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Materials Science and Engineering A272 (1999) 90 – 98 www.elsevier.com/locate/msea
Basic aspects of the co-deformation of bcc/fcc materials C.W. Sinclair *, J.D. Embury, G.C. Weatherly Department of Materials Science and Engineering, McMaster Uni6ersity, Hamilton, Ont., Canada L8S 4L7
Abstract Materials consisting of a hard phase embedded within a softer, ductile matrix represent an industrially important class of materials. Recent work has shown that in many cases such two-phase materials can be co-deformed to produce alloys with strengths far surpassing those that would be predicted from a simple ‘rule of mixtures’ estimate. Alloys comprising a bcc phase embedded within an fcc matrix are representative of materials exhibiting this type of behaviour and may be considered as excellent model systems for the study of the co-deformation process. Here, the important processes associated with co-deformation will be reviewed and new results from the co-deformation of a Cu – Cr and a Ni – W alloy will be described. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Co-deformation; Cu–Cr alloy; In situ composites; Multifilamentary composites; Ni – W alloy
1. Introduction The deformation of a hard phase embedded in a softer, more ductile phase is the basis of many industrial processes for example the production of a number of ultra high strength materials in the form of wires [1]. In order to provide an overview of the problem of co-deformation we consider the case of a bcc phase embedded within a fcc matrix. This provides a valuable reference system because the deformation of the individual bcc and fcc structures is well understood, both in terms of operating slip systems, stored dislocation structures and more macroscopic descriptions such as texture formation [2]. Further, because of the marked temperature dependence of the Peierls stress in many bcc systems, the relative strength of the bcc and fcc phases can be varied by performing the co-deformation studies over the temperature range 77 – 298 K. A number of authors have examined the macroscopic aspects of co-deformation in terms of the occurrence of multiple necking and shape instabilities for a
Dedicated to Professor Herbert Herman on the occasion of his 65th birthday. * Corresponding author. Tel.: + 1-905-525-9140/27226; fax: +1905-528-9295. E-mail address: [email protected] (C.W. Sinclair)
variety of stress states [3–5]. However in the current work emphasis is given to delineating a variety of factors that influence the ability to enforce co-deformation of the embedded bcc phase including some new results on the Ni–W and Cu–Cr systems. The problem of the deformation of a bcc phase embedded in an fcc phase is central to such industrial problems as the deformation of a/b brasses, the deformation of two-phase (ferrite–austenite) stainless steels during warm rolling and the co-deformation of copper with a variety of embedded bcc phases such as Fe, Nb, and Cr drawn to produce ultra high strength multifilamentary conductors [6–8].
2. Formal description of co-deformation We can consider the operative slip systems in fcc and bcc crystals and the requirements for compatible deformation in the manner outlined in Fig. 1. Clearly the transmission of slip from the fcc to the bcc phase can be considered analogous to that occurring at a grain boundary in a single-phase alloy and will be influenced by the relative orientation of slip systems in the two phases. This may in turn be controlled either by the initial orientation of the phases (e.g. the solidification
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structure) or the orientation of the phases after strains sufficient to develop texture. Extensive experimental work on slip transmission across grain boundaries in single-phase materials has defined some general criteria necessary for predicting the slip system(s) which are the most likely to be activated in one grain due to the action of a slip system in an adjacent grain [9–11]. These criteria may be generally stated as [9]; 1. the resolved shear stress from the dislocations approaching the boundary should be a maximum on the activated system [10]; 2. the angle between the lines of intersection for the slip planes of the incident and activated systems should be a minimum at the boundary; and 3. the resulting configuration at the boundary should be one of minimum energy. A variety of processes may occur to reduce the energy of the system, one possibility being the selection of a transmitted slip system in the bcc phase that produces the smallest residual dislocation content at the boundary. One difficulty in applying these rules to the co-deformation of two-phase materials is that they have been verified only under low strain (single slip) conditions in single-phase materials. However, it appears that these conditions do apply for the case of fcc – bcc materials that exhibit the Kurdjumov – Sachs orientation relationship. Work on two-phase stainless steels has shown unambiguously that the slip system activated in ferrite is that which is exactly coincident to an fcc slip system ({111}B 11( 0\) via the Kurdjumov – Sachs orientation relationship [12].
Fig. 1. Schematic diagram illustrating the parameters which are important in the compatible plastic deformation of two-phase fcc/bcc alloys by slip transmission. The selection of a bcc slip system due to slip on an fcc system depends on (i) the system with the highest resultant stress due to the fcc system; (ii) the minimum angle u; and (iii) least energy consumption, i.e. minimum residual Burgers vector at the boundary bb r = bb 2 − bb 1.
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The process of slip transmission as a mechanism for achieving plasticity in embedded bcc phases may be possible only for those embedded phases that lie within a certain size range. The embedded bcc phase produced via eutectic solidification often consists of defect-free single crystals separated by fcc matrix [13–15]. When extracted, such bcc crystals exhibit whisker-like behaviour reaching strengths approaching E/50 when tested in tension [16,17]. Load transfer to the harder phase can be accommodated by either plastic flow or fracture. Langford [18] has illustrated this for the case of pearlitic steel wire. In this system he found that for cementite lamella B10 nm thick, extensive plasticity was exhibited in the carbide, while for lamella \ 100 nm in thickness plasticity was inhibited and fracture of the carbide resulted. Thus the competition between plasticity and fracture is scale dependent and may involve the size of surface irregularities on the embedded phase, the local hydrostatic component of the stress state and the details of transmission of plasticity at the interface.
