated at the BL-15A station [12], Photon Factory (PF) at High
Fig. 1. Photograph of the deformed specimen with grain size of 170 m after the tensile test. : part of uniform deformation, : part of growth of local necking, and : part of onset of initial necking.
Energy Accelerator Research Organization (KEK), Tsukuba. X-ray beam was monochromatized to = 1.50 Å (: wavelength) and focused to 0.7 mm × 0.7 mm at a detector (at a sample position: 0.8 mm × 0.8 mm) by a bent monochromator and by a bent mirror. A position-sensitive proportional counter (PSPC) was used for a detector. The camera length was set in 2300 mm. The observable s-region in the present measurements was from 0.01 to 0.17 Å−1 , where the scattering parameter, s, is defined as 4 sin / (2: scattering angle). The accumulation time of each run was 600 s. In the same experimental condition, the scattering intensities were measured for the case that the sample was set and for the case that the sample was not set. The net intensity of the sample from the observed intensity, Iobs (s), and the background intensity, Iback (s), is give by I(s) =
Iobs (s) − Iback (s) · exp(− l) · exp( l),
(2)
where is the linear absorption coefficient for aluminum and l is the sample thickness. Accuracy of the correction defined by Eq. (2) affects seriously the precision of the corrected SAXS intensity, I(s), since this exponentially depends on the absorption factor, l. Therefore, the precise determination of the absorption factor is inevitable. In the present measurement, system for in situ determination of the factor was applied to the SAXS experiment by use of a beam stopper with a pinhole which is covered by a thin metal foil. According to the Babinet’s theorem [8], the scattering intensity from particle with the average electron density e in void becomes the same as that from vacancies in the matrix with the average electron density e . Judging from densities of the samples, it is possible to consider that vacancies, i.e. cavities, are the “scatters” in the experiment. 3. Results and discussion 3.1. Stress–strain behavior Fig. 2 shows true stress–true strain behaviors at a true strain rate of 1 × 10−2 /s and 553 K for specimens of grain size of 170 and 346 m. Each arrow indicates a true strain at which the onset of local necking was detected on continuously taken photographs of gauge part [6]. They were 0.36 and 0.46 for grain size of 170 and 346 m, respectively. However, as may be expected, it was fairly hard to determine exactly the stage of deformation at which the macroscopic necking had occurred. Therefore, the value may
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Fig. 2. True stress t –true strain εt behaviors for tensile tests made at a true strain rate of 1 × 10−2 /s and 553 K for specimens with grain size of (a) 170 m and (b) 346 m. Each arrow indicates the strain at which onset of local necking was macroscopically observed.
possibly change in a range of ±0.05 in maximum. The local nominal strain (εn )local estimated from the well-known relation εn = eεt − 1 (εt : true strain) were 43 and 58%, respectively, which were properly nearly equal to those obtained from the diameter of the sectioned samples. Further, the strain for the onset of local necking tended to be larger for larger grain size.
Fig. 4. SAXS intensities at various local nominal strain (εn )local for specimen of grain size of 170 m (a) and of 346 m (b). The value of (εn )local for the onset of local necking is 43 and 58% for grain sizes 170 and 346 m, respectively.
3.2. Effects of strain and grain size on SAXS Fig. 3 shows the SAXS intensities at (εn )local = 0 and = 27%, which correspond to “before deformation” and “uniform deformation”, respectively. Little small-angle scattering is observed at (εn )local = 0. This evidences that the correction of absorption and subtraction
Fig. 3. SAXS intensities of polycrystalline pure aluminum with grain size of 170 m at local nominal strain (εn )local = 0 (䊉, before deformation) and (εn )local = 27% (, uniform deformation).
