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Multi-scale model for the ductility of multiple phase materials
Mechanics of Materials 41 (2009) 622–633
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Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat
Multi-scale model for the ductility of multiple phase materials Min Song *, Yue-hui He, Zheng-gang Wu, Bai-yun Huang State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Article history: Received 8 April 2008 Received in revised form 5 November 2008
a b s t r a c t A multi-scale model for the ductility of multiple phase materials has been developed in this paper. The model can be used to quantify the effects of various types of the second phase particles (including cracking formation phases and non-cracking formation phases) on the ductility of a multiple phase material. The model calculation indicates that the volume fraction, cracking fraction and shape factor of the cracking formation phases (CFPs), and the volume fraction and size of the non-cracking formation phases (NCFPs) have important effects on the ductility of a multiple phase material. It has been shown that the model predictions are in good agreement with the experimental data when applying the model to an Al–Zn–Mg alloy, in which the ductility of the material decreases with aging time until the alloy reaches the peak-aged stage, afterwards the ductility increases. The most important feature of the model is that calculation of the macroscopic ductility of a multiple phase material requires only a small amount of time and cost, compared to other models based on finite element method (FEM). Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Macroscopic properties, such as yield stress, ductility and toughness, are the key criteria in choosing a material for engineering applications. In principle, the macroscopic properties of a material are determined by its microstructure. For a single phase material, the overall behavior is mainly determined by the matrix material, solute atoms and grain size. But for a specific multiple phase material, the overall behavior strongly depends on the type, size and distribution of various phases in the material, as well as the grain size. Thus, building a link between the microstructure and the macroscopic properties of a multiple phase material is essential to design a preferred microstructure, which contains the desired properties for specific engineering applications. Most multiple phase alloys (including two phase alloys) work-harden faster than do those consisting of a single phase because the multiple phases are not equally easy to deform (Ashby, 1970). For example, in a typical Al or Cu alloy, the matrix phase deforms plastically much more * Corresponding author. Tel.: +86 731 8877880; fax: +86 731 8710855. E-mail address: [email protected] (M. Song). 0167-6636/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2008.11.003
than the second phase particles (in most conditions the hard second phase particles are assumed to be unshearable). Thus, the deformation gradients build up, with a wavelength similar to the spacing between the two neighboring particles. Similar to the multiphase alloys, the particulate/whisker reinforced metal matrix composites (such as SiC/Al composites) also exhibit non-homogeneous deformation, since the reinforced particles deform much less than the matrix. Even for a single phase polycrystalline material, the deformation gradients build up due to the strength difference between the boundary and matrix, while the wavelength is the size of the grain. Ashby (1970) showed that the gradients of deformation require that dislocations be stored. These dislocations are called geometrically-necessary dislocations, and their arrangement and density can be calculated (Ashby, 1970). It is believed that geometrically-necessary dislocations contribute significantly to the work-hardening of the materials. In a very recent paper, Vernerey et al. (2007a) developed a multi-scale micromorphic model for hierarchical materials, which has been successfully used on the multiscale character of damage and failure in steel possessing particles distributions at two distinct scales (Vernerey
M. Song et al. / Mechanics of Materials 41 (2009) 622–633
et al., 2007b). The model can be used to describe a material containing an arbitrary number of scales of microstructure and opens the door to the modeling of a range of materials possessing a hierarchical microstructure, and it captures deformation history after onset of macroscopic instability. However, when implementing the model in the finite element method, the element size is required to be on the order of the smallest microstructural length scale, which can result in very expensive (time and cost) simulation on large domains (Vernerey et al., 2007b). Hutchinson (1968), and Rice and Rosengren (1968) first studied crack-tip singularities for both plane stress and plane strain with the aid of an energy line integral, which indicated that all contours surrounding a crack tip in a two-dimensional deformation field exhibit path independence. Their work set up a fundamental basis for the further study of the fracture (ductility and toughness) of ductile materials. Based on the pioneer work of Hutchinson (1968), and Rice and Rosengren (1968), Chan (1995) developed a fracture model for hydride-induced embrittlement by considering the hydrides to crack readily under tensile loading so that an array of microcracks forms in the microstructure. In Chan’s work, the microstructure has been divided into two types of phases – microcrack formation phase (hydride) and plastic deformed phase (matrix). Interaction of the plastic fields of the microcracks leads to fracture of the matrix ligaments, and a loss in the tensile ductility (Chan, 1995). The predicted values are in reasonable agreement with the experimental observations when applying the model to zircaloys. It has been shown that the hydride-induced embrittlement in zircaloys depends on the hydride size, morphology and distribution. For most other multiple phase materials with various types of the second phase particles distributed in the matrix, the second phase particles can be divided into cracking formation phases (CFPs) and non-cracking formation phases (NCFPs). Such as in aluminum alloy, constituents are normally treated as CFP, while dispersoids and precipitates are treated as NCFPs. During deformation, high stress concentration will cause fracture of the constituents due to the low fracture strength and large size, while dispersoids and precipitates keep their shape unchanged due to the high fracture strength and small size (Thompson, 1975; Hahn and Rosenfield, 1975). To maintain the deformation continuously, geometrically-necessary dislocations must be stored in the matrix to make it compatible with the dispersoids and precipitates in shape. Based on the above discussion, Liu et al. (2003a, 2004, 2005, 2007) developed models for the ductility and toughness of Al alloys by taking constituents as inner cracks and deducing the specific expression of microscopic plastic strain of the ligament. The models have been successfully used on 2XXX and 6XXX aluminum alloys. The effects of the solution treatment, quenching and aging procedures (change the sizes and volume fractions of the constituents and precipitates) on the ductility and toughness can be determined when incorporating the aging hardening behavior of the aluminum alloy (Liu et al., 2003b) into the models (Liu et al., 2005, 2007). Similar to the method used by Liu et al. (2003a, 2004, 2005, 2007), Song and Huang (2007) developed a model for the ductility of SiC reinforced aluminum
623
alloy composites. By incorporating the aging hardening behavior of the composites during aging (Song et al., 2007), the model can be used to calculate the evolution of the ductility with aging time and aging temperature. The model includes the fact that the geometrically-necessary dislocations are stored in the matrix due to the large difference in thermal expansion between matrix and SiC particles. The generation of dislocations is generally caused by quenching from the recrystallization or solution treatment temperature. In this paper, we extend the analytic expressions of the ductility from Al alloys (Liu et al., 2004, 2005, 2007) and SiC reinforced Al alloy composites (Song and Huang, 2007) to arbitrary multiple phase materials, in which discontinuously distributed second phases (including both CFPs and NCFPs) are distributed in the continuously distributed matrix phase. The prediction of the macroscopic ductility of the multiple phase materials based on the microstructures requires only a small amount of time and cost, compared to the model by Vernerey et al. (2007a,b).
2. Model 2.1. Ductile fracture mechanisms In most multiple phase materials with a continuously distributed matrix, plastic deformation starts from the matrix. Then, the load is transferred from the ductile matrix to the discontinuously distributed second phase particles, which are surrounded by the matrix. The second phase particles can be divided into two types: large-sized CFPs and small-sized NCFPs. In general, the large sized CFPs have low fracture strength such as constituents in aluminum alloys (Hahn and Rosenfield, 1975; Thompson, 1975) or low particle-matrix interface bonding strength such as SiC particles in SiC/Al metal matrix composites (Fang et al., 2006). Microcracks are initiated due to the stress concentration that results if the load continues to increase. For NCFPs, their shapes are kept unchanged when the ligaments deform. Thus, geometrically-necessary dislocations must be stored around these phases due to nonhomogeneous deformation as described by Ashby (1970). The geometrically-necessary dislocations contribute to the work-hardening of the materials, and inevitably affect their ductility since voids will be initiated around the NCFPs if the density of the dislocations reaches the saturated situation. In general, it is believed that both the microscopic plastic strain of ligaments between two microcracks and the crack-opening displacement reach their critical values when the density of the dislocations reaches the saturated value. At that time, fracture in an unstable manner occurs ahead of the microcrack tips and voids start to grow and coalesce. Fig. 1 illustrates the mechanism of the ductile fracture of a typical multiple phase material, showing that six stages exist from the beginning of the deformation to the final fracture. It should be noted that in a multiple phase material, sometimes more than one type of the CFP exist in the ductile matrix (such as in SiC/Al composites, both
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Fig. 1. Schematic of the fracture process of multiple phase materials during deformation.
constituents and SiC particles are CFPs), and more than one type of the NCFP with various shapes and sizes exist in the matrix (such as in Al–Zn–Mg–Cu–Zr alloy, dispersoids of Al3Zr and precipitates of MgZn2 are NCFPs). It should also be noted from stage 4 that the unstable deformation of the materials starts at the onset of the voids initiation around the NCFPs. At that time, necking will be observed during tensile deformation for most ductile materials. 2.2. Multi-scale model for the ductility of multiple phase materials If we assume that the distribution of the second phase particles in the matrix is in a cubic array, the one-dimensional example of the multiple phase materials is illus-
Fig. 2. Geometrical description of the model for the ductility of multiple phase materials.
trated in Fig. 2, in which various types of NCFPs distribute between two CFP particles. In Fig. 2, e1ij and e2ij are the strain caused by these two microcracks during deformation, respectively, while eij is the sum of e1ij and e2ij . For simplicity, we assume that the size and interspacing of the two neighboring microcracks (caused by CFPs) are 2a and k, respectively. Thus, at a distance, r (r is the distance between two neighboring microcracks), the strain caused by these two microcracks can be written as Hutchinson (1968), Rice and Rosengren (1968) and Kanninen and Popelar (1985)
1
I
ecp ¼ ~ en ðhÞ 0:405ph
n 1þn
kf 1 2r f
n 1þn
ec 2
ð1Þ
where ~en ðhÞ is the effective value of reduced coefficient ~ eij ðhÞ when h = 0, ecp is the macroscopic fracture strain (ductility), ec is the critical value of the microscopic plastic strain in the ligament, e, n is the inverse of the strain hardradius of ening exponent, kf and rf are the spacing and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the inner microcracks, respectively, I ¼ 10:3 0:13 þ n and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ¼ 3=2 1 þ 3n. Eq. (1) was first developed to describe the strain distribution around a microcrack in a ductile material during deformation. Thus, the use of the equation is for ductile materials or materials with ductile matrix, instead of brittle materials. One should be noted that the application of the equation to a ductile material (or a material with ductile matrix) with more than one microcracks requires that the distance between two neighboring microcracks is not less than the length (usually the size of CFPs) of the microcracks (in general, the volume fraction of CFPs is not more than 50%). Since the development of the equation does not depend on deformation style, thus, using the equation for both tensile and compressive loading is a good first approximation. It should be noted that Eq. (1) has also been used in Liu et al. (2003a, 2004, 2005, 2007) for aluminum alloys and Song and Huang (2007) for SiC reinforced metal matrix composites.