3. Relative plastic strain in the two phases From the onset of deformation in most two-phase alloys, the imposed plastic strains are not distributed equally between the phases. This strain partitioning is greatest for systems containing constituents with very different flow stresses and rates of work hardening [19]. Typically, the harder embedded phase initially accommodates less of the applied strain than the softer matrix. This is illustrated with reference to Fig. 2 where the partitioning of strain in a Ni–W alloy deformed by rolling at room temperature is illustrated. Here the Ni–W alloy contains isolated single crystal spheroids of tungsten, approximately 1 mm in diameter, embedded within a polycrystalline nickel matrix. The change in the aspect ratio of the tungsten phase was followed by quantitative metallography as a function of the applied plastic strain and used as a measure the average plastic strain in the tungsten phase. Physically, strain partitioning between two phases must result in one phase undergoing additional deformation so as to accommodate the more rigid phase. The resulting complicated flow pattern in the softer phase involves both local work hardening due to the large density of geometrically necessary dislocations and changes in flow pattern due to local stress concentrations [20]. As the flow stress of matrix and embedded phase approach one another due to work hardening of the softer matrix phase and local stresses at the interface, the partitioning of strains tends to become more homogeneous. This is particularly true for processes such as wire drawing where it is observed that the partitioning of strain becomes more homogeneous for
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Fig. 2. Partitioning of strain within a spherodized Ni – W alloy deformed by rolling. The tungsten phase is initially in the form of equiaxed single crystal spheroids of approximately 1 mm diameter. The aspect ratio of the tungsten phase has been used as a measure of the strain in that phase with the strains computed using the minimum, mean and maximum aspect ratios in the distribution. Here, the strain is given by the Von Mises equivalent ovm =
)
)
2 ln(1+ r) ,
3
where r is the rolling reduction.
phase population (i.e. how similar the strain undergone by each crystallite is) should be expected to be heterogeneous with it becoming more homogeneous as the scale of the slip approaches the scale of the embedded phase. In the limiting case where the distribution of slip is much finer than the scale of the embedded phase, relatively homogeneous deformation throughout the embedded phase population should be expected. Fig. 3 shows the fraction of tungsten crystallites which have undergone small or negligible plastic strains (as measured by their aspect ratio) as a function of applied rolling strain for the same spherodized Ni–W alloy as that discussed earlier with reference to Fig. 2. Two distinct regimes of deformation may be seen under these conditions. For applied strains1 of less than 0.90 it is observed in Fig. 3 that the fraction of tungsten crystallites that have undergone appreciable plastic deformation increases with increasing applied strain. Over the same range of strain it is observed by optical microscopy (see legend of Fig. 4 for details of sample preparation) that the distribution of slip within the matrix becomes more homogeneous with increasing applied strain. For an imposed strain of 0.18 the spacing between slip traces on the surface of the sample is significantly larger (5–10 mm) than the diameter of the tungsten crystallites with some regions exhibiting much finer slip than others (e.g. region marked by an arrow in Fig. 4a). At an imposed strain of 0.46 the scale of slip is reduced to the point where it is of the same order as the diameter of the tungsten crystallites (Fig. 4b),
applied strains (o) \1.0 [21]. However, as can be seen in Fig. 2, it appears that in processes such as rolling, the partitioning of strain becomes more, not less, heterogeneous with applied strains above o = 1.0. If this is the case then individual crystallites may or may not lie within regions that are undergoing plastic deformation.
4. Effect of slip distribution An implicit assumption often made in the analysis of deformation is that it occurs homogeneously throughout the microstructure. This description allows for simple models to be developed to describe the process of deformation in single-phase materials. The distribution of both slip and the spatial distribution of particles has important consequences in co-deformation if the deformation of the embedded phase depends on slip transmission. When the scale of the embedded phase becomes of the order of 1 mm it is quite possible for it to become finer than the scale of slip within the matrix. If this is the case then it is possible to imagine that individual crystallites may lie within regions of the matrix which have undergone plastic strains far less than those which are macroscopically imposed. In such a situation the strain distribution within the embedded
Fig. 3. Measured fraction of tungsten crystallites exhibiting aspect ratios between 1 and 2 versus the Von Mises equivalent applied strain. 1 All strains quoted for rolling and plane strain compression are given as Von Mises equivalent.
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Fig. 5. BSE SEM image of a spherodized Ni – W alloy deformed by rolling to ovm =3.0. The tungsten crystallite in the center of the image has been cut by a macroscopic shear band. Other shear bands are also obvious in the image.
Fig. 4. Optical micrographs (under conditions of Nomarski contrast) of spherodized Ni – W eutectic alloy. Samples were polished and given an incremental plane strain compression of 0.5% after larger plane strain deformations of (a) ovm = 0.19 and (b) ovm = 0.46. Observations made in the plane normal to the zero strain direction.