given by Eq. (2) is made appropriately, and significant scatters for SAXS does not exist originally in the sample material. The SAXS intensity at (εn )local = 27% slightly increases in a small s-region compared with that before deformation. This suggests that the cavity formation occurs even during the uniform deformation, though local necking is not observable macroscopically. Fig. 4 shows the SAXS intensities as a function of local nominal strain (εn )local . As shown in Fig. 4(a) for grain size of 170 m, the intensities significantly increases at (εn )local = 48%, that is just above the onset strain of local necking, 43%. Further, the intensities continue to increase up to (εn )local = 147% and then decrease slightly. Similar tendency in the change of SAXS intensities with strain can be observed for larger grain size of 346 m (Fig. 4(b)), too. In comparison with the SAXS intensity at (εn )local = 87% for grain size of 170 m, the intensity at (εn )local = 106% for grain size of 346 m is smaller especially in the small s-region, though the strain increment beyond the onset of local necking is larger for the latter. Further, for 346 m, the intensity firstly reaches to the same level of that for 170 m at (εn )local = 87% when the deformation proceeds up to (εn )local = 268%. These observations of Figs. 3 and 4 indicate the following aspects: (1) Cavitation starts during uniform deformation, namely cavities form in advance of onset of local necking. (2) Cavitation rapidly progresses in a gauge part where local necking proceeds. This may be due to the occurrence of so-called three-axial tensile stress condition in the necked part. (3) Because cavities become elongated longitudinally by local deformation as strain approaches to that of fracture, the level of
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cavitation becomes apparently no substantial from a viewpoint of SAXS data of the sectioned sample. (4) Increase in SAXS intensity during deformation occurs more rapidly for smaller grain size both in the small and the middle s-regions. Since the increase in small and middle s-regions corresponds respectively to the increase in size and in number of cavities, the above fact indicates that both the formation and the growth of cavities occur more rapidly in a specimen with smaller grain size. 3.3. Cavity size and size distribution For the common approximation to evaluate size of cavities, the Guinier approximation [8] was applied for the SAXS signals. SAXS intensity, I(s), is represented by the approximation as follows [8]:
2
I(s) = NIinc exp
−
Rg2 3
s
2
,
(3)
where N is the number of particles, Iinc the intensity of the incident X-rays, the number of excess electrons between matrix and void, and Rg the Guinier gyration radius. By taking logarithms of Eq. (3), the following relation is obtained: ln(I(s)) = ln(NIinc 2 ) −
Rg2 3
s2 .
(4)
The Guinier plot, i.e. ln(I(s)) vs. s2 , derives a value of Rg , from which the cavity size is estimated quantitatively. If the shape of cavity is supposed to be sphere, the diameter of cavity, RD , is calculated by [8]:
RD = 2
5 Rg . 3
(5)
Fig. 5 shows the Guinier plots at typical strains for grain size of 170 m. If the plot forms a straight line in the appropriate s-region, in the so-called Guinier region, the value of Rg is evaluated from the slope of the straight line. However, the plots of the present SAXS signals do not form straight lines. It is considered that cavities with a wide range of size exist especially at high strains. Size distribution of the formed cavities could be investigated by the measured SAXS spectra with the Fankuchen’s method [13]. According to the method, zero-angle scattering intensity, I(0), is divided by the following relation [13]: 2
I(0) = I1 (0) + I2 (0) + · · · + In (0) = N1 V1 2 ( 1 ) + N2 V2 2 ( 2 ) 2
+· · · + Nn Vn 2 ( n ) ,
Fig. 5. The Guinier plots of the measured SAXS intensities at typical strains for grain size of 170 m.