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As described above, microcracks are initiated from the CFPs. Since not all of the CFPs will initiate microcracks during deformation, the cracking fraction of a phase (i) is defined as Fi. Thus, the volume fraction of the microcracks (ui) caused by one type of the CFP is with the form of
ui ¼ F i uci ¼ 2pK i r3i =k3i
ð2Þ
where uci is the volume fraction of one type of the CFP (i) and Ki is the shape factor (aspect ratio) and equals 1 for spherical particles. According to Ashby (1970), the microscopic plastic strain (e) is characterized by a geometric slip distance, kg, analogous to the slip distance in pure crystals, but determined only by the microstructure. The density of the geometrically-necessary dislocations (qg) is then given by (Ashby, 1970)
qg ¼
4e kg b
ð3Þ
where b is the Burgers vector of the dislocations. Ashby (1970) showed that for alloys containing more or less spherical particles, kg = rs/fs, where rs is the particle radius forffi alloys containing plate/ and fs the volume fraction; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rod-like particles, kg ¼ 3 pr 2s hs =fs (equals the particle spacing), where hs is the height of the plate-like particles or length of the rod-like particles; for pure polycrystals, it is
ec1 ¼ ec2 ¼ ¼ eci ¼ ¼ ecx ¼
ec ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðec1 Þ2 þ ðec2 Þ2 þ þ ðeci Þ2 þ þ ðecx Þ2 ¼
qcg ¼
4ec kg b
1
I
n 1þn
qg ¼ q1 þ q2 þ þ qi þ þ qx ¼
x X
qi
kf 1 2r f
n 1þn
From the aforementioned analyses, Eq. (4) can be rewritten as
ð7Þ
The critical value of the microscopic plastic strain, ec, can be expressed by the combination of all the single values, eci , in the form of Pythagorean theorem, not the linear addition due to the existence of the overlap effect, with the form of
ð8Þ
So the strain to fracture or the ductility of the multiple materials can be obtained by combining Eqs. (1) and (8) as
pffiffiffi bk1 k2 ki kx qcg x 8ðk2 k3 kx þ k1 k3 . . . kx þ þ k1 k2 ki1 kiþ1 kx þ . . . þ k1 k2 kx1 Þ
For multiple phase materials with more than one type of the NCFPs, the density of the partial geometrically-necessary dislocations due to the non-homogeneous deformation around each type of the non-cracking formation particles has the relation of (Ashby, 1970)
ð6Þ
i¼1
pffiffiffi bk1 k2 ki kx qcg x 4ðk2 k3 kx þ k1 k3 . . . kx þ þ k1 k2 ki1 kiþ1 kx þ . . . þ k1 k2 kx1 Þ
ð4Þ
ð5Þ
where Ci is the scaling factor and considered as unity here, and subscribe x is the number of the types of the NCFPs. Eq. (5) was originally developed by Ashby (1970), and was then subsequently used by Liu et al. (2004) when dealing with the ductility of aluminum alloy. The equation indicates that the ratio of the partial geometrically-necessary dislocation densities is associated with the ratio among particle spacings of the various populations of the particles. Liu et al. (2004) showed that for materials containing continuously distributed ductile matrix and non-continuously distributed second phase particles, the equation is a very good first approximation. One should be noted that the scaling factor Ci does not generally equals unity since the degree of the inhomogeneous microscopic deformation in the ligaments depends on the size of the particles. Here in this paper, for the first approximation, Ci is assumed to be unity for simplicity. The density of the geometricallynecessary dislocations (qg) is assumed to be the sum of the density of all the partial geometrically-necessary dislocations by
bk1 k2 k3 . . . kx qcg 4ðk2 k3 kx þ k1 k3 . . . kx þ þ k1 k2 ki1 kiþ1 kx þ . . . þ k1 k2 kx1 Þ
proportional to the grain size. When the density of the geometrically-necessary dislocations (qg) reaches the critical value, qcg , the microscopic strain (e) reaches the critical value, ec, and macroscopic fracture appears. At that time, Eq. (3) changes to
ecp ¼ ~ en ðhÞ 0:405ph
C 1 q1 k1 ¼ C 2 q2 k2 ¼ ¼ C i qi ki ¼ ¼ C x qx kx
ð9Þ
If we define ðecp ÞR as the reference ductility, the normalized ductility of a multiple phase material, R, can be expressed as
R ¼ ecp =ðecp ÞR
ð10Þ
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Eqs. (9) and (10) indicate that if the ductility of a multiple phase material under a specific condition (volume fraction and size of the phases) is known, the ductility under other conditions can be calculated.
3. Prediction of the normalized ductility using the present model 3.1. Effects of the CFPs on the ductility In this subsection, we will study the effects of CFPs on ductility of multiple phase materials. First, a single type of the CFP, which is distributed uniformly in the ductile matrix, will be considered in the calculation. Many metals and alloys belong to this system, such as a commercial pure aluminum alloy, in which brittle constituents (formed by impurities with aluminum) are distributed in the Al matrix. To calculate the normalized ductility, Eq. (9) can be rewritten as
c p
e
n "sffiffiffiffiffiffiffiffiffiffiffiffi #1þn n 1þn 1 I ec 3 p K ¼ 1 ~en ðhÞ 0:405ph 4 Fu 2
ð11Þ
Fig. 3. Effects of the volume fraction of the CFP on the normalized ductility with (a) various cracking fractions (K = 1) and (b) various shape factors (F = 1).
Fig. 3 shows the effect of the CFP on the normalized ductility of the materials. The reference was defined as the volume fraction of the CFP of 0.01% (it should be noted that the reference can not be set as zero, otherwise a singularity will appear when applying Eq. (10) during calculation). The effects of the cracking fraction F (Fig. 3a) and the shape factor K (Fig. 3b) on the normalized ductility have been considered in the calculation. It can be seen that no matter what values of the cracking fraction and shape factor are, the normalized ductility decreases as the volume fraction of the CFP increases. This indicates that an increase in the CFP will degrade the ductility of the material. It should be noted that the increase in the cracking fraction will also degrade the normalized ductility, since the increase in the cracking fraction increases the real fraction of the microcracks as well. In contrast to the cracking fraction, the increase in the shape factor will improve the ductility of the material. This is not surprising, since a sharp crack can cause a much higher stress concentration at the crack tip if disk-like cracks are perpendicular to the stress direction (low shape factor value) rather than along the stress direction (high shape factor value). Thus, the higher the shape factor value, the higher the ductility. From Fig. 3b, we can notice that when the shape factor is very low, the normalized ductility will decrease very quickly when the volume fraction of the CFP reaches a critical value (for example, when the shape factor is 0.1, the critical volume fraction is about 0.1). This is caused by the unstable propagation of the microcracks and sudden fracture of the materials under externally applied stress due to the high stress concentration and overlapping. Another important system is a material containing two types of the CFPs. Many alloys and metal matrix composites belong to this system, such as constituents and SiC particles in the SiC reinforced aluminum alloy composites. Since the effect of a single type of the CFP on the normalized ductility has been studied in Fig. 3, here we concentrate on the effect of the second type of the CFP on the normalized ductility. Fig. 4 shows the effect of the second type of the CFP on the normalized ductility. The reference was defined as the volume fraction of the first type of the CFP of 10%, and two types of the CFPs are assumed to have
Fig. 4. Effects of the volume fraction of the second type of the CFP on the normalized ductility.
M. Song et al. / Mechanics of Materials 41 (2009) 622–633
627
Fig. 6. Effects of the volume fraction of the second type of the CFP on the normalized ductility with various shape factors.
Fig. 5. Effects of the volume fraction of the second type of the CFP on the normalized ductility with various cracking fractions of (a) the first type of CFP and (b) the second type of CFP.
the similar size, and the cracking fraction and shape factor are both 1. It can be seen that the normalized ductility decreases with the increase in the volume fraction of the first type of the CFP, which has also been shown in Fig. 3. Most importantly, the normalized ductility decreases with the increase in the volume fraction of the second type of the CFP. The higher the volume fraction of the second type of the CFP, the higher the volume fraction of the microcracks and the lower the normalized ductility. It should be noted from Fig. 4 that the normalized ductility decreases slightly quicker with the volume fraction of the second type of the CFP if the volume fraction of the first type of the CFP is low. For a material containing two types of the CFPs, the fraction of the microcracks due to the second type of the CFP decreases as the volume fraction of the first type of the CFP increases. Thus, the effect of the second type of the CFP on the normalized ductility decreases with the increase in the volume fraction of the first type of the CFP. The effects of the cracking fraction of the two types of the CFPs have been illustrated in Fig. 5. The reference was defined as the volume fraction of the first type of the CFP of 10% with its corresponding cracking fractions. The two types of the CFPs are assumed to have the similar size, and shape factor are both 1. Similar to Fig. 4, the ductility decreases as the volume fraction of the second type of the CFP increases, no matter what cracking fraction of
the first or second type of the CFP is. It can be seen that the cracking fractions of both types of the CFPs have important effects on the normalized ductility of the materials. Increasing the cracking fraction of the first type of the CFP can obviously improve the normalized ductility if the volume fraction of the second type of the CFP remains constant (Fig. 5a). This is not surprising, since the microcracks are caused by both types of the CFPs, the increase in the cracking fraction of the first type of the CFP inevitably increases its fraction of the total micocracks, thus decreases the fraction of the microcracks due to the second type of the CFP, and the effect on the normalized ductility. This is essentially the case for SiC reinforced aluminum composites, in which the increases in the cracking fraction of the SiC particles will decrease the effect of the constituents on the normalized ductility (Song and Huang, 2007). Contrary to Fig. 5a, increasing the cracking fraction of the second type of the CFP obviously degrades the normalized ductility if the volume fraction of the first type of the CFP remains constant (Fig. 5b), since the increase in the cracking fraction of the second type of the CFP increases its fraction of the total micocracks, thus increases the effect on the normalized ductility. Another important calculation is the effect of the shape factor of both types of the CFPs on the normalized ductility, which has been illustrated in Fig. 6. The reference was defined as the volume fraction of the first type of the CFP of 10%. The cracking fractions of two types of the CFPs are assumed to be 1. It can be seen that the increase in the shape factor will improve the normalized ductility of the material. Similar to the discussion for Fig. 3b, a sharp crack can cause a much higher stress concentration at the crack tip if disk-like cracks are perpendicular to the stress direction rather than along the stress direction. Thus, the higher the shape factor value, the higher the ductility. 3.2. Effects of the NCFPs on the ductility In this subsection, we will study the effects of the NCFPs on the ductility of multiple phase materials. From Section 2, we know that the NCFPs influence the ductility of the materials by changing the geometric slip distance (liga-
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M. Song et al. / Mechanics of Materials 41 (2009) 622–633
Fig. 8. Effects of the second type of the NCFP on the normalized ductility.