again with the scale of slip varying spatially throughout the sample. The observations made in Figs. 3 and 4 are consistent with the simple arguments made above. However, above strains of 1.0 this argument does not seem to hold as it appears that the fraction of crystallites that have undergone negligible plastic deformation plateaus. The strain at which this occurs coincides approximately with the observation of the initiation of macroscopic shear bands within the nickel phase. Fig. 5 shows a shear band intersecting a tungsten crystallite in a sample deformed in rolling to a strain of 3.07. At a lower magnification (Fig. 6) the extreme variation in the strain between individual tungsten crystallites may be seen in a sample deep etched to reveal the tungsten phase. The result of shear banding is to effectively increase the spacing of slip to the spacing between shear bands (of the order of 10 – 50 mm), thus increasing the chance that crystallites will lie within regions undergo-
ing little or no plastic strain. Similar observations of inhomogeneous deformation of the embedded bcc phase have been made in a/b brass [22], Cu–Cr [14], and Cu–Nb [23] deformed by rolling. A second example of the effect of slip distribution can be seen in a directionally solidified Cu–Cr eutectic alloy fabricated by means of a modified Bridgeman technique with a growth rate of 4 mm/h. In this alloy the average diameter of the Cr fibres is approximately 1
Fig. 6. BSE SEM image of the same sample as in Fig. 5 after deep etching to reveal the tungsten crystallites. The inhomogeneous nature of the deformation is clearly evidenced by the morphology of the tungsten crystallites.
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5. Texture and morphological development
Fig. 7. BF TEM image of a Cr fibre chemically extracted from a directionally solidified Cu–Cr alloy deformed in plane strain compression by ovm = 0.42 with the fibres parallel to the direction of tensile strain. These fibres are characterized by alternating regions which are dislocated (i) and defect free (ii). Prior to deformation these fibres are defect free.
mm, while the lengths may be as much as 1000 times this. In the as-cast state the Cr fibres are essentially defect-free single crystals. If the Cr fibres are extracted from the alloy after it has been deformed it is possible to image them in the transmission electron microscope so as to reveal their dislocation structure. The substructure developed within the fibres mimics the scale of deformation in the matrix with relatively large regions (of the order of a few microns in length) of perfect single crystal along the fibre length bounded by regions of relatively high dislocation content. Fig. 7 illustrates this for a Cr fibre extracted from a Cu – Cr alloy deformed to a strain of 0.42 in plane strain compression with the fibres lying parallel to the rolling direction.
There has been less work, both theoretically and experimentally, aimed at determining the textures developed in two-phase materials compared with that in single-phase materials. However, there is evidence to suggest that in a given deformation process, the fcc and bcc phases develop the same texture components they would in isolation but the rate of texture development may be different [24]. The results of a detailed texture analysis on rolled a/b brass may be used to illustrate this point. Engler et al. [22] found that texture in the a phase was enhanced at medium strains by the presence of the harder b phase and suggested that this was a result of the extra deformation accommodated by the a phase. At higher strains it was found that texture evolution became retarded due to the b phase acting as barriers to the formation of shear bands. The b phase was found to develop a typical bcc fibre rolling texture, although it was weaker than that observed for singlephase b brass. Modeling of the texture evolution led to the conclusion that the b phase deformed under partially relaxed Taylor conditions at low strains while at higher strains the deformation was better described by a Sachs condition (fully relaxed constraints). In other words, conditions of strain homogeneity appears to dominate at lower applied plastic strains, while at higher applied strains, homogeneity of deformation is sacrificed in favour of stress homogeneity. This suggests that at low strains the deformation of the embedded phase is closely linked to that in the matrix while at higher strains the embedded phase is able to deform independently of the matrix. Despite the apparent simplicity of the development of texture in co-deformation, the morphological development of the embedded phase may be complex. At low strains it is often possible to observe steps on the surface of the bcc phase corresponding to slip transmission across the phase interface. However, at larger strains the morphology of the embedded phase may become complicated and individual slip steps may no longer be obvious. Fig. 8 illustrates this for the case of the directionally solidified Cu–Cr eutectic alloy discussed previously with reference to Fig. 7. Fig. 8a shows the morphology of the as-grown Cr fibres after chemical extraction from the Cu matrix, while Fig. 8b and c shows the morphology of extracted fibres after deformation of the alloy by plane strain compression to strains of 0.42 and 0.80, respectively. Embedded bcc single crystals have been found to adopt particularly convoluted cross-sectional shapes upon deformation to large strains (\ 1.0) by wire drawing. This observation is closely related to the one made by Hosford [25], who noted the curling of grains about one another in drawn single-phase bcc metals. Bcc crystals with [110] drawing texture have slip sys-
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Fig. 8. Surface morphology of Cr fibres chemically extracted from a directionally solidified Cu – Cr eutectic alloy. (a) Fibres extracted in as cast condition showing the morphology of the Cr phase. (b) Slip steps on the surface of Cr fibres deformed by plane strain compression at room temperature. (c) Irregular surface morphology of Cr fibres extracted from an alloy rolled to ovm =2.66.
tems oriented in such a way that it is energetically favorable for plane strain deformation to be preferred over the imposed axisymmetric state of strain [25]. The end result, in the case of interest here, is that compatibility between axisymmetric deformation of the matrix and plane strain deformation of the embedded phase is accomplished by the bcc crystals undergoing redundant deformation by curling about their axes [26]. The role of deformation bands which traverse the bcc phase on
the curling phenomena in fcc/bcc alloys has been described by Malzahn Kampe and Courtney [27]. If the embedded bcc phase is polycrystalline and exhibits some ductility at room temperature in the bulk form, the fcc/bcc composite may also show significant ductility. However, in this situation the bcc phase appears to take up a shape intermediate between the curled microstructure observed in embedded single crystals and the applied axisymmetric deformation.