2
(6)
where n is the number of divisions, In (0) the intensity for the nth division, Nn the number of particles in the nth division, Vn the volume of a particle in the nth division, n the electron density difference in the nth division, respectively. Fig. 6 shows the result of deconvolution of the SAXS pattern at (εn )local = 147% for grain size of 170 m. At first, a tangent line was drawn from the Guinier plot given by Eq. (4) and then I1 (0) was estimated by the intersection with I(s)-axis. Subsequently, the tangent line was subtracted from I(s) curve, and the procedure was repeated. Volume of cavities is estimated by values of RD given by Eq. (5), and is considered to be a uniform value, namely, it is supposed that all the cavities consist of vacancies under vacuum. Fig. 7 shows the size distribution of the cavities as a function of strain in specimens of grain size of 170 and 346 m. The distribution is defined as volume ratio of the cavities, that is the ratio of total volume of cavities with a certain diameter Vn to that of summation for all cavities divided in Eq. (6), where the summation of Vn in Eq. (6) is equal to 1. For both grain sizes, the ratio of the larger size (e.g. >100 Å) increases with strain, especially beyond the onset
of localized neck formation. The amount of increase is larger for 170 m than for 346 m. Further, it is seen that most of cavities are concentrated from 10 to 25 Å in diameter in specimens of both grain size, and for the grown cavities with size of 300–400 Å is the ratio in order of 10−1 to 10−2 for grain size of 170 m while in order of 10−2 to 10−3 for 346 m. Let us consider here the effect of grain size on the early stage of cavitation. As was observed previously Al–Mg solid solution alloy [7], the serration of grain boundary and it’s sliding cause the cavity formation at the cusps of the serrated boundary. Further, the serration of grain boundary is found to be caused by the interaction with sub-boundaries which have formed by the plastic deformation due to dislocation motion in the vicinity of grain boundary. Here, the localized deformation there is certainly constrained by the existence of neighboring grains. This indicates that the deformation behavior near grain boundaries is different from that in the grain interior. Taking into consideration a fact that the wavelength of the serration (a several m) is much smaller than the grain size (a few 100 m), the deformation behavior near the grain boundaries and hence the process of grain boundary cavitation too is reasonably considered not to be largely influenced from the grain size itself.
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the deformation stage during the growth of local necking, the process of cavitation may be affected largely by the three-axial tensile stress condition formed in the necked part. 4. Concluding remarks
Fig. 6. Deconvolution of the SAXS spectrum by the Fankuchen’s method at (εn )local = 147% for grain size of 170 m. The symbol of represents the measured SAXS data. The solid curve and dashed solid lines show fitting curve and the tangent lines drawn by procedure of the analysis, respectively. Zero-angle scattering intensities, In (0), shown in Eq. (6) and the diameters of cavities are evaluated by the intersections of the tangent lines with I(s)-axis and the slope of the each line, respectively.
(1) It is proved that SAXS method is applicable to the evaluation of cavitation on grain boundaries during high-temperature tensile deformation of polycrystalline pure aluminum. (2) Judging from the change of SAXS intensities with straining, the cavitation occurs slightly even in the early uniform deformation. And then, it proceeds more remarkably at strains beyond the onset of local necking. (3) The formation and growth of cavities progresses more rapidly with straining for smaller grain size. Larger number of cavities for smaller grain size may simply be due to larger number of possible site of the grain boundary cavitation. Reason for more remarkable growth of cavities for smaller grain size must be the subject of future research. (4) The degree of cavitation may differ from place to place even in the uniform deformation, and the macroscopic local necking may occur in the gauge part where the cavitation has occurred more severely in comparison with neighboring parts. This may not contradict the fact that the onset of local necking occurs at smaller strain in specimen with smaller grain size. Acknowledgements The authors are grateful to Professor K. Okazaki for helpful discussion of cavitation, and to Professor K. Nishikawa and Dr. K. Fukuyama for helpful suggestion in analysis of SAXS data. They also thank to PF at KEK for giving them the opportunity to perform the SAXS experiments. This work was partially supported by the Light Materials Educational Foundation of Japan. One of the authors (T. M.) thanks for the support by the Iketani Science and Technology Foundation Award. References [1] [2] [3] [4] [5] [6] [7] [8]
Fig. 7. Volume ratio of the cavities vs. the cavity diameter in specimens of grain size of 170 and 346 m. Numbers on the right side show the local nominal strain (εn )local .