increases linearly with the size of the NCFP if the volume fraction remains constant, which indicated that largesized particles can substantially improve the ductility of the materials. This is reasonable by taking into account the fact that the geometric slip distance increases with the particle size when the volume fraction of the NCFP remains constant. In many other materials, two types of the NCFPs are distributed in the matrix, such as Al–Zn–Mg–Cu, in which meso-stable g0 phase and T0 phase are distributed uniformly in the Al matrix. To calculate the normalized ductility of this system, Eq. (9) can be rewritten as Fig. 7. Effects of the (a) volume fraction and (b) size of the NCFP on the normalized ductility. Note the volume fraction remains constant in Fig. 7(b).
ment size, interspacing between two neighboring particles) through volume fraction and size variations. First, a single type of the NCFP, which is distributed uniformly in the ductile matrix, will be considered in the calculation. Many materials belong to this simple system, such as Al–Sc, Al–Zr and Al–Cr alloy, in which a single type of dispersoids (Al3Sc, Al3Zr or Al3Cr) is distributed in the Al matrix. To calculate the normalized ductility, Eq. (9) can be rewritten as
1
I
ecp ¼ ~ en ðhÞ 0:405ph
n 1þn
kf 1 2r f
n 1þn
bkqcg 8
ð12Þ
Fig. 7 shows the effects of the NCFP on the normalized ductility of the materials. The reference was defined as the volume fraction of the NCFP of 0.0001%, and the NCFP is assumed to be in round shape. It can be seen from Fig. 7a that the normalized ductility of the material decreases with the increase in the volume fraction of the NCFP, with a very steep trend when the NCFP begins to appear. The trend becomes much milder when the volume fraction of the NCFP continues to increase. The decrease in the normalized ductility is due to the decrease in the geometric slip distance with the increase in the volume fraction of the particles. It can also be seen from Fig. 7b that the normalized ductility of the material
1
I
ecp ¼ ~ en ðhÞ 0:405ph
n 1þn
kf 1 2r f
pffiffiffi n 1þn 2bk1 k2 qcg 8ðk2 þ k1 Þ
ð13Þ
Since the effect of a single type of the NCFP on the normalized ductility has been studied in Fig. 7, here we concentrate on the effect of the second type of the NCFP on the normalized ductility, as illustrated in Fig. 8. The reference was defined as the volume fraction of the first type of the NCFP of 0.0001%, and two types of the CFPs are assumed to be in round shape. It can be seen that the normalized ductility decreases with the increase in the volume fraction of the second type of the NCFP, with the similar trend as illustrated in Fig. 7a. That is, it decreases very quickly with the volume fraction of the second type of the NCFP at the very beginning, and then the decreasing speed slows down. It can also be seen that the normalized ductility increases with the increase in the volume fraction of the first type of the NCFP if the volume fraction of the second type of the NCFP remains constant. This does not mean that the real ductility of the material increases with the volume fraction of the first type of the NCFP, but only indicates that the influence fraction of the first type of the particles increases when the volume fraction increases. This is very similar to Fig. 5, in which the influence fraction of the first type of the CFP also increases with the volume fraction. A further extension of the calculation is using the model to calculate the normalized ductility of the materials containing three types of the NCFPs. In this case, Eq. (9) can be rewritten as
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M. Song et al. / Mechanics of Materials 41 (2009) 622–633
4.1. Experimental methodology
Fig. 9. Effects of the third type of the NCFP on the normalized ductility. Note the total volume fraction of the first and second types of the NCFPs remains constant.
c p
e
pffiffiffi n n 1þn 1þn 3bk1 k2 k3 qcg kf 1 I ¼ 1 ~en ðhÞ 0:405ph 2r f 8ðk2 k3 þ k1 k3 þ k1 k2 Þ ð14Þ
A typical system containing three types of the NCFPs is Al–Zn–Mg–Cu–Sc alloy, in which meso-stable g0 phase and T0 phase, and stable Al3Sc dispersoids are distributed uniformly in the matrix. Fig. 9 shows the effect of the third type of the NCFP on the normalized ductility. The reference was defined as the volume fraction of the first and second types of the NCFPs of 0.0001% and 0, respectively, and all three types of the NCFPs are assumed to be in round shape. It can be seen that the normalized ductility decreases with the increase in the volume fraction of the third type of the NCFP. Since the total volume fraction of the first and second types of the non-cracking phases remains unchanged during calculation, the effect of the volume fraction ratio of the first and second types of the NCFPs can be illustrated by comparing the curves in Fig. 9. It can be seen that the larger the difference between the volume fractions of the first and second types of the particles, the lower the normalized ductility. However, the effect of the difference between the volume fractions is not so important, compared to the effect of the volume fraction of the third type of the NCFP. 4. Application of the model to a real material system In Section 3, we simulated the effects of the CFPs and NCFPs on the ductility of multiple phase materials. To further prove the accuracy of the model, we apply the model to a real material system – Al–Zn–Mg alloy. In Al–Zn–Mg alloys, several types of the particles exist in the matrix. These particles include CFP (constituents) and NCFP (precipitates). Brittle constituents are resulted from the impurities or excess micro-alloying elements with Al matrix during solidification process, while precipitates (MgZn2) are resulted from the micro-alloying elements during aging procedure. In the following subsection, we will describe the experiment method to obtain a distribution of the volume fraction of the constituents and precipitates in Al–Zn–Mg alloy.
The nominal composition of the alloy used in the present study has been shown in Table 1. The alloy was prepared in an induction furnace in an argon atmosphere. The as-cast ingot was homogenized at 455 °C for 10 h, followed by air cooling to room temperature. Then the ingot was converted into rods by hot extrusion at 450 °C, with an extrusion ratio of 9:1. The extruded rods were solution treated under a series conditions, followed by cold water quenching (room temperature, 20 °C) and aging treatments (120 °C). The solution treatments include traditional solution treatment and enhanced solution treatments (EST). Detailed description of the solution treatment can be found in Table 2. It should be noted that the EST uses an increasing temperature method instead of the isothermal treatment since some multi phase eutectic has a low melting point. The prior dissolution of the multi phase eutectic during the EST enhances the melting point temperature and, thus allows a further high temperature solution treatment to be used. Aging treatment with various times (under a constant temperature of 120 °C) was subsequently applied in order to produce variously sized and volume-fractioned precipitates. The tensile ductility was tested at room temperature using a smooth dog-bone-shaped tensile specimen that had a gage size of 6 mm in diameter and 40 mm in length at a constant strain rate of 5104 s1 by an Instron-8802 testing machine. All the specimens have an axis along the extrusion direction. The sizes and volume fractions of the constituents and precipitates at different stages (various solution treatments and aging treatments) have been measured using optical microscopy (OM) and transmission electron microscopy (TEM). The TEM specimens (thin foils) were prepared by twin-jet electro-polishing in a 30% nitric acid and 70% methanol solution at 35 °C and examined in a Tecnai G2 20 microscopy operating at 200 kV. The size and volume fraction of the precipitates were determined from at least 100 random precipitates seen edge on. Since the radius of the precipitates in the matrix was in the same
Table 1 Nominal composition of the tested Al–Zn–Mg alloy (mass fraction, %). Zn
Mg
Cu
Mn
Al
4.0
3.2
0.15
0.15
Bal.
Table 2 The procedures of the solution treatments. Solution type
Traditional solution treatment
Enhanced solution treatment (EST)
S1 S2
450 °C/2 h 450 °C/2 h
S3
450 °C/2 h
S4
450 °C/2 h
– Increasing temperature to 460 °C with a rate of 5 °C per hour Increasing temperature to 470 °C with a rate of 5 °C per hour Increasing temperature to 480 °C with a rate of 5 °C per hour
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order as and even greater than the thickness of the thin foil, a correction was employed for differentiating the real dimension from the observed one (Crompton et al., 1966) and the volume fraction of the precipitates in a thin foil projection was determined by (Gilmore and Starker, 1997)
fv ¼
2pr lnð1 AÞ pr þ 4t
ð15Þ
where r is the half radius of the precipitates, t is the foil thickness, and A is the project area fraction of the precipitates, determined by the point count method. The foil thickness was easily obtained by utilizing a grain boundary fringes technique (Edington, 1976). 4.2. Microstructures and tensile ductility Fig. 10 shows the optical microstructures of the Al–Zn–Mg alloy after various solution treatments. It can be seen that a large number of the constituents exist in the matrix after traditional solution treatment (S1), while the number and the aggregation of the constituents have substantially been decreased after the EST (S2, S3 and S4). It can be seen that the higher the ending temperature of the EST, the less the number of the constituents. These results indicate that the application of the EST facilitates the dissolution of the soluble constituents and hence the decrease in the volume fraction of the constituents. It should be noted that after S4 solution treatment, the
volume fraction of the constituents is very small, and almost no constituents can be found from the optical microscopy. Table 3 illustrates the volume fraction of the constituents and the tensile ductility of the specimens after various solution treatments. It can be seen that the EST substantially improves the tensile ductility and decreases the volume fraction of the soluble constituents. Previous studies (Hahn and Rosenfield, 1975; Thompson, 1975) indicated that the coarse constituents are brittle and have low fracture strength and, thus, generate microcracks during deformation. The larger the volume fraction of the constituents, the lower the tensile ductility. The constituents are normally resulted from the presence of the Fe and Si impurities or excess amounts of major alloying elements such as Mg, Zn and Cu (Hahn and Rosenfield, 1975; Thompson, 1975). The EST can substantially improve the tensile ductility of the alloy by decreasing the number of the microcrack sources (volume fraction of the constituents).