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Such is the case for multifilamentary composites of Cu–Nb formed using the technique of bundling polycrystalline niobium with copper as shown in Fig. 9 (J.T. Wood, unpublished work). On the other hand, if the bcc phase is brittle at room temperature in the bulk form, fracture of the polycrystalline embedded bcc phase appears to be common. In a hypereutectic Cu– Cr alloy it was found that the dendritic Cr phase was ductile as long as the volume fraction was sufficiently low that the dendrites remained isolated [28]. The behaviour of these systems may be attributed to the added constraint associated with the polycrystalline embedded phase. The sources of this added constraint may include the necessity of compatible deformation between bcc grains and the possible modification of the stress state associated with a percolated microstructure.
6. Internal stresses A consequence of the non-uniform plasticity in twophase materials is the development of internal stresses. These internal stresses may result in unexpected mechanical behaviour [29] as well as other properties (e.g. magnetic) that deviate from those observed in the undeformed state [30]. The effect of internal stresses may be to significantly modify the stress state acting on the co-deforming phases compared to the applied stress state. The development of internal stresses due to the necessary compatibility of plastic and elastic strains between two phases may considered in terms of an Eshelby model [31–33]. In this case the Eshelby transformation strain may simply be considered as the difference in plastic strain between the two phases as illustrated in Fig. 10. Such Eshelby type models have been applied to various two-phase materials where both phases are ductile in an attempt to model the development of internal stresses
Fig. 9. Morphology of polycrystalline Nb fibres extracted from a drawn Cu – Nb alloy [26].
Fig. 10. Schematic illustration of the Eshelby method as applied to the case of co-deformation. In this situation the Eshelby transformation strain, o T* may be considered as the difference in the plastic strain between the two phases (as in (a)). Alternatively, it is possible to consider the internal stresses developed by the inclusion alone undergoing a strain of o T* (as in (b)). Using the method of Brown and Clarke [32] the internal stresses due to the deformation may be written as [33], o Iija =fbgijo T* o Iijb =fagijo T* where gij consists of the Eshelby matrix and the elastic properties of the matrix and inclusion [32].
and the overall stress/strain response of the material [34,35]. One manifestation of internal stresses in two-phase materials is the strain-dependent Bauschinger effect observed in fcc/bcc two-phase materials. Verhaeghe et al. [36] have observed such effects in percolated austenoferritic stainless steels. At low plastic tensile strains (B 0.5%), where the ferritic phase is loaded purely elastically, the internal stresses are found to develop rapidly at a linear rate. Above 0.5% strain, localized plastic flow in the ferrite allows for the relaxation of the internal stresses resulting in a reduction of the rate of internal stress accumulation with plastic strain. It is important to note that, although plastic flow in the ferrite acts as a relaxation mechanism, at strains be-
C.W. Sinclair et al. / Materials Science and Engineering A272 (1999) 90–98
yond yielding of the ferrite the internal stresses continue to rise but at a lower rate to that observed prior to yielding. A more direct method for the measurement of the volume-averaged internal strains (those elastic strains associated with the internal stresses) within two-phase materials during co-deformation is neutron diffraction. However, to date there have been few references in the literature to the use of this technique for the measurement of internal stresses during co-deformation. Experiments with Cu – Nb and Cu – Ag two-phase wires deformed to a drawing strain of 2.0 show large internal strains of the order of 0.2 – 0.5% [37]. These internal strains (and thus internal stresses) are of short wavelength in the radial direction with the Cu – Nb composite showing larger internal stresses than Cu–Ag. Like the Bauschinger test described above, the internal stresses are found to be strain dependent in these materials, with the internal stresses rising with applied strain even at large strains.
7. Processes at interfaces In co-deformation the events at the interphase interfaces are of paramount importance and may change as a consequence both of the scale of the structure and the amount of plastic strain. At low strains the interface acts as a barrier to dislocation motion in the softer phase and as a source of dislocations in the embedded phase. Slip transmission across the interface produces residual dislocations at the interface that may rearrange by glide or climb within the interface [38]. The detailed atomic fit between the two lattices can often be described in terms of interface dislocations [39,40]. In many heavily deformed materials the scale of the constituent phases is such that little dislocation – dislocation interaction occurs within the phases but both dislocation production and storage occurs at the interfaces. This has two important consequences: (a) slip occurs by dislocations shuttling between the interfaces and (b) energy storage occurs by the production of additional interfacial area with plastic strain [41]. Models for such deformation mechanisms have been described by Gil Sevillano et al. [41] and Embury and Hirth [42], and have been applied to describe the behaviour of nanoscaled multilayers by Misra et al. [43]. The events at the interfaces may be complicated by chemical effects that can include mixing at the interface, increased solubility due to the fine scale of the structure [44] and the possibility of local amorphisation [45]. Additional effects can arise from coherency stresses or stresses due to the difference in elastic constants across the interface. The detailed delineation of these events requires careful study both by HRTEM and other techniques. However, it is also important to note that
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the scale of events at the interfaces lends itself to atomic simulation and direct comparison of the results of simulation and experiment as illustrated by the recent work of Hoagland and Voter (private communication).
8. Summary The co-deformation of fcc/bcc materials is a problem that extends over a variety of length scales and depends on the specific systems chosen, the scale and morphology of the structure and the mode of deformation. From the brief survey presented above, it is clear that the influence of both the scale of the structure and the distribution of plasticity is of great importance. As deformation increases and the scale of the structure is reduced, the role of the interfaces becomes dominant in the description of co-deformation and the study of the detailed mechanisms of plasticity, dislocation accumulation and strain distribution in the interfacial regions becomes crucial. This clearly represents a rich area for future experimentation and atomic simulation.