[9] [10]
The larger increase in the SAXS intensities and in the volume ratio of cavities (Figs. 4 and 7), and also the smaller strain at the onset of local necking for smaller grain size may be due to a larger number of site for grain boundary cavitation per unit volume of specimen. In
[11] [12] [13]
E.W. Hart, Acta Metall. 15 (1967) 351. T. Hasegawa, K. Okazaki, Mater. Sci. Eng. A297 (2001) 266. T. Hasegawa, T. Takahashi, K. Okazaki, Acta Mater. 48 (2000) 1789. T. Watanabe, Met. Trans. 14A (1983) 531. H.E. Evans, Mechanisms of Creep Fracture, Elsevier Appl. Sci. Pub., 1983, p. 25. T. Hasegawa, Y. Sakurai, K. Okazaki, Mater. Sci. Eng. A346 (2003) 34. T. Hasegawa, K. Kawaguchi, T. Murata, T. Yakou, Mater. Trans. JIM 32 (1991) 244. A. Guinier, G. Fournet (translation by C.B. Walker), Small-angle Scattering of X-rays, Wiley, New York, 1955. O. Glatter, O. Kratky (Eds.), Small Angle X-ray Scattering, Academic Press, London, 1982. K. Fukuyama, Y. Kasahara, N. Kasahara, A. Oya, K. Nishikawa, Carbon 39 (2001) 287. K. Nishikawa, Pore Structure Analyses of Carbons by Small-angle X-ray Scattering, in Carbon Alloys, Elsevier, Oxford, 2003, p. 175. Y. Amemiya, K. Wakabayashi, T. Hamanaka, T. Wakabayashi, T. Matsushita, H. Hashizume, Nucl. Instrum. Methods 208 (1983) 471. M.H. Jellinek, E. Solomon, I. Fankuchen, Ind. Eng. Chem. 18 (1946) 172.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Relation between local necking and cavitation during high-temperature tensile deformation of polycrystalline pure aluminum T. Morita a,∗ , Y. Sakurai b , Y. Tanaka c , T. Hasegawa b a b c
Department of Chemistry, Faculty of Education, Aichi University of Education, Kariya, Aichi 448-8542, Japan Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan Division of Diversity Science, Graduate School of Science and Technology, Chiba University, Yayoi, Inage-ku, Chiba 263-8522, Japan
a r t i c l e
i n f o
Article history: Received 1 April 2009 Received in revised form 1 July 2009 Accepted 6 July 2009
Keywords: Cavitation Local necking Pure aluminum High temperature tensile deformation Grain boundary Small-angle X-ray scattering
a b s t r a c t For polycrystalline aluminum with grain size of 170 and 346 m, tensile deformation was carried out at a true strain rate of 1 × 10−2 /s and a temperature of 553 K (=0.59Tm , Tm : melting temperature in K). Onset and progress of local necking during the deformation were investigated by measurements of smallangle X-ray scattering (SAXS) from a viewpoint of cavity formation on grain boundaries. The cavitation was observed in the change of the SAXS intensities as a function of local strain along the gauge part of specimen. The intensities increase rather slowly during the beginning uniform deformation. However, it increases rapidly in the gauge part where the local necking has occurred. The degree of cavitation is found to be more remarkable for a specimen with smaller grain size. Cavity size and size distribution are evaluated from the SAXS profiles. The local necking phenomenon is discussed from a viewpoint of cavitation. © 2009 Elsevier B.V. All rights reserved.
1. Introduction There have been proposed the following two aspects of localized neck formation during tensile deformation of metallic materials, though correlations between them have not been discussed. One aspect is related to the deformation parameters such as strain ˙ and strain hardening parameter rate sensitivity m (=∂ ln /∂ ln ε) (=∂ ln /∂ε). Hart proposed the relation for uniform deformation as follows [1]: m + ≥ 1.