Table 3 Volume fraction of the constituents and tensile ductility of the alloy after various solution treatments (before aging treatment). Aging time (h)
S1
S2
S3
S4
Volume fraction of the constituents (%) Tensile ductility (%)*
4.7 11.5
3.2 11.8
1.9 12.4
0 16.5
*
All the data are the average values from five tests.
Fig. 10. OM images of the tested Al–Zn–Mg alloy after (a) S1, (b) S2, (c) S3 and (d) S4 treatments.
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Fig. 11. TEM images of the tested Al–Zn–Mg alloy at (a) under-aged stage (10 h), (b) peak-aged stage (24 h), (c) over-aged stage (50 h) and (d) the SADPs of the precipitates at peak-aged stage.
Fig. 11 shows the TEM microstructures of the Al–Zn–Mg alloy after aging treatment for various length of time. It can be seen that a large number of the precipitates exist in the matrix after aging treatment, while the size of the precipitates increases with aging time. These precipitates are the typical strengthening particles (MgZn2) in Al–Zn–Mg alloy. The evolutions of the volume fraction of the precipitates and the tensile ductility of the alloy with aging time have been shown in Table 4. Note that all the specimens were applied S4 solution treatment before aging to minimize the volume fraction of the constituents and increase the super-solution degree. It can be seen that the volume fraction of the precipitates increases with the aging time until it reaches a constant value after 24 h. At that time, the alloy reaches the peak-aged stage, and the volume fraction of the precipitates remains constant, while the size increases by the growth of larger precipitates and the dissolving of smaller precipitates. It can be seen that as the aging time increases, the tensile ductility decreases until the aging time reaches 24 h. Then the tensile ductility increases as
the aging time continues to increase. The decrease in the tensile ductility at under-aged stage is due to the increase in the volume fraction of the precipitates, which inevitably decreases the geometric slip distance, while the increase in the tensile ductility after aging to peak-aged stage is due to the increase in the precipitate size under constant volume fraction, which inevitably increases the geometric slip distance. 4.3. Comparison between the calculated results and the experimental data In this subsection, the experimental data listed in Tables 3 and 4 have been used to verify the model accuracy. In the calculation, both constituents and precipitates are assumed to be in round shape. In addition, the constituents are assumed to be totally fractured. Fig. 12 shows the model predicted relation between the normalized tensile ductility and volume fraction of the constituents. The experimental data listed in Table 3 has also been included.
Table 4 Volume fraction of the precipitates and tensile ductility of the alloy after aging treatment (the specimen was applied S4 solution treatment before aging to minimum the volume fraction of the constituents and increase the super-solution degree). Aging time, s (h)
0 (0)
Volume fraction of precipitates (%) Tensile ductility (%)a
0
a b
16.5
1800 (0.5) 0.035 10.4
18,000 (5)
36,000 (10)
64,800 (18)
86,400 (24)b
180,000 (50)
288,000 (80)
540,000 (150)
0.068
0.082
0.094
0.1
0.103
0.102
0.101
9.1
8.6
7.8
7.9
8.4
8.9
9.1
All the data are the average values from five tests. The alloy reaches the peak-aged stage after aging for 24 h.
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Fig. 12. Measured and predicted normalized tensile ductility of the tested alloy with the volume fraction of the constituents.
It can be seen that model prediction agrees well with the experimental data, in which the normalized tensile ductility decreases with the increase in the volume fraction of the constituents. Fig. 13 shows the relation between the experimental determined volume fraction of the precipitates and model predicted normalized tensile ductility and aging time. It should be noted that the calculation of the normalized tensile ductility requires the volume fraction of the precipitates and the geometric slip distance (distance between two neighboring precipitates). In principle, the volume
fractions of the precipitates in Table 3 and Fig. 13a have been used in the model calculation with the reference being defined as the volume fraction of the precipitates of 0.0001%, while the geometric slip distance can be easily obtained since the volume fraction and size of the precipitates can be determined experimentally. The experimental data of the tensile ductility listed in Table 4 has also been included in Fig. 13b. It can be seen that the normalized tensile ductility decreases with the aging time until the alloy reaches the peak-aged stage. Then the normalized tensile ductility increases with the aging time. This is essentially the case for aluminum alloys, in which the tensile ductility and fracture toughness of aluminum alloys decrease with aging time at under-aged stage, but increase with aging time at over-aged stage. During under-aged stage, the volume fraction of the precipitates increases and the geometric slip distance decreases with aging time, and thus the tensile ductility decreases with aging time. However, when the alloy reaches the peak-aged stage, the excess solute atoms are totally exhausted and the growth stage of the precipitates is terminated. At that time, the volume fraction of the precipitates remains constant, and the precipitates size and the geometric slip distance increase through the growth of the larger precipitates and the dissolving of the smaller precipitates, and thus the tensile ductility increases. It can also be seen from Figs. 12 and 13 that the model calculations are in good agreement with the experimental counterparts, which indicated that the model can be used to quantify the ductility of a material based on its microstructure. An important feature of the model is that calculation of the ductility of a multiple phase material can be run on a personal computer and requires only a small amount of time, compared to the other models based on FEM, which may require long time calculation and high cost. 5. Conclusions In this paper, we developed a multi-scale model for the ductility of multiple phase materials. The effects of various types of the CFPs and NCFPs on the ductility can be calculated based on the model calculation. It has been shown that the model predictions are in good agreement with the experimental data when testing on an Al–Zn–Mg alloy. Based on model predictions and experimental data, the following conclusions are drawn.
Fig. 13. (a) Experimentally determined volume fraction of precipitates and (b) measured and predicted normalized tensile ductility of the tested alloy as a function of aging time.
(1) The volume fraction, fracture fraction and shape factor of the CFPs have important effects on the ductility of a multiple phase material. The normalized ductility decreases as the volume fraction and fracture fraction of the CFPs increase, but increases as the shape factor of the CFPs increases. (2) The volume fraction and size of the NCFPs have important effects on the ductility of a multiple phase material. The normalized ductility decreases as the volume fraction of the NCFPs increases. Under the condition of the constant volume fraction, the normalized ductility increases linearly with the size of the NCFPs.
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(3) For a typical material with precipitation hardening during aging, the ductility of the material decreases with aging time until the material reaches the peakaged stage, and then the ductility increases with aging time. The decrease in the ductility at the early stage of aging is due to the increase in the volume fraction of the precipitates and decrease in the geometric slip distance, while the increase in the ductility after peak-aged stage is due to increase in the geometric slip distance since the volume fraction of the precipitates remains constant but the precipitate size increases by the growth of the larger precipitates and the dissolving of the smaller precipitates.
Acknowledgements This work was supported by National Natural Science Foundation of China (No. 50801068), Creative Research Group of National Natural Science Foundation of China (No. 50721003) and Hunan Provincial Natural Science Foundation of China (No. 07JJ3117). One of the authors would also like to thank the support of Chinese Postdoctoral Science Foundation (No. 20070410303). References Ashby, M.F., 1970. The deformation of plastically non-homogeneous materials. Philosophical Magazine 21, 399–424. Chan, K.S., 1995. A fracture model for hydride-induced embrittlement. Acta Metallurgical Materialia 43, 4325–4335. Crompton, J.M.G., Waghorne, R.M., Brook Jr, G.B., 1966. The estimation of size distribution and density of precipitates from electron micrographs of thin foils. British Journal of Applied Physics 17, 1301–1305. Edington, J.W., 1976. Practical Electron Microscopy in Materials Science. Von Norstrand Reinhold Company, London. p. 207. Fang, J., Xiao, B., Zuo, T., Sang, J., Zhang, W., Xu, J., Shi, L., 2006. Effect of heat treatment on strength and ductility of SiCp/Al composites. The Chinese Journal of Nonferrous Metals 16, 228–234.