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Basic aspects of the co-deformation of bcc/fcc materials C.W. Sinclair *, J.D. Embury, G.C. Weatherly Department of Materials Science and Engineering, McMaster Uni6ersity, Hamilton, Ont., Canada L8S 4L7
Abstract Materials consisting of a hard phase embedded within a softer, ductile matrix represent an industrially important class of materials. Recent work has shown that in many cases such two-phase materials can be co-deformed to produce alloys with strengths far surpassing those that would be predicted from a simple ‘rule of mixtures’ estimate. Alloys comprising a bcc phase embedded within an fcc matrix are representative of materials exhibiting this type of behaviour and may be considered as excellent model systems for the study of the co-deformation process. Here, the important processes associated with co-deformation will be reviewed and new results from the co-deformation of a Cu – Cr and a Ni – W alloy will be described. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Co-deformation; Cu–Cr alloy; In situ composites; Multifilamentary composites; Ni – W alloy
1. Introduction The deformation of a hard phase embedded in a softer, more ductile phase is the basis of many industrial processes for example the production of a number of ultra high strength materials in the form of wires [1]. In order to provide an overview of the problem of co-deformation we consider the case of a bcc phase embedded within a fcc matrix. This provides a valuable reference system because the deformation of the individual bcc and fcc structures is well understood, both in terms of operating slip systems, stored dislocation structures and more macroscopic descriptions such as texture formation [2]. Further, because of the marked temperature dependence of the Peierls stress in many bcc systems, the relative strength of the bcc and fcc phases can be varied by performing the co-deformation studies over the temperature range 77 – 298 K. A number of authors have examined the macroscopic aspects of co-deformation in terms of the occurrence of multiple necking and shape instabilities for a
Dedicated to Professor Herbert Herman on the occasion of his 65th birthday. * Corresponding author. Tel.: + 1-905-525-9140/27226; fax: +1905-528-9295. E-mail address: [email protected] (C.W. Sinclair)
variety of stress states [3–5]. However in the current work emphasis is given to delineating a variety of factors that influence the ability to enforce co-deformation of the embedded bcc phase including some new results on the Ni–W and Cu–Cr systems. The problem of the deformation of a bcc phase embedded in an fcc phase is central to such industrial problems as the deformation of a/b brasses, the deformation of two-phase (ferrite–austenite) stainless steels during warm rolling and the co-deformation of copper with a variety of embedded bcc phases such as Fe, Nb, and Cr drawn to produce ultra high strength multifilamentary conductors [6–8].
2. Formal description of co-deformation We can consider the operative slip systems in fcc and bcc crystals and the requirements for compatible deformation in the manner outlined in Fig. 1. Clearly the transmission of slip from the fcc to the bcc phase can be considered analogous to that occurring at a grain boundary in a single-phase alloy and will be influenced by the relative orientation of slip systems in the two phases. This may in turn be controlled either by the initial orientation of the phases (e.g. the solidification
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structure) or the orientation of the phases after strains sufficient to develop texture. Extensive experimental work on slip transmission across grain boundaries in single-phase materials has defined some general criteria necessary for predicting the slip system(s) which are the most likely to be activated in one grain due to the action of a slip system in an adjacent grain [9–11]. These criteria may be generally stated as [9]; 1. the resolved shear stress from the dislocations approaching the boundary should be a maximum on the activated system [10]; 2. the angle between the lines of intersection for the slip planes of the incident and activated systems should be a minimum at the boundary; and 3. the resulting configuration at the boundary should be one of minimum energy. A variety of processes may occur to reduce the energy of the system, one possibility being the selection of a transmitted slip system in the bcc phase that produces the smallest residual dislocation content at the boundary. One difficulty in applying these rules to the co-deformation of two-phase materials is that they have been verified only under low strain (single slip) conditions in single-phase materials. However, it appears that these conditions do apply for the case of fcc – bcc materials that exhibit the Kurdjumov – Sachs orientation relationship. Work on two-phase stainless steels has shown unambiguously that the slip system activated in ferrite is that which is exactly coincident to an fcc slip system ({111}B 11( 0\) via the Kurdjumov – Sachs orientation relationship [12].
Fig. 1. Schematic diagram illustrating the parameters which are important in the compatible plastic deformation of two-phase fcc/bcc alloys by slip transmission. The selection of a bcc slip system due to slip on an fcc system depends on (i) the system with the highest resultant stress due to the fcc system; (ii) the minimum angle u; and (iii) least energy consumption, i.e. minimum residual Burgers vector at the boundary bb r = bb 2 − bb 1.
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The process of slip transmission as a mechanism for achieving plasticity in embedded bcc phases may be possible only for those embedded phases that lie within a certain size range. The embedded bcc phase produced via eutectic solidification often consists of defect-free single crystals separated by fcc matrix [13–15]. When extracted, such bcc crystals exhibit whisker-like behaviour reaching strengths approaching E/50 when tested in tension [16,17]. Load transfer to the harder phase can be accommodated by either plastic flow or fracture. Langford [18] has illustrated this for the case of pearlitic steel wire. In this system he found that for cementite lamella B10 nm thick, extensive plasticity was exhibited in the carbide, while for lamella \ 100 nm in thickness plasticity was inhibited and fracture of the carbide resulted. Thus the competition between plasticity and fracture is scale dependent and may involve the size of surface irregularities on the embedded phase, the local hydrostatic component of the stress state and the details of transmission of plasticity at the interface.