(1)
In fact, in a previous paper [2], in MM Al–1.1 at.% Mg–1.2 at.% Cu alloy, m and have been descriptively represented by several deformation parameters that have been obtained from computer simulation of stress–strain curves on basis of a dislocation dynamics viewpoint [3]. Further, the shape of the gauge part of the specimen has been continuously recorded by use of a high speed camera. It was observed that local necking starts when the Hart’s criterion given by Eq. (1) breaks down [2]. Another aspect of necking is the formation of cavities or voids during tensile deformation. It has been widely observed that, in practical metallic materials, voids are formed at the boundaries of inclusions or the second phase particles, and this leads to the local necking in the specimen
∗ Corresponding author. Tel.: +81 566 26 2352; fax: +81 566 26 2352. E-mail address: [email protected] (T. Morita). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.07.011
gauge part [4,5]. However, even in polycrystalline pure aluminum (99.99 mass%) in which few inclusions or few second phase particles were observed, local necking certainly occurs during tensile deformation at high temperatures. Further, it is concluded that the thermal recovery due to the annihilation of crystal lattice dislocations into grain boundaries strongly affects the stress–strain behavior of pure aluminum [6]. Such absorption of crystal lattice dislocations into grain boundaries may cause the grain boundary sliding to reduce local strains along grain boundaries. Since grain boundary becomes wavy during hightemperature deformation due to the interaction of grain boundaries and sub-boundaries formed near grain boundaries [7], the sliding of such wavy boundaries should cause the cavitation at grain boundaries. From works mentioned above, we believe that, even in pure aluminum, cavitation occurs at grain boundaries during high-temperature deformation, and such cavity formation should be related to the onset of local necking in tensile deformation. The present paper is concerned with measurements of cavitation and population of cavities as a function of strain by small-angle X-ray scattering (SAXS) method. The method is a powerful tool for deriving quantitative information on cavities such as size, size distribution, number and shape [8–11]. The method endures for structural investigations of voids or vacancies in addition to particles, based on the Babinet’s theorem [8]. Further, the information is comprehensively obtained from the interior of a material, as well as from the surface. A crystal defect and dislocation itself are negligible for change of SAXS signals, because the signal corresponds
T. Morita et al. / Materials Science and Engineering A 526 (2009) 118–122
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to scattering from inhomogeneous region in electron density of the order from 10 to 103 Å. Pure aluminum used for the present sample is the only metallic material of which absorption coefficient for the used X-rays is adequate for accurate measurement, and in which other unconformities except for the grain boundary almost do not exist originally. Here, we report the relation between the cavitation and the onset and growth of local necking during high-temperature tensile deformation of polycrystalline pure aluminum, studied by the SAXS method. 2. Experimental procedure 2.1. Material preparation Pure aluminum of 99.99 mass% was melted in air and cast into a mold after degassing. It was then forged and finally swaged at ambient temperature to a bar in 12 mm diameter and the swaged bar was then machined on lathe to tensile specimens with a gauge length and diameter of 20 and 8 mm, respectively. The machined specimens were then annealed at temperatures higher than the deformation temperature 553 K, for preventing the grain growth during deformation. The resultant grain sizes were 170 and 346 m. The grain was equi-axial, and the grain size was measured on an optical microscope by using a linear intercept method. The main impurities were Si, Fe and Cu of the order of 0.001 mass%. 2.2. Tensile test Tensile tests were conducted at 553 K (the ratio to the melting temperature: 0.59) on a hydraulic testing machine in which the true strain rate of 1 × 10−2 /s was maintained constant throughout straining by continuously increasing the crosshead speed for compensating the instantaneous gauge length change. During the test, the load and extension were recorded on a chart recorder and/or a storage oscilloscope, and the nominal stress n and nominal strain εn were converted into the true stress t and true strain εt by using the well-known relations, t = n (1 + εn ) and εt = ln(1 + εn ). Further, during each test, the shape change of the gauge part was continuously recorded by a high speed camera, and hence the onset of local necking could be detected from a series of photographs taken during deformation. 2.3. Evaluation of cavitation 2.3.1. Sample preparation for SAXS experiment The deformed specimen was sectioned to disks of 0.6 mm in thickness by using a wire-electrical discharge machine. The sectioned disks were mechanically polished to about 0.16 mm in thickness. Finally, the sample thickness was reduced to almost uniform thickness of 0.10 mm by the use of electrochemical polishing in a solution of 90 vol.% methanol and 10 vol.% perchloric acid. This thickness was deliberately optimized to meet the requirements for balance between sufficient intensity of scattering from cavities formed on grain boundaries and consideration of the large X-ray absorption coefficient of aluminum for used X-rays. The samples were prepared from several gauge part, such as parts of uniform deformation, onset of local necking, growth of the necking and grip of the specimen for zero strain. Fig. 1 shows an example of specimen deformed beyond the onset of local necking. Strain in each part was determined from the diameter of that part, and is defined here as local nominal strain, (εn )local . 2.3.2. Measurements of SAXS SAXS measurements were carried out using the apparatus situated at the BL-15A station [12], Photon Factory (PF) at High
Fig. 1. Photograph of the deformed specimen with grain size of 170 m after the tensile test. : part of uniform deformation, : part of growth of local necking, and : part of onset of initial necking.