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Gilmore, D.L., Starker Jr., R.M., 1997. Trace element effects on precipitation processes and mechanical properties in an Al–Cu–Li alloy. Metallurgical and Materials Transactions A 28, 1399–1415. Hahn, G.T., Rosenfield, A.R., 1975. Metallurgical factors affecting fracture toughness of aluminum alloys. Metallurgical Transactions A 6, 653– 668. Hutchinson, J.W., 1968. Singular behavior at the end of a tensile crack in a hardening material. Journal of the Mechanics and Physics of Solids 16, 13–31. Kanninen, M.F., Popelar, C.H., 1985. Advanced Fracture Mechanics. Oxford University Press, New York, NY. p. 300. Liu, G., Zhang, G.J., Ding, X.D., Sun, J., Chen, K.H., 2003a. Dependence of fracture toughness on multi-scale second phase particles in high strength Al alloys. Materials Science and Technology 19, 887–896. Liu, G., Zhang, G.J., Ding, X.D., Sun, J., Chen, K.H., 2003b. Modeling the strengthening response to aging process of heat-treatable aluminum alloy containing plate/disc- or rod/needle-shaped precipitates. Materials Science and Engineering A 344, 113–124. Liu, G., Zhang, G.J., Ding, X.D., Sun, J., Chen, K.H., 2004. The influence of multiscale-sized second-phase particles on ductility of aged aluminum alloys. Metallurgical and Materials Transactions A 35, 1725–1734. Liu, G., Sun, J., Nan, C.W., Chen, K.H., 2005. Experimental and multiscale modeling of the coupled influence of constituents and precipitates on the ductile fracture of heat-treatable aluminum alloys. Acta Materialia 53, 3459–3468. Liu, G., Zhang, G.J., Wang, R.W., Hu, W., Sun, J., Chen, K.H., 2007. Heat treatment-modulated coupling effect of multi-scale second-phase particles on the ductile fracture of aged aluminum alloys. Acta Materialia 55, 273–284. Rice, J.R., Rosengren, G.F., 1968. Plane strain deformation near a crack tip in a power-law hardening material. Journal of the Mechanics and Physics of Solids 16, 1–12. Song, M., Huang, D., 2007. Experimental and modeling of the coupled influences of variously sized particles on the tensile ductility of SiCp/ Al metal matrix composites. Metallurgical and Materials Transactions A 38, 2127–2137. Song, M., Li, X., Chen, K.H., 2007. Modeling the ageing hardening behavior of SiC/Al metal matrix composites. Metallurgical and Materials Transactions A 38, 638–648. Thompson, D.S., 1975. Metallurgical factors affecting high strength aluminum alloy production. Metallurgical Transactions A 6, 671–683. Vernerey, F., Liu, W.K., Moran, B., 2007a. Multi-Scale micromorphic theory for hierarchical materials. Journal of the Mechanics and Physics of Solids 55, 2603–2651. Vernerey, F., Liu, W.K., Moran, B., Olson, G., 2007b. A micromorphic model for the multiple scale failure of heterogeneous materials. Journal of the Mechanics and Physics of Solids 56, 1320–1347.
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Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat
Multi-scale model for the ductility of multiple phase materials Min Song *, Yue-hui He, Zheng-gang Wu, Bai-yun Huang State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Article history: Received 8 April 2008 Received in revised form 5 November 2008
a b s t r a c t A multi-scale model for the ductility of multiple phase materials has been developed in this paper. The model can be used to quantify the effects of various types of the second phase particles (including cracking formation phases and non-cracking formation phases) on the ductility of a multiple phase material. The model calculation indicates that the volume fraction, cracking fraction and shape factor of the cracking formation phases (CFPs), and the volume fraction and size of the non-cracking formation phases (NCFPs) have important effects on the ductility of a multiple phase material. It has been shown that the model predictions are in good agreement with the experimental data when applying the model to an Al–Zn–Mg alloy, in which the ductility of the material decreases with aging time until the alloy reaches the peak-aged stage, afterwards the ductility increases. The most important feature of the model is that calculation of the macroscopic ductility of a multiple phase material requires only a small amount of time and cost, compared to other models based on finite element method (FEM). Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Macroscopic properties, such as yield stress, ductility and toughness, are the key criteria in choosing a material for engineering applications. In principle, the macroscopic properties of a material are determined by its microstructure. For a single phase material, the overall behavior is mainly determined by the matrix material, solute atoms and grain size. But for a specific multiple phase material, the overall behavior strongly depends on the type, size and distribution of various phases in the material, as well as the grain size. Thus, building a link between the microstructure and the macroscopic properties of a multiple phase material is essential to design a preferred microstructure, which contains the desired properties for specific engineering applications. Most multiple phase alloys (including two phase alloys) work-harden faster than do those consisting of a single phase because the multiple phases are not equally easy to deform (Ashby, 1970). For example, in a typical Al or Cu alloy, the matrix phase deforms plastically much more * Corresponding author. Tel.: +86 731 8877880; fax: +86 731 8710855. E-mail address: [email protected] (M. Song). 0167-6636/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2008.11.003
than the second phase particles (in most conditions the hard second phase particles are assumed to be unshearable). Thus, the deformation gradients build up, with a wavelength similar to the spacing between the two neighboring particles. Similar to the multiphase alloys, the particulate/whisker reinforced metal matrix composites (such as SiC/Al composites) also exhibit non-homogeneous deformation, since the reinforced particles deform much less than the matrix. Even for a single phase polycrystalline material, the deformation gradients build up due to the strength difference between the boundary and matrix, while the wavelength is the size of the grain. Ashby (1970) showed that the gradients of deformation require that dislocations be stored. These dislocations are called geometrically-necessary dislocations, and their arrangement and density can be calculated (Ashby, 1970). It is believed that geometrically-necessary dislocations contribute significantly to the work-hardening of the materials. In a very recent paper, Vernerey et al. (2007a) developed a multi-scale micromorphic model for hierarchical materials, which has been successfully used on the multiscale character of damage and failure in steel possessing particles distributions at two distinct scales (Vernerey
M. Song et al. / Mechanics of Materials 41 (2009) 622–633
et al., 2007b). The model can be used to describe a material containing an arbitrary number of scales of microstructure and opens the door to the modeling of a range of materials possessing a hierarchical microstructure, and it captures deformation history after onset of macroscopic instability. However, when implementing the model in the finite element method, the element size is required to be on the order of the smallest microstructural length scale, which can result in very expensive (time and cost) simulation on large domains (Vernerey et al., 2007b). Hutchinson (1968), and Rice and Rosengren (1968) first studied crack-tip singularities for both plane stress and plane strain with the aid of an energy line integral, which indicated that all contours surrounding a crack tip in a two-dimensional deformation field exhibit path independence. Their work set up a fundamental basis for the further study of the fracture (ductility and toughness) of ductile materials. Based on the pioneer work of Hutchinson (1968), and Rice and Rosengren (1968), Chan (1995) developed a fracture model for hydride-induced embrittlement by considering the hydrides to crack readily under tensile loading so that an array of microcracks forms in the microstructure. In Chan’s work, the microstructure has been divided into two types of phases – microcrack formation phase (hydride) and plastic deformed phase (matrix). Interaction of the plastic fields of the microcracks leads to fracture of the matrix ligaments, and a loss in the tensile ductility (Chan, 1995). The predicted values are in reasonable agreement with the experimental observations when applying the model to zircaloys. It has been shown that the hydride-induced embrittlement in zircaloys depends on the hydride size, morphology and distribution. For most other multiple phase materials with various types of the second phase particles distributed in the matrix, the second phase particles can be divided into cracking formation phases (CFPs) and non-cracking formation phases (NCFPs). Such as in aluminum alloy, constituents are normally treated as CFP, while dispersoids and precipitates are treated as NCFPs. During deformation, high stress concentration will cause fracture of the constituents due to the low fracture strength and large size, while dispersoids and precipitates keep their shape unchanged due to the high fracture strength and small size (Thompson, 1975; Hahn and Rosenfield, 1975). To maintain the deformation continuously, geometrically-necessary dislocations must be stored in the matrix to make it compatible with the dispersoids and precipitates in shape. Based on the above discussion, Liu et al. (2003a, 2004, 2005, 2007) developed models for the ductility and toughness of Al alloys by taking constituents as inner cracks and deducing the specific expression of microscopic plastic strain of the ligament. The models have been successfully used on 2XXX and 6XXX aluminum alloys. The effects of the solution treatment, quenching and aging procedures (change the sizes and volume fractions of the constituents and precipitates) on the ductility and toughness can be determined when incorporating the aging hardening behavior of the aluminum alloy (Liu et al., 2003b) into the models (Liu et al., 2005, 2007). Similar to the method used by Liu et al. (2003a, 2004, 2005, 2007), Song and Huang (2007) developed a model for the ductility of SiC reinforced aluminum
623
alloy composites. By incorporating the aging hardening behavior of the composites during aging (Song et al., 2007), the model can be used to calculate the evolution of the ductility with aging time and aging temperature. The model includes the fact that the geometrically-necessary dislocations are stored in the matrix due to the large difference in thermal expansion between matrix and SiC particles. The generation of dislocations is generally caused by quenching from the recrystallization or solution treatment temperature. In this paper, we extend the analytic expressions of the ductility from Al alloys (Liu et al., 2004, 2005, 2007) and SiC reinforced Al alloy composites (Song and Huang, 2007) to arbitrary multiple phase materials, in which discontinuously distributed second phases (including both CFPs and NCFPs) are distributed in the continuously distributed matrix phase. The prediction of the macroscopic ductility of the multiple phase materials based on the microstructures requires only a small amount of time and cost, compared to the model by Vernerey et al. (2007a,b).
2. Model 2.1. Ductile fracture mechanisms In most multiple phase materials with a continuously distributed matrix, plastic deformation starts from the matrix. Then, the load is transferred from the ductile matrix to the discontinuously distributed second phase particles, which are surrounded by the matrix. The second phase particles can be divided into two types: large-sized CFPs and small-sized NCFPs. In general, the large sized CFPs have low fracture strength such as constituents in aluminum alloys (Hahn and Rosenfield, 1975; Thompson, 1975) or low particle-matrix interface bonding strength such as SiC particles in SiC/Al metal matrix composites (Fang et al., 2006). Microcracks are initiated due to the stress concentration that results if the load continues to increase. For NCFPs, their shapes are kept unchanged when the ligaments deform. Thus, geometrically-necessary dislocations must be stored around these phases due to nonhomogeneous deformation as described by Ashby (1970). The geometrically-necessary dislocations contribute to the work-hardening of the materials, and inevitably affect their ductility since voids will be initiated around the NCFPs if the density of the dislocations reaches the saturated situation. In general, it is believed that both the microscopic plastic strain of ligaments between two microcracks and the crack-opening displacement reach their critical values when the density of the dislocations reaches the saturated value. At that time, fracture in an unstable manner occurs ahead of the microcrack tips and voids start to grow and coalesce. Fig. 1 illustrates the mechanism of the ductile fracture of a typical multiple phase material, showing that six stages exist from the beginning of the deformation to the final fracture. It should be noted that in a multiple phase material, sometimes more than one type of the CFP exist in the ductile matrix (such as in SiC/Al composites, both
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M. Song et al. / Mechanics of Materials 41 (2009) 622–633
Fig. 1. Schematic of the fracture process of multiple phase materials during deformation.