3. Relative plastic strain in the two phases From the onset of deformation in most two-phase alloys, the imposed plastic strains are not distributed equally between the phases. This strain partitioning is greatest for systems containing constituents with very different flow stresses and rates of work hardening [19]. Typically, the harder embedded phase initially accommodates less of the applied strain than the softer matrix. This is illustrated with reference to Fig. 2 where the partitioning of strain in a Ni–W alloy deformed by rolling at room temperature is illustrated. Here the Ni–W alloy contains isolated single crystal spheroids of tungsten, approximately 1 mm in diameter, embedded within a polycrystalline nickel matrix. The change in the aspect ratio of the tungsten phase was followed by quantitative metallography as a function of the applied plastic strain and used as a measure the average plastic strain in the tungsten phase. Physically, strain partitioning between two phases must result in one phase undergoing additional deformation so as to accommodate the more rigid phase. The resulting complicated flow pattern in the softer phase involves both local work hardening due to the large density of geometrically necessary dislocations and changes in flow pattern due to local stress concentrations [20]. As the flow stress of matrix and embedded phase approach one another due to work hardening of the softer matrix phase and local stresses at the interface, the partitioning of strains tends to become more homogeneous. This is particularly true for processes such as wire drawing where it is observed that the partitioning of strain becomes more homogeneous for
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Fig. 2. Partitioning of strain within a spherodized Ni – W alloy deformed by rolling. The tungsten phase is initially in the form of equiaxed single crystal spheroids of approximately 1 mm diameter. The aspect ratio of the tungsten phase has been used as a measure of the strain in that phase with the strains computed using the minimum, mean and maximum aspect ratios in the distribution. Here, the strain is given by the Von Mises equivalent ovm =
)
)
2 ln(1+ r) ,
3
where r is the rolling reduction.
phase population (i.e. how similar the strain undergone by each crystallite is) should be expected to be heterogeneous with it becoming more homogeneous as the scale of the slip approaches the scale of the embedded phase. In the limiting case where the distribution of slip is much finer than the scale of the embedded phase, relatively homogeneous deformation throughout the embedded phase population should be expected. Fig. 3 shows the fraction of tungsten crystallites which have undergone small or negligible plastic strains (as measured by their aspect ratio) as a function of applied rolling strain for the same spherodized Ni–W alloy as that discussed earlier with reference to Fig. 2. Two distinct regimes of deformation may be seen under these conditions. For applied strains1 of less than 0.90 it is observed in Fig. 3 that the fraction of tungsten crystallites that have undergone appreciable plastic deformation increases with increasing applied strain. Over the same range of strain it is observed by optical microscopy (see legend of Fig. 4 for details of sample preparation) that the distribution of slip within the matrix becomes more homogeneous with increasing applied strain. For an imposed strain of 0.18 the spacing between slip traces on the surface of the sample is significantly larger (5–10 mm) than the diameter of the tungsten crystallites with some regions exhibiting much finer slip than others (e.g. region marked by an arrow in Fig. 4a). At an imposed strain of 0.46 the scale of slip is reduced to the point where it is of the same order as the diameter of the tungsten crystallites (Fig. 4b),
applied strains (o) \1.0 [21]. However, as can be seen in Fig. 2, it appears that in processes such as rolling, the partitioning of strain becomes more, not less, heterogeneous with applied strains above o = 1.0. If this is the case then individual crystallites may or may not lie within regions that are undergoing plastic deformation.
4. Effect of slip distribution An implicit assumption often made in the analysis of deformation is that it occurs homogeneously throughout the microstructure. This description allows for simple models to be developed to describe the process of deformation in single-phase materials. The distribution of both slip and the spatial distribution of particles has important consequences in co-deformation if the deformation of the embedded phase depends on slip transmission. When the scale of the embedded phase becomes of the order of 1 mm it is quite possible for it to become finer than the scale of slip within the matrix. If this is the case then it is possible to imagine that individual crystallites may lie within regions of the matrix which have undergone plastic strains far less than those which are macroscopically imposed. In such a situation the strain distribution within the embedded
Fig. 3. Measured fraction of tungsten crystallites exhibiting aspect ratios between 1 and 2 versus the Von Mises equivalent applied strain. 1 All strains quoted for rolling and plane strain compression are given as Von Mises equivalent.
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Fig. 5. BSE SEM image of a spherodized Ni – W alloy deformed by rolling to ovm =3.0. The tungsten crystallite in the center of the image has been cut by a macroscopic shear band. Other shear bands are also obvious in the image.
Fig. 4. Optical micrographs (under conditions of Nomarski contrast) of spherodized Ni – W eutectic alloy. Samples were polished and given an incremental plane strain compression of 0.5% after larger plane strain deformations of (a) ovm = 0.19 and (b) ovm = 0.46. Observations made in the plane normal to the zero strain direction.