Energy Accelerator Research Organization (KEK), Tsukuba. X-ray beam was monochromatized to = 1.50 Å (: wavelength) and focused to 0.7 mm × 0.7 mm at a detector (at a sample position: 0.8 mm × 0.8 mm) by a bent monochromator and by a bent mirror. A position-sensitive proportional counter (PSPC) was used for a detector. The camera length was set in 2300 mm. The observable s-region in the present measurements was from 0.01 to 0.17 Å−1 , where the scattering parameter, s, is defined as 4 sin / (2: scattering angle). The accumulation time of each run was 600 s. In the same experimental condition, the scattering intensities were measured for the case that the sample was set and for the case that the sample was not set. The net intensity of the sample from the observed intensity, Iobs (s), and the background intensity, Iback (s), is give by I(s) =
Iobs (s) − Iback (s) · exp(− l) · exp( l),
(2)
where is the linear absorption coefficient for aluminum and l is the sample thickness. Accuracy of the correction defined by Eq. (2) affects seriously the precision of the corrected SAXS intensity, I(s), since this exponentially depends on the absorption factor, l. Therefore, the precise determination of the absorption factor is inevitable. In the present measurement, system for in situ determination of the factor was applied to the SAXS experiment by use of a beam stopper with a pinhole which is covered by a thin metal foil. According to the Babinet’s theorem [8], the scattering intensity from particle with the average electron density e in void becomes the same as that from vacancies in the matrix with the average electron density e . Judging from densities of the samples, it is possible to consider that vacancies, i.e. cavities, are the “scatters” in the experiment. 3. Results and discussion 3.1. Stress–strain behavior Fig. 2 shows true stress–true strain behaviors at a true strain rate of 1 × 10−2 /s and 553 K for specimens of grain size of 170 and 346 m. Each arrow indicates a true strain at which the onset of local necking was detected on continuously taken photographs of gauge part [6]. They were 0.36 and 0.46 for grain size of 170 and 346 m, respectively. However, as may be expected, it was fairly hard to determine exactly the stage of deformation at which the macroscopic necking had occurred. Therefore, the value may
120
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Fig. 2. True stress t –true strain εt behaviors for tensile tests made at a true strain rate of 1 × 10−2 /s and 553 K for specimens with grain size of (a) 170 m and (b) 346 m. Each arrow indicates the strain at which onset of local necking was macroscopically observed.
possibly change in a range of ±0.05 in maximum. The local nominal strain (εn )local estimated from the well-known relation εn = eεt − 1 (εt : true strain) were 43 and 58%, respectively, which were properly nearly equal to those obtained from the diameter of the sectioned samples. Further, the strain for the onset of local necking tended to be larger for larger grain size.
Fig. 4. SAXS intensities at various local nominal strain (εn )local for specimen of grain size of 170 m (a) and of 346 m (b). The value of (εn )local for the onset of local necking is 43 and 58% for grain sizes 170 and 346 m, respectively.
3.2. Effects of strain and grain size on SAXS Fig. 3 shows the SAXS intensities at (εn )local = 0 and = 27%, which correspond to “before deformation” and “uniform deformation”, respectively. Little small-angle scattering is observed at (εn )local = 0. This evidences that the correction of absorption and subtraction
Fig. 3. SAXS intensities of polycrystalline pure aluminum with grain size of 170 m at local nominal strain (εn )local = 0 (䊉, before deformation) and (εn )local = 27% (, uniform deformation).