constituents and SiC particles are CFPs), and more than one type of the NCFP with various shapes and sizes exist in the matrix (such as in Al–Zn–Mg–Cu–Zr alloy, dispersoids of Al3Zr and precipitates of MgZn2 are NCFPs). It should also be noted from stage 4 that the unstable deformation of the materials starts at the onset of the voids initiation around the NCFPs. At that time, necking will be observed during tensile deformation for most ductile materials. 2.2. Multi-scale model for the ductility of multiple phase materials If we assume that the distribution of the second phase particles in the matrix is in a cubic array, the one-dimensional example of the multiple phase materials is illus-
Fig. 2. Geometrical description of the model for the ductility of multiple phase materials.
trated in Fig. 2, in which various types of NCFPs distribute between two CFP particles. In Fig. 2, e1ij and e2ij are the strain caused by these two microcracks during deformation, respectively, while eij is the sum of e1ij and e2ij . For simplicity, we assume that the size and interspacing of the two neighboring microcracks (caused by CFPs) are 2a and k, respectively. Thus, at a distance, r (r is the distance between two neighboring microcracks), the strain caused by these two microcracks can be written as Hutchinson (1968), Rice and Rosengren (1968) and Kanninen and Popelar (1985)
1
I
ecp ¼ ~ en ðhÞ 0:405ph
n 1þn
kf 1 2r f
n 1þn
ec 2
ð1Þ
where ~en ðhÞ is the effective value of reduced coefficient ~ eij ðhÞ when h = 0, ecp is the macroscopic fracture strain (ductility), ec is the critical value of the microscopic plastic strain in the ligament, e, n is the inverse of the strain hardradius of ening exponent, kf and rf are the spacing and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the inner microcracks, respectively, I ¼ 10:3 0:13 þ n and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ¼ 3=2 1 þ 3n. Eq. (1) was first developed to describe the strain distribution around a microcrack in a ductile material during deformation. Thus, the use of the equation is for ductile materials or materials with ductile matrix, instead of brittle materials. One should be noted that the application of the equation to a ductile material (or a material with ductile matrix) with more than one microcracks requires that the distance between two neighboring microcracks is not less than the length (usually the size of CFPs) of the microcracks (in general, the volume fraction of CFPs is not more than 50%). Since the development of the equation does not depend on deformation style, thus, using the equation for both tensile and compressive loading is a good first approximation. It should be noted that Eq. (1) has also been used in Liu et al. (2003a, 2004, 2005, 2007) for aluminum alloys and Song and Huang (2007) for SiC reinforced metal matrix composites.
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As described above, microcracks are initiated from the CFPs. Since not all of the CFPs will initiate microcracks during deformation, the cracking fraction of a phase (i) is defined as Fi. Thus, the volume fraction of the microcracks (ui) caused by one type of the CFP is with the form of
ui ¼ F i uci ¼ 2pK i r3i =k3i
ð2Þ
where uci is the volume fraction of one type of the CFP (i) and Ki is the shape factor (aspect ratio) and equals 1 for spherical particles. According to Ashby (1970), the microscopic plastic strain (e) is characterized by a geometric slip distance, kg, analogous to the slip distance in pure crystals, but determined only by the microstructure. The density of the geometrically-necessary dislocations (qg) is then given by (Ashby, 1970)
qg ¼
4e kg b
ð3Þ
where b is the Burgers vector of the dislocations. Ashby (1970) showed that for alloys containing more or less spherical particles, kg = rs/fs, where rs is the particle radius forffi alloys containing plate/ and fs the volume fraction; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rod-like particles, kg ¼ 3 pr 2s hs =fs (equals the particle spacing), where hs is the height of the plate-like particles or length of the rod-like particles; for pure polycrystals, it is
ec1 ¼ ec2 ¼ ¼ eci ¼ ¼ ecx ¼
ec ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðec1 Þ2 þ ðec2 Þ2 þ þ ðeci Þ2 þ þ ðecx Þ2 ¼
qcg ¼
4ec kg b
1
I
n 1þn
qg ¼ q1 þ q2 þ þ qi þ þ qx ¼
x X
qi
kf 1 2r f
n 1þn
From the aforementioned analyses, Eq. (4) can be rewritten as
ð7Þ
The critical value of the microscopic plastic strain, ec, can be expressed by the combination of all the single values, eci , in the form of Pythagorean theorem, not the linear addition due to the existence of the overlap effect, with the form of
ð8Þ
So the strain to fracture or the ductility of the multiple materials can be obtained by combining Eqs. (1) and (8) as
pffiffiffi bk1 k2 ki kx qcg x 8ðk2 k3 kx þ k1 k3 . . . kx þ þ k1 k2 ki1 kiþ1 kx þ . . . þ k1 k2 kx1 Þ
For multiple phase materials with more than one type of the NCFPs, the density of the partial geometrically-necessary dislocations due to the non-homogeneous deformation around each type of the non-cracking formation particles has the relation of (Ashby, 1970)
ð6Þ
i¼1
pffiffiffi bk1 k2 ki kx qcg x 4ðk2 k3 kx þ k1 k3 . . . kx þ þ k1 k2 ki1 kiþ1 kx þ . . . þ k1 k2 kx1 Þ
ð4Þ
ð5Þ
where Ci is the scaling factor and considered as unity here, and subscribe x is the number of the types of the NCFPs. Eq. (5) was originally developed by Ashby (1970), and was then subsequently used by Liu et al. (2004) when dealing with the ductility of aluminum alloy. The equation indicates that the ratio of the partial geometrically-necessary dislocation densities is associated with the ratio among particle spacings of the various populations of the particles. Liu et al. (2004) showed that for materials containing continuously distributed ductile matrix and non-continuously distributed second phase particles, the equation is a very good first approximation. One should be noted that the scaling factor Ci does not generally equals unity since the degree of the inhomogeneous microscopic deformation in the ligaments depends on the size of the particles. Here in this paper, for the first approximation, Ci is assumed to be unity for simplicity. The density of the geometricallynecessary dislocations (qg) is assumed to be the sum of the density of all the partial geometrically-necessary dislocations by
bk1 k2 k3 . . . kx qcg 4ðk2 k3 kx þ k1 k3 . . . kx þ þ k1 k2 ki1 kiþ1 kx þ . . . þ k1 k2 kx1 Þ
proportional to the grain size. When the density of the geometrically-necessary dislocations (qg) reaches the critical value, qcg , the microscopic strain (e) reaches the critical value, ec, and macroscopic fracture appears. At that time, Eq. (3) changes to
ecp ¼ ~ en ðhÞ 0:405ph
C 1 q1 k1 ¼ C 2 q2 k2 ¼ ¼ C i qi ki ¼ ¼ C x qx kx
ð9Þ
If we define ðecp ÞR as the reference ductility, the normalized ductility of a multiple phase material, R, can be expressed as
R ¼ ecp =ðecp ÞR
ð10Þ
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Eqs. (9) and (10) indicate that if the ductility of a multiple phase material under a specific condition (volume fraction and size of the phases) is known, the ductility under other conditions can be calculated.
3. Prediction of the normalized ductility using the present model 3.1. Effects of the CFPs on the ductility In this subsection, we will study the effects of CFPs on ductility of multiple phase materials. First, a single type of the CFP, which is distributed uniformly in the ductile matrix, will be considered in the calculation. Many metals and alloys belong to this system, such as a commercial pure aluminum alloy, in which brittle constituents (formed by impurities with aluminum) are distributed in the Al matrix. To calculate the normalized ductility, Eq. (9) can be rewritten as
c p
e
n "sffiffiffiffiffiffiffiffiffiffiffiffi #1þn n 1þn 1 I ec 3 p K ¼ 1 ~en ðhÞ 0:405ph 4 Fu 2
ð11Þ
Fig. 3. Effects of the volume fraction of the CFP on the normalized ductility with (a) various cracking fractions (K = 1) and (b) various shape factors (F = 1).
Fig. 3 shows the effect of the CFP on the normalized ductility of the materials. The reference was defined as the volume fraction of the CFP of 0.01% (it should be noted that the reference can not be set as zero, otherwise a singularity will appear when applying Eq. (10) during calculation). The effects of the cracking fraction F (Fig. 3a) and the shape factor K (Fig. 3b) on the normalized ductility have been considered in the calculation. It can be seen that no matter what values of the cracking fraction and shape factor are, the normalized ductility decreases as the volume fraction of the CFP increases. This indicates that an increase in the CFP will degrade the ductility of the material. It should be noted that the increase in the cracking fraction will also degrade the normalized ductility, since the increase in the cracking fraction increases the real fraction of the microcracks as well. In contrast to the cracking fraction, the increase in the shape factor will improve the ductility of the material. This is not surprising, since a sharp crack can cause a much higher stress concentration at the crack tip if disk-like cracks are perpendicular to the stress direction (low shape factor value) rather than along the stress direction (high shape factor value). Thus, the higher the shape factor value, the higher the ductility. From Fig. 3b, we can notice that when the shape factor is very low, the normalized ductility will decrease very quickly when the volume fraction of the CFP reaches a critical value (for example, when the shape factor is 0.1, the critical volume fraction is about 0.1). This is caused by the unstable propagation of the microcracks and sudden fracture of the materials under externally applied stress due to the high stress concentration and overlapping. Another important system is a material containing two types of the CFPs. Many alloys and metal matrix composites belong to this system, such as constituents and SiC particles in the SiC reinforced aluminum alloy composites. Since the effect of a single type of the CFP on the normalized ductility has been studied in Fig. 3, here we concentrate on the effect of the second type of the CFP on the normalized ductility. Fig. 4 shows the effect of the second type of the CFP on the normalized ductility. The reference was defined as the volume fraction of the first type of the CFP of 10%, and two types of the CFPs are assumed to have
Fig. 4. Effects of the volume fraction of the second type of the CFP on the normalized ductility.
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Fig. 6. Effects of the volume fraction of the second type of the CFP on the normalized ductility with various shape factors.