again with the scale of slip varying spatially throughout the sample. The observations made in Figs. 3 and 4 are consistent with the simple arguments made above. However, above strains of 1.0 this argument does not seem to hold as it appears that the fraction of crystallites that have undergone negligible plastic deformation plateaus. The strain at which this occurs coincides approximately with the observation of the initiation of macroscopic shear bands within the nickel phase. Fig. 5 shows a shear band intersecting a tungsten crystallite in a sample deformed in rolling to a strain of 3.07. At a lower magnification (Fig. 6) the extreme variation in the strain between individual tungsten crystallites may be seen in a sample deep etched to reveal the tungsten phase. The result of shear banding is to effectively increase the spacing of slip to the spacing between shear bands (of the order of 10 – 50 mm), thus increasing the chance that crystallites will lie within regions undergo-
ing little or no plastic strain. Similar observations of inhomogeneous deformation of the embedded bcc phase have been made in a/b brass [22], Cu–Cr [14], and Cu–Nb [23] deformed by rolling. A second example of the effect of slip distribution can be seen in a directionally solidified Cu–Cr eutectic alloy fabricated by means of a modified Bridgeman technique with a growth rate of 4 mm/h. In this alloy the average diameter of the Cr fibres is approximately 1
Fig. 6. BSE SEM image of the same sample as in Fig. 5 after deep etching to reveal the tungsten crystallites. The inhomogeneous nature of the deformation is clearly evidenced by the morphology of the tungsten crystallites.
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5. Texture and morphological development
Fig. 7. BF TEM image of a Cr fibre chemically extracted from a directionally solidified Cu–Cr alloy deformed in plane strain compression by ovm = 0.42 with the fibres parallel to the direction of tensile strain. These fibres are characterized by alternating regions which are dislocated (i) and defect free (ii). Prior to deformation these fibres are defect free.
mm, while the lengths may be as much as 1000 times this. In the as-cast state the Cr fibres are essentially defect-free single crystals. If the Cr fibres are extracted from the alloy after it has been deformed it is possible to image them in the transmission electron microscope so as to reveal their dislocation structure. The substructure developed within the fibres mimics the scale of deformation in the matrix with relatively large regions (of the order of a few microns in length) of perfect single crystal along the fibre length bounded by regions of relatively high dislocation content. Fig. 7 illustrates this for a Cr fibre extracted from a Cu – Cr alloy deformed to a strain of 0.42 in plane strain compression with the fibres lying parallel to the rolling direction.
There has been less work, both theoretically and experimentally, aimed at determining the textures developed in two-phase materials compared with that in single-phase materials. However, there is evidence to suggest that in a given deformation process, the fcc and bcc phases develop the same texture components they would in isolation but the rate of texture development may be different [24]. The results of a detailed texture analysis on rolled a/b brass may be used to illustrate this point. Engler et al. [22] found that texture in the a phase was enhanced at medium strains by the presence of the harder b phase and suggested that this was a result of the extra deformation accommodated by the a phase. At higher strains it was found that texture evolution became retarded due to the b phase acting as barriers to the formation of shear bands. The b phase was found to develop a typical bcc fibre rolling texture, although it was weaker than that observed for singlephase b brass. Modeling of the texture evolution led to the conclusion that the b phase deformed under partially relaxed Taylor conditions at low strains while at higher strains the deformation was better described by a Sachs condition (fully relaxed constraints). In other words, conditions of strain homogeneity appears to dominate at lower applied plastic strains, while at higher applied strains, homogeneity of deformation is sacrificed in favour of stress homogeneity. This suggests that at low strains the deformation of the embedded phase is closely linked to that in the matrix while at higher strains the embedded phase is able to deform independently of the matrix. Despite the apparent simplicity of the development of texture in co-deformation, the morphological development of the embedded phase may be complex. At low strains it is often possible to observe steps on the surface of the bcc phase corresponding to slip transmission across the phase interface. However, at larger strains the morphology of the embedded phase may become complicated and individual slip steps may no longer be obvious. Fig. 8 illustrates this for the case of the directionally solidified Cu–Cr eutectic alloy discussed previously with reference to Fig. 7. Fig. 8a shows the morphology of the as-grown Cr fibres after chemical extraction from the Cu matrix, while Fig. 8b and c shows the morphology of extracted fibres after deformation of the alloy by plane strain compression to strains of 0.42 and 0.80, respectively. Embedded bcc single crystals have been found to adopt particularly convoluted cross-sectional shapes upon deformation to large strains (\ 1.0) by wire drawing. This observation is closely related to the one made by Hosford [25], who noted the curling of grains about one another in drawn single-phase bcc metals. Bcc crystals with [110] drawing texture have slip sys-
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Fig. 8. Surface morphology of Cr fibres chemically extracted from a directionally solidified Cu – Cr eutectic alloy. (a) Fibres extracted in as cast condition showing the morphology of the Cr phase. (b) Slip steps on the surface of Cr fibres deformed by plane strain compression at room temperature. (c) Irregular surface morphology of Cr fibres extracted from an alloy rolled to ovm =2.66.
tems oriented in such a way that it is energetically favorable for plane strain deformation to be preferred over the imposed axisymmetric state of strain [25]. The end result, in the case of interest here, is that compatibility between axisymmetric deformation of the matrix and plane strain deformation of the embedded phase is accomplished by the bcc crystals undergoing redundant deformation by curling about their axes [26]. The role of deformation bands which traverse the bcc phase on
the curling phenomena in fcc/bcc alloys has been described by Malzahn Kampe and Courtney [27]. If the embedded bcc phase is polycrystalline and exhibits some ductility at room temperature in the bulk form, the fcc/bcc composite may also show significant ductility. However, in this situation the bcc phase appears to take up a shape intermediate between the curled microstructure observed in embedded single crystals and the applied axisymmetric deformation.