given by Eq. (2) is made appropriately, and significant scatters for SAXS does not exist originally in the sample material. The SAXS intensity at (εn )local = 27% slightly increases in a small s-region compared with that before deformation. This suggests that the cavity formation occurs even during the uniform deformation, though local necking is not observable macroscopically. Fig. 4 shows the SAXS intensities as a function of local nominal strain (εn )local . As shown in Fig. 4(a) for grain size of 170 m, the intensities significantly increases at (εn )local = 48%, that is just above the onset strain of local necking, 43%. Further, the intensities continue to increase up to (εn )local = 147% and then decrease slightly. Similar tendency in the change of SAXS intensities with strain can be observed for larger grain size of 346 m (Fig. 4(b)), too. In comparison with the SAXS intensity at (εn )local = 87% for grain size of 170 m, the intensity at (εn )local = 106% for grain size of 346 m is smaller especially in the small s-region, though the strain increment beyond the onset of local necking is larger for the latter. Further, for 346 m, the intensity firstly reaches to the same level of that for 170 m at (εn )local = 87% when the deformation proceeds up to (εn )local = 268%. These observations of Figs. 3 and 4 indicate the following aspects: (1) Cavitation starts during uniform deformation, namely cavities form in advance of onset of local necking. (2) Cavitation rapidly progresses in a gauge part where local necking proceeds. This may be due to the occurrence of so-called three-axial tensile stress condition in the necked part. (3) Because cavities become elongated longitudinally by local deformation as strain approaches to that of fracture, the level of
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cavitation becomes apparently no substantial from a viewpoint of SAXS data of the sectioned sample. (4) Increase in SAXS intensity during deformation occurs more rapidly for smaller grain size both in the small and the middle s-regions. Since the increase in small and middle s-regions corresponds respectively to the increase in size and in number of cavities, the above fact indicates that both the formation and the growth of cavities occur more rapidly in a specimen with smaller grain size. 3.3. Cavity size and size distribution For the common approximation to evaluate size of cavities, the Guinier approximation [8] was applied for the SAXS signals. SAXS intensity, I(s), is represented by the approximation as follows [8]:
2
I(s) = NIinc exp
−
Rg2 3
s
2
,
(3)
where N is the number of particles, Iinc the intensity of the incident X-rays, the number of excess electrons between matrix and void, and Rg the Guinier gyration radius. By taking logarithms of Eq. (3), the following relation is obtained: ln(I(s)) = ln(NIinc 2 ) −
Rg2 3
s2 .
(4)
The Guinier plot, i.e. ln(I(s)) vs. s2 , derives a value of Rg , from which the cavity size is estimated quantitatively. If the shape of cavity is supposed to be sphere, the diameter of cavity, RD , is calculated by [8]:
RD = 2
5 Rg . 3
(5)
Fig. 5 shows the Guinier plots at typical strains for grain size of 170 m. If the plot forms a straight line in the appropriate s-region, in the so-called Guinier region, the value of Rg is evaluated from the slope of the straight line. However, the plots of the present SAXS signals do not form straight lines. It is considered that cavities with a wide range of size exist especially at high strains. Size distribution of the formed cavities could be investigated by the measured SAXS spectra with the Fankuchen’s method [13]. According to the method, zero-angle scattering intensity, I(0), is divided by the following relation [13]: 2
I(0) = I1 (0) + I2 (0) + · · · + In (0) = N1 V1 2 ( 1 ) + N2 V2 2 ( 2 ) 2
+· · · + Nn Vn 2 ( n ) ,
Fig. 5. The Guinier plots of the measured SAXS intensities at typical strains for grain size of 170 m.