Fig. 5. Effects of the volume fraction of the second type of the CFP on the normalized ductility with various cracking fractions of (a) the first type of CFP and (b) the second type of CFP.
the similar size, and the cracking fraction and shape factor are both 1. It can be seen that the normalized ductility decreases with the increase in the volume fraction of the first type of the CFP, which has also been shown in Fig. 3. Most importantly, the normalized ductility decreases with the increase in the volume fraction of the second type of the CFP. The higher the volume fraction of the second type of the CFP, the higher the volume fraction of the microcracks and the lower the normalized ductility. It should be noted from Fig. 4 that the normalized ductility decreases slightly quicker with the volume fraction of the second type of the CFP if the volume fraction of the first type of the CFP is low. For a material containing two types of the CFPs, the fraction of the microcracks due to the second type of the CFP decreases as the volume fraction of the first type of the CFP increases. Thus, the effect of the second type of the CFP on the normalized ductility decreases with the increase in the volume fraction of the first type of the CFP. The effects of the cracking fraction of the two types of the CFPs have been illustrated in Fig. 5. The reference was defined as the volume fraction of the first type of the CFP of 10% with its corresponding cracking fractions. The two types of the CFPs are assumed to have the similar size, and shape factor are both 1. Similar to Fig. 4, the ductility decreases as the volume fraction of the second type of the CFP increases, no matter what cracking fraction of
the first or second type of the CFP is. It can be seen that the cracking fractions of both types of the CFPs have important effects on the normalized ductility of the materials. Increasing the cracking fraction of the first type of the CFP can obviously improve the normalized ductility if the volume fraction of the second type of the CFP remains constant (Fig. 5a). This is not surprising, since the microcracks are caused by both types of the CFPs, the increase in the cracking fraction of the first type of the CFP inevitably increases its fraction of the total micocracks, thus decreases the fraction of the microcracks due to the second type of the CFP, and the effect on the normalized ductility. This is essentially the case for SiC reinforced aluminum composites, in which the increases in the cracking fraction of the SiC particles will decrease the effect of the constituents on the normalized ductility (Song and Huang, 2007). Contrary to Fig. 5a, increasing the cracking fraction of the second type of the CFP obviously degrades the normalized ductility if the volume fraction of the first type of the CFP remains constant (Fig. 5b), since the increase in the cracking fraction of the second type of the CFP increases its fraction of the total micocracks, thus increases the effect on the normalized ductility. Another important calculation is the effect of the shape factor of both types of the CFPs on the normalized ductility, which has been illustrated in Fig. 6. The reference was defined as the volume fraction of the first type of the CFP of 10%. The cracking fractions of two types of the CFPs are assumed to be 1. It can be seen that the increase in the shape factor will improve the normalized ductility of the material. Similar to the discussion for Fig. 3b, a sharp crack can cause a much higher stress concentration at the crack tip if disk-like cracks are perpendicular to the stress direction rather than along the stress direction. Thus, the higher the shape factor value, the higher the ductility. 3.2. Effects of the NCFPs on the ductility In this subsection, we will study the effects of the NCFPs on the ductility of multiple phase materials. From Section 2, we know that the NCFPs influence the ductility of the materials by changing the geometric slip distance (liga-
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Fig. 8. Effects of the second type of the NCFP on the normalized ductility.
increases linearly with the size of the NCFP if the volume fraction remains constant, which indicated that largesized particles can substantially improve the ductility of the materials. This is reasonable by taking into account the fact that the geometric slip distance increases with the particle size when the volume fraction of the NCFP remains constant. In many other materials, two types of the NCFPs are distributed in the matrix, such as Al–Zn–Mg–Cu, in which meso-stable g0 phase and T0 phase are distributed uniformly in the Al matrix. To calculate the normalized ductility of this system, Eq. (9) can be rewritten as Fig. 7. Effects of the (a) volume fraction and (b) size of the NCFP on the normalized ductility. Note the volume fraction remains constant in Fig. 7(b).
ment size, interspacing between two neighboring particles) through volume fraction and size variations. First, a single type of the NCFP, which is distributed uniformly in the ductile matrix, will be considered in the calculation. Many materials belong to this simple system, such as Al–Sc, Al–Zr and Al–Cr alloy, in which a single type of dispersoids (Al3Sc, Al3Zr or Al3Cr) is distributed in the Al matrix. To calculate the normalized ductility, Eq. (9) can be rewritten as
1
I
ecp ¼ ~ en ðhÞ 0:405ph
n 1þn
kf 1 2r f
n 1þn
bkqcg 8
ð12Þ
Fig. 7 shows the effects of the NCFP on the normalized ductility of the materials. The reference was defined as the volume fraction of the NCFP of 0.0001%, and the NCFP is assumed to be in round shape. It can be seen from Fig. 7a that the normalized ductility of the material decreases with the increase in the volume fraction of the NCFP, with a very steep trend when the NCFP begins to appear. The trend becomes much milder when the volume fraction of the NCFP continues to increase. The decrease in the normalized ductility is due to the decrease in the geometric slip distance with the increase in the volume fraction of the particles. It can also be seen from Fig. 7b that the normalized ductility of the material
1
I
ecp ¼ ~ en ðhÞ 0:405ph
n 1þn
kf 1 2r f
pffiffiffi n 1þn 2bk1 k2 qcg 8ðk2 þ k1 Þ
ð13Þ
Since the effect of a single type of the NCFP on the normalized ductility has been studied in Fig. 7, here we concentrate on the effect of the second type of the NCFP on the normalized ductility, as illustrated in Fig. 8. The reference was defined as the volume fraction of the first type of the NCFP of 0.0001%, and two types of the CFPs are assumed to be in round shape. It can be seen that the normalized ductility decreases with the increase in the volume fraction of the second type of the NCFP, with the similar trend as illustrated in Fig. 7a. That is, it decreases very quickly with the volume fraction of the second type of the NCFP at the very beginning, and then the decreasing speed slows down. It can also be seen that the normalized ductility increases with the increase in the volume fraction of the first type of the NCFP if the volume fraction of the second type of the NCFP remains constant. This does not mean that the real ductility of the material increases with the volume fraction of the first type of the NCFP, but only indicates that the influence fraction of the first type of the particles increases when the volume fraction increases. This is very similar to Fig. 5, in which the influence fraction of the first type of the CFP also increases with the volume fraction. A further extension of the calculation is using the model to calculate the normalized ductility of the materials containing three types of the NCFPs. In this case, Eq. (9) can be rewritten as
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4.1. Experimental methodology
Fig. 9. Effects of the third type of the NCFP on the normalized ductility. Note the total volume fraction of the first and second types of the NCFPs remains constant.
c p
e
pffiffiffi n n 1þn 1þn 3bk1 k2 k3 qcg kf 1 I ¼ 1 ~en ðhÞ 0:405ph 2r f 8ðk2 k3 þ k1 k3 þ k1 k2 Þ ð14Þ
A typical system containing three types of the NCFPs is Al–Zn–Mg–Cu–Sc alloy, in which meso-stable g0 phase and T0 phase, and stable Al3Sc dispersoids are distributed uniformly in the matrix. Fig. 9 shows the effect of the third type of the NCFP on the normalized ductility. The reference was defined as the volume fraction of the first and second types of the NCFPs of 0.0001% and 0, respectively, and all three types of the NCFPs are assumed to be in round shape. It can be seen that the normalized ductility decreases with the increase in the volume fraction of the third type of the NCFP. Since the total volume fraction of the first and second types of the non-cracking phases remains unchanged during calculation, the effect of the volume fraction ratio of the first and second types of the NCFPs can be illustrated by comparing the curves in Fig. 9. It can be seen that the larger the difference between the volume fractions of the first and second types of the particles, the lower the normalized ductility. However, the effect of the difference between the volume fractions is not so important, compared to the effect of the volume fraction of the third type of the NCFP. 4. Application of the model to a real material system In Section 3, we simulated the effects of the CFPs and NCFPs on the ductility of multiple phase materials. To further prove the accuracy of the model, we apply the model to a real material system – Al–Zn–Mg alloy. In Al–Zn–Mg alloys, several types of the particles exist in the matrix. These particles include CFP (constituents) and NCFP (precipitates). Brittle constituents are resulted from the impurities or excess micro-alloying elements with Al matrix during solidification process, while precipitates (MgZn2) are resulted from the micro-alloying elements during aging procedure. In the following subsection, we will describe the experiment method to obtain a distribution of the volume fraction of the constituents and precipitates in Al–Zn–Mg alloy.
The nominal composition of the alloy used in the present study has been shown in Table 1. The alloy was prepared in an induction furnace in an argon atmosphere. The as-cast ingot was homogenized at 455 °C for 10 h, followed by air cooling to room temperature. Then the ingot was converted into rods by hot extrusion at 450 °C, with an extrusion ratio of 9:1. The extruded rods were solution treated under a series conditions, followed by cold water quenching (room temperature, 20 °C) and aging treatments (120 °C). The solution treatments include traditional solution treatment and enhanced solution treatments (EST). Detailed description of the solution treatment can be found in Table 2. It should be noted that the EST uses an increasing temperature method instead of the isothermal treatment since some multi phase eutectic has a low melting point. The prior dissolution of the multi phase eutectic during the EST enhances the melting point temperature and, thus allows a further high temperature solution treatment to be used. Aging treatment with various times (under a constant temperature of 120 °C) was subsequently applied in order to produce variously sized and volume-fractioned precipitates. The tensile ductility was tested at room temperature using a smooth dog-bone-shaped tensile specimen that had a gage size of 6 mm in diameter and 40 mm in length at a constant strain rate of 5104 s1 by an Instron-8802 testing machine. All the specimens have an axis along the extrusion direction. The sizes and volume fractions of the constituents and precipitates at different stages (various solution treatments and aging treatments) have been measured using optical microscopy (OM) and transmission electron microscopy (TEM). The TEM specimens (thin foils) were prepared by twin-jet electro-polishing in a 30% nitric acid and 70% methanol solution at 35 °C and examined in a Tecnai G2 20 microscopy operating at 200 kV. The size and volume fraction of the precipitates were determined from at least 100 random precipitates seen edge on. Since the radius of the precipitates in the matrix was in the same
Table 1 Nominal composition of the tested Al–Zn–Mg alloy (mass fraction, %). Zn
Mg
Cu
Mn
Al
4.0
3.2
0.15
0.15
Bal.