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Such is the case for multifilamentary composites of Cu–Nb formed using the technique of bundling polycrystalline niobium with copper as shown in Fig. 9 (J.T. Wood, unpublished work). On the other hand, if the bcc phase is brittle at room temperature in the bulk form, fracture of the polycrystalline embedded bcc phase appears to be common. In a hypereutectic Cu– Cr alloy it was found that the dendritic Cr phase was ductile as long as the volume fraction was sufficiently low that the dendrites remained isolated [28]. The behaviour of these systems may be attributed to the added constraint associated with the polycrystalline embedded phase. The sources of this added constraint may include the necessity of compatible deformation between bcc grains and the possible modification of the stress state associated with a percolated microstructure.
6. Internal stresses A consequence of the non-uniform plasticity in twophase materials is the development of internal stresses. These internal stresses may result in unexpected mechanical behaviour [29] as well as other properties (e.g. magnetic) that deviate from those observed in the undeformed state [30]. The effect of internal stresses may be to significantly modify the stress state acting on the co-deforming phases compared to the applied stress state. The development of internal stresses due to the necessary compatibility of plastic and elastic strains between two phases may considered in terms of an Eshelby model [31–33]. In this case the Eshelby transformation strain may simply be considered as the difference in plastic strain between the two phases as illustrated in Fig. 10. Such Eshelby type models have been applied to various two-phase materials where both phases are ductile in an attempt to model the development of internal stresses
Fig. 9. Morphology of polycrystalline Nb fibres extracted from a drawn Cu – Nb alloy [26].
Fig. 10. Schematic illustration of the Eshelby method as applied to the case of co-deformation. In this situation the Eshelby transformation strain, o T* may be considered as the difference in the plastic strain between the two phases (as in (a)). Alternatively, it is possible to consider the internal stresses developed by the inclusion alone undergoing a strain of o T* (as in (b)). Using the method of Brown and Clarke [32] the internal stresses due to the deformation may be written as [33], o Iija =fbgijo T* o Iijb =fagijo T* where gij consists of the Eshelby matrix and the elastic properties of the matrix and inclusion [32].
and the overall stress/strain response of the material [34,35]. One manifestation of internal stresses in two-phase materials is the strain-dependent Bauschinger effect observed in fcc/bcc two-phase materials. Verhaeghe et al. [36] have observed such effects in percolated austenoferritic stainless steels. At low plastic tensile strains (B 0.5%), where the ferritic phase is loaded purely elastically, the internal stresses are found to develop rapidly at a linear rate. Above 0.5% strain, localized plastic flow in the ferrite allows for the relaxation of the internal stresses resulting in a reduction of the rate of internal stress accumulation with plastic strain. It is important to note that, although plastic flow in the ferrite acts as a relaxation mechanism, at strains be-
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yond yielding of the ferrite the internal stresses continue to rise but at a lower rate to that observed prior to yielding. A more direct method for the measurement of the volume-averaged internal strains (those elastic strains associated with the internal stresses) within two-phase materials during co-deformation is neutron diffraction. However, to date there have been few references in the literature to the use of this technique for the measurement of internal stresses during co-deformation. Experiments with Cu – Nb and Cu – Ag two-phase wires deformed to a drawing strain of 2.0 show large internal strains of the order of 0.2 – 0.5% [37]. These internal strains (and thus internal stresses) are of short wavelength in the radial direction with the Cu – Nb composite showing larger internal stresses than Cu–Ag. Like the Bauschinger test described above, the internal stresses are found to be strain dependent in these materials, with the internal stresses rising with applied strain even at large strains.
7. Processes at interfaces In co-deformation the events at the interphase interfaces are of paramount importance and may change as a consequence both of the scale of the structure and the amount of plastic strain. At low strains the interface acts as a barrier to dislocation motion in the softer phase and as a source of dislocations in the embedded phase. Slip transmission across the interface produces residual dislocations at the interface that may rearrange by glide or climb within the interface [38]. The detailed atomic fit between the two lattices can often be described in terms of interface dislocations [39,40]. In many heavily deformed materials the scale of the constituent phases is such that little dislocation – dislocation interaction occurs within the phases but both dislocation production and storage occurs at the interfaces. This has two important consequences: (a) slip occurs by dislocations shuttling between the interfaces and (b) energy storage occurs by the production of additional interfacial area with plastic strain [41]. Models for such deformation mechanisms have been described by Gil Sevillano et al. [41] and Embury and Hirth [42], and have been applied to describe the behaviour of nanoscaled multilayers by Misra et al. [43]. The events at the interfaces may be complicated by chemical effects that can include mixing at the interface, increased solubility due to the fine scale of the structure [44] and the possibility of local amorphisation [45]. Additional effects can arise from coherency stresses or stresses due to the difference in elastic constants across the interface. The detailed delineation of these events requires careful study both by HRTEM and other techniques. However, it is also important to note that
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the scale of events at the interfaces lends itself to atomic simulation and direct comparison of the results of simulation and experiment as illustrated by the recent work of Hoagland and Voter (private communication).
8. Summary The co-deformation of fcc/bcc materials is a problem that extends over a variety of length scales and depends on the specific systems chosen, the scale and morphology of the structure and the mode of deformation. From the brief survey presented above, it is clear that the influence of both the scale of the structure and the distribution of plasticity is of great importance. As deformation increases and the scale of the structure is reduced, the role of the interfaces becomes dominant in the description of co-deformation and the study of the detailed mechanisms of plasticity, dislocation accumulation and strain distribution in the interfacial regions becomes crucial. This clearly represents a rich area for future experimentation and atomic simulation.
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