2
(6)
where n is the number of divisions, In (0) the intensity for the nth division, Nn the number of particles in the nth division, Vn the volume of a particle in the nth division, n the electron density difference in the nth division, respectively. Fig. 6 shows the result of deconvolution of the SAXS pattern at (εn )local = 147% for grain size of 170 m. At first, a tangent line was drawn from the Guinier plot given by Eq. (4) and then I1 (0) was estimated by the intersection with I(s)-axis. Subsequently, the tangent line was subtracted from I(s) curve, and the procedure was repeated. Volume of cavities is estimated by values of RD given by Eq. (5), and is considered to be a uniform value, namely, it is supposed that all the cavities consist of vacancies under vacuum. Fig. 7 shows the size distribution of the cavities as a function of strain in specimens of grain size of 170 and 346 m. The distribution is defined as volume ratio of the cavities, that is the ratio of total volume of cavities with a certain diameter Vn to that of summation for all cavities divided in Eq. (6), where the summation of Vn in Eq. (6) is equal to 1. For both grain sizes, the ratio of the larger size (e.g. >100 Å) increases with strain, especially beyond the onset
of localized neck formation. The amount of increase is larger for 170 m than for 346 m. Further, it is seen that most of cavities are concentrated from 10 to 25 Å in diameter in specimens of both grain size, and for the grown cavities with size of 300–400 Å is the ratio in order of 10−1 to 10−2 for grain size of 170 m while in order of 10−2 to 10−3 for 346 m. Let us consider here the effect of grain size on the early stage of cavitation. As was observed previously Al–Mg solid solution alloy [7], the serration of grain boundary and it’s sliding cause the cavity formation at the cusps of the serrated boundary. Further, the serration of grain boundary is found to be caused by the interaction with sub-boundaries which have formed by the plastic deformation due to dislocation motion in the vicinity of grain boundary. Here, the localized deformation there is certainly constrained by the existence of neighboring grains. This indicates that the deformation behavior near grain boundaries is different from that in the grain interior. Taking into consideration a fact that the wavelength of the serration (a several m) is much smaller than the grain size (a few 100 m), the deformation behavior near the grain boundaries and hence the process of grain boundary cavitation too is reasonably considered not to be largely influenced from the grain size itself.
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the deformation stage during the growth of local necking, the process of cavitation may be affected largely by the three-axial tensile stress condition formed in the necked part. 4. Concluding remarks
Fig. 6. Deconvolution of the SAXS spectrum by the Fankuchen’s method at (εn )local = 147% for grain size of 170 m. The symbol of represents the measured SAXS data. The solid curve and dashed solid lines show fitting curve and the tangent lines drawn by procedure of the analysis, respectively. Zero-angle scattering intensities, In (0), shown in Eq. (6) and the diameters of cavities are evaluated by the intersections of the tangent lines with I(s)-axis and the slope of the each line, respectively.
(1) It is proved that SAXS method is applicable to the evaluation of cavitation on grain boundaries during high-temperature tensile deformation of polycrystalline pure aluminum. (2) Judging from the change of SAXS intensities with straining, the cavitation occurs slightly even in the early uniform deformation. And then, it proceeds more remarkably at strains beyond the onset of local necking. (3) The formation and growth of cavities progresses more rapidly with straining for smaller grain size. Larger number of cavities for smaller grain size may simply be due to larger number of possible site of the grain boundary cavitation. Reason for more remarkable growth of cavities for smaller grain size must be the subject of future research. (4) The degree of cavitation may differ from place to place even in the uniform deformation, and the macroscopic local necking may occur in the gauge part where the cavitation has occurred more severely in comparison with neighboring parts. This may not contradict the fact that the onset of local necking occurs at smaller strain in specimen with smaller grain size. Acknowledgements The authors are grateful to Professor K. Okazaki for helpful discussion of cavitation, and to Professor K. Nishikawa and Dr. K. Fukuyama for helpful suggestion in analysis of SAXS data. They also thank to PF at KEK for giving them the opportunity to perform the SAXS experiments. This work was partially supported by the Light Materials Educational Foundation of Japan. One of the authors (T. M.) thanks for the support by the Iketani Science and Technology Foundation Award. References [1] [2] [3] [4] [5] [6] [7] [8]
Fig. 7. Volume ratio of the cavities vs. the cavity diameter in specimens of grain size of 170 and 346 m. Numbers on the right side show the local nominal strain (εn )local .
[9] [10]
The larger increase in the SAXS intensities and in the volume ratio of cavities (Figs. 4 and 7), and also the smaller strain at the onset of local necking for smaller grain size may be due to a larger number of site for grain boundary cavitation per unit volume of specimen. In
[11] [12] [13]
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