Table 2 The procedures of the solution treatments. Solution type
Traditional solution treatment
Enhanced solution treatment (EST)
S1 S2
450 °C/2 h 450 °C/2 h
S3
450 °C/2 h
S4
450 °C/2 h
– Increasing temperature to 460 °C with a rate of 5 °C per hour Increasing temperature to 470 °C with a rate of 5 °C per hour Increasing temperature to 480 °C with a rate of 5 °C per hour
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order as and even greater than the thickness of the thin foil, a correction was employed for differentiating the real dimension from the observed one (Crompton et al., 1966) and the volume fraction of the precipitates in a thin foil projection was determined by (Gilmore and Starker, 1997)
fv ¼
2pr lnð1 AÞ pr þ 4t
ð15Þ
where r is the half radius of the precipitates, t is the foil thickness, and A is the project area fraction of the precipitates, determined by the point count method. The foil thickness was easily obtained by utilizing a grain boundary fringes technique (Edington, 1976). 4.2. Microstructures and tensile ductility Fig. 10 shows the optical microstructures of the Al–Zn–Mg alloy after various solution treatments. It can be seen that a large number of the constituents exist in the matrix after traditional solution treatment (S1), while the number and the aggregation of the constituents have substantially been decreased after the EST (S2, S3 and S4). It can be seen that the higher the ending temperature of the EST, the less the number of the constituents. These results indicate that the application of the EST facilitates the dissolution of the soluble constituents and hence the decrease in the volume fraction of the constituents. It should be noted that after S4 solution treatment, the
volume fraction of the constituents is very small, and almost no constituents can be found from the optical microscopy. Table 3 illustrates the volume fraction of the constituents and the tensile ductility of the specimens after various solution treatments. It can be seen that the EST substantially improves the tensile ductility and decreases the volume fraction of the soluble constituents. Previous studies (Hahn and Rosenfield, 1975; Thompson, 1975) indicated that the coarse constituents are brittle and have low fracture strength and, thus, generate microcracks during deformation. The larger the volume fraction of the constituents, the lower the tensile ductility. The constituents are normally resulted from the presence of the Fe and Si impurities or excess amounts of major alloying elements such as Mg, Zn and Cu (Hahn and Rosenfield, 1975; Thompson, 1975). The EST can substantially improve the tensile ductility of the alloy by decreasing the number of the microcrack sources (volume fraction of the constituents).
Table 3 Volume fraction of the constituents and tensile ductility of the alloy after various solution treatments (before aging treatment). Aging time (h)
S1
S2
S3
S4
Volume fraction of the constituents (%) Tensile ductility (%)*
4.7 11.5
3.2 11.8
1.9 12.4
0 16.5
*
All the data are the average values from five tests.
Fig. 10. OM images of the tested Al–Zn–Mg alloy after (a) S1, (b) S2, (c) S3 and (d) S4 treatments.
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Fig. 11. TEM images of the tested Al–Zn–Mg alloy at (a) under-aged stage (10 h), (b) peak-aged stage (24 h), (c) over-aged stage (50 h) and (d) the SADPs of the precipitates at peak-aged stage.
Fig. 11 shows the TEM microstructures of the Al–Zn–Mg alloy after aging treatment for various length of time. It can be seen that a large number of the precipitates exist in the matrix after aging treatment, while the size of the precipitates increases with aging time. These precipitates are the typical strengthening particles (MgZn2) in Al–Zn–Mg alloy. The evolutions of the volume fraction of the precipitates and the tensile ductility of the alloy with aging time have been shown in Table 4. Note that all the specimens were applied S4 solution treatment before aging to minimize the volume fraction of the constituents and increase the super-solution degree. It can be seen that the volume fraction of the precipitates increases with the aging time until it reaches a constant value after 24 h. At that time, the alloy reaches the peak-aged stage, and the volume fraction of the precipitates remains constant, while the size increases by the growth of larger precipitates and the dissolving of smaller precipitates. It can be seen that as the aging time increases, the tensile ductility decreases until the aging time reaches 24 h. Then the tensile ductility increases as
the aging time continues to increase. The decrease in the tensile ductility at under-aged stage is due to the increase in the volume fraction of the precipitates, which inevitably decreases the geometric slip distance, while the increase in the tensile ductility after aging to peak-aged stage is due to the increase in the precipitate size under constant volume fraction, which inevitably increases the geometric slip distance. 4.3. Comparison between the calculated results and the experimental data In this subsection, the experimental data listed in Tables 3 and 4 have been used to verify the model accuracy. In the calculation, both constituents and precipitates are assumed to be in round shape. In addition, the constituents are assumed to be totally fractured. Fig. 12 shows the model predicted relation between the normalized tensile ductility and volume fraction of the constituents. The experimental data listed in Table 3 has also been included.
Table 4 Volume fraction of the precipitates and tensile ductility of the alloy after aging treatment (the specimen was applied S4 solution treatment before aging to minimum the volume fraction of the constituents and increase the super-solution degree). Aging time, s (h)
0 (0)
Volume fraction of precipitates (%) Tensile ductility (%)a
0
a b
16.5
1800 (0.5) 0.035 10.4
18,000 (5)
36,000 (10)
64,800 (18)
86,400 (24)b
180,000 (50)
288,000 (80)
540,000 (150)
0.068
0.082
0.094
0.1
0.103
0.102
0.101
9.1
8.6
7.8
7.9
8.4
8.9
9.1
All the data are the average values from five tests. The alloy reaches the peak-aged stage after aging for 24 h.
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Fig. 12. Measured and predicted normalized tensile ductility of the tested alloy with the volume fraction of the constituents.
It can be seen that model prediction agrees well with the experimental data, in which the normalized tensile ductility decreases with the increase in the volume fraction of the constituents. Fig. 13 shows the relation between the experimental determined volume fraction of the precipitates and model predicted normalized tensile ductility and aging time. It should be noted that the calculation of the normalized tensile ductility requires the volume fraction of the precipitates and the geometric slip distance (distance between two neighboring precipitates). In principle, the volume
fractions of the precipitates in Table 3 and Fig. 13a have been used in the model calculation with the reference being defined as the volume fraction of the precipitates of 0.0001%, while the geometric slip distance can be easily obtained since the volume fraction and size of the precipitates can be determined experimentally. The experimental data of the tensile ductility listed in Table 4 has also been included in Fig. 13b. It can be seen that the normalized tensile ductility decreases with the aging time until the alloy reaches the peak-aged stage. Then the normalized tensile ductility increases with the aging time. This is essentially the case for aluminum alloys, in which the tensile ductility and fracture toughness of aluminum alloys decrease with aging time at under-aged stage, but increase with aging time at over-aged stage. During under-aged stage, the volume fraction of the precipitates increases and the geometric slip distance decreases with aging time, and thus the tensile ductility decreases with aging time. However, when the alloy reaches the peak-aged stage, the excess solute atoms are totally exhausted and the growth stage of the precipitates is terminated. At that time, the volume fraction of the precipitates remains constant, and the precipitates size and the geometric slip distance increase through the growth of the larger precipitates and the dissolving of the smaller precipitates, and thus the tensile ductility increases. It can also be seen from Figs. 12 and 13 that the model calculations are in good agreement with the experimental counterparts, which indicated that the model can be used to quantify the ductility of a material based on its microstructure. An important feature of the model is that calculation of the ductility of a multiple phase material can be run on a personal computer and requires only a small amount of time, compared to the other models based on FEM, which may require long time calculation and high cost. 5. Conclusions In this paper, we developed a multi-scale model for the ductility of multiple phase materials. The effects of various types of the CFPs and NCFPs on the ductility can be calculated based on the model calculation. It has been shown that the model predictions are in good agreement with the experimental data when testing on an Al–Zn–Mg alloy. Based on model predictions and experimental data, the following conclusions are drawn.
Fig. 13. (a) Experimentally determined volume fraction of precipitates and (b) measured and predicted normalized tensile ductility of the tested alloy as a function of aging time.
(1) The volume fraction, fracture fraction and shape factor of the CFPs have important effects on the ductility of a multiple phase material. The normalized ductility decreases as the volume fraction and fracture fraction of the CFPs increase, but increases as the shape factor of the CFPs increases. (2) The volume fraction and size of the NCFPs have important effects on the ductility of a multiple phase material. The normalized ductility decreases as the volume fraction of the NCFPs increases. Under the condition of the constant volume fraction, the normalized ductility increases linearly with the size of the NCFPs.
M. Song et al. / Mechanics of Materials 41 (2009) 622–633
(3) For a typical material with precipitation hardening during aging, the ductility of the material decreases with aging time until the material reaches the peakaged stage, and then the ductility increases with aging time. The decrease in the ductility at the early stage of aging is due to the increase in the volume fraction of the precipitates and decrease in the geometric slip distance, while the increase in the ductility after peak-aged stage is due to increase in the geometric slip distance since the volume fraction of the precipitates remains constant but the precipitate size increases by the growth of the larger precipitates and the dissolving of the smaller precipitates.
Acknowledgements This work was supported by National Natural Science Foundation of China (No. 50801068), Creative Research Group of National Natural Science Foundation of China (No. 50721003) and Hunan Provincial Natural Science Foundation of China (No. 07JJ3117). One of the authors would also like to thank the support of Chinese Postdoctoral Science Foundation (No. 20070410303). References Ashby, M.F., 1970. The deformation of plastically non-homogeneous materials. Philosophical Magazine 21, 399–424. Chan, K.S., 1995. A fracture model for hydride-induced embrittlement. Acta Metallurgical Materialia 43, 4325–4335. Crompton, J.M.G., Waghorne, R.M., Brook Jr, G.B., 1966. The estimation of size distribution and density of precipitates from electron micrographs of thin foils. British Journal of Applied Physics 17, 1301–1305. Edington, J.W., 1976. Practical Electron Microscopy in Materials Science. Von Norstrand Reinhold Company, London. p. 207. Fang, J., Xiao, B., Zuo, T., Sang, J., Zhang, W., Xu, J., Shi, L., 2006. Effect of heat treatment on strength and ductility of SiCp/Al composites. The Chinese Journal of Nonferrous Metals 16, 228–234.
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