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Microstructure and flow stress of deformed polycrystalline metals
Scripta METALLURGICA et MATERIALIA
Vol. 27, pp. 969-974, 1992 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
VIEWPOINT
SET No. 20
MICROSTRUCTURE AND FLOW STRESS OF DEFORMED POLYCRYSTALLINE METALS D. A. Hughes Center for Materials and Applied Mechanics Sandia National Laboratories Livermore, CA 94550, USA (Received July 27, 1992)
Introduction The end tom of many microstructural studies is to relate the microstructure to flow stress behavior. This relation is not always straightforward, since most deformation mlcrostructures are quite complex and rich in detail, encompassing many more structures than the simple cell structure. Thus it is first necessary to have a simplifying framework to identify and classify the important rnicrostructural components and their functions. A key example of this classification is the idea of geometrically necessary boundaries and incidental boundaries (1). These two types of boundaries have separate roles in deformation strengthening and in accommodating the imposed shape change during deformation. Secondly, it is necessary to consider each microstructural component's relative strength contribution. The relative strength contribution will depend on such parameters as the distribution of that structure, volume fractions, morphology (i.e. equiaxed or planar), macroscopic orientation to the deformation axis, the misorientations and dislocation densities of boundaries. The morphology and macroscopic orientation of deformation structures have further implications in calculating flow stress anisotropy. The following sections will first describe a general framework for deformation microstructures based on several experimental observations of grain subdivision. These observations apply mainly to microstructures which develop at low homologous temperatures, below 0.35 of the melting temperature. However, the evolutionary framework can also be applied to warm and hot working processes occurring at high strain rates. Secondly it will discuss the role of these microstructures during deformation and present some reasons for their formation. The effects of deformation mode and strain level on the observed structures will also be noted. Lastly, a method of incorporating complex microstructures into flow strength calculations is reviewed and its limitations discussed. This method considers that geometrically necessary boundaries and cells provide separate and additive strength contributions. Cell strength is based on the classical empirical inverse relationship between flow stress and cell size and the strength of geometrically necessary boundaries is estimated by using a Hall-Petch type equation. These equations are of the general form: o- = tr o + K G ( b / D )
rt
(1)
~r is the flow stress; ~0 is the friction stress; K is a constant; G is the shear modulus; b is the Burgers vector; D is the microstructural size, i.e cell diameter. Framework for Microstructural Evolution During deformation grains are subdivided into rotated volume elements at two levels for all deformation modes (Figure 1) (1,2). The smallest subdivision includes incidental boundaries, principally dislocation cells, which arise from the statistical trapping of glide dislocations together with forest dislocations. Cell walls contain small lattice misorientations. At a larger subdivision scale, geometrically necessary boundaries, principally dense dislocation wails and microbands form. These geometrically necessary boundaries incorporate the lattice rotations which arise from the geometrical requirements of strain accommodation. Consequently, there is a larger misorientation change across their boundaries than across cell walls. More familiar examples of geometrically necessary boundaries would include subgrains, twin boundaries and grain boundaries. 969 0 9 S 6 - 7 1 6 X / 9 2 $ 5 . 0 0 + .00 C o p y r i g h t ( c ) 1992 P e r g a m o n P r e s s
Ltd.
970
MICROSTRUCTURE
AND FLOW STRESS
Vol.
27, No.
The dense dislocation walls and microbands are large planar boundaries which delineate misoriented regions of cell blocks in which different slip systems operate (Fig. 1) (2). The development of these dislocation structures (boundaries), is caused by the conditions of polyslip and the driving force to minimize the energy of the structure per unit of dislocation line. Slip which occurs on a finite number of crystallographic slip planes and directions must occur in a combination which also meets the requirements of the imposed conditions on strain and/or stress. A minimum of five slip systems is needed to make any arbitrary shape change and achieve strain accommodation, the Taylor criterion (3). However activation of so many slip systems raises the flow stress, so that the crystal would prefer to deform on less than five systems. To achieve strain accommodation on fewer systems, grain subdivision occurs creating regions in which different slip systems operate. The boundaries between these regions are geometrically necessary boundaries (Figure 1). Together these regions act to achieve strain accommodation. Dense dislocation walls and microbands are distributed throughout the volume of a grain indicating that the strain accommodation occurs throughout a grain rather than in a narrow region around the grain boundary. Since strain compatibility will also be needed at the interface between the differently deforming regions, the morphology and macroscopic orientation of the geometrically necessary boundaries at the interface will evolve to maximize the strain accommodation between the regions. Since neighboring cell blocks slip on different slip systems, continued slip within each cell block will increase the misorientations across the geometrically necessary boundaries surrounding each cell block. These diverging orientations create the need for further strain accommodation with increasing strain. The additional strain accommodation is achieved by further grain subdivision through the creation of microbands at dense dislocation walls and new dense dislocation walls within cell blocks. Thus the grain subdivision refines in size scaie with increasing strain as illustrated by the widely spaced dense dislocation walls in Figure 2 at small strains compared to the closer spacing of dense dislocation walls and microbands in Figure 3. Misorientations will also accummulate at the level of individual cells with increasing deformation. These increased misorientations will cause the slip systems across individual cell walls within a single ceU block to diverge. This slip divergence will result in the evolution of an individual cell (incidental boundary) into a subgrain (geometrically necessary boundary). Since geometrically necessary boundaries are interfaces between misoriented and differently deforming regions, they take on a morphology and macroscopic orientation to the deformation axis which facilitates the imposed deformation. This macroscopic orientation is fairly constant for a given deformation mode and within a particular strain regime (2). For example, for rolling deformation at small to intermediate strains, ~ef/ -- 0.05 to 0.5, the majority of geometrically necessary boundaries are platelike and are on average oriented parallel to the transverse direction and +40 deg or - 40 deg to the rolling direction (Fig. 4). For torsion within this strain range, one set of planar geometrically necessary boundaries is within 10 deg of the shear plane, while the other set is 70 deg (Fig. 2). These macroscopic orientations are general observations for fcc polycrystals. At very large strains ~ / / ~ 2, the geometrically necessary boundaries include equiaxed subgrains and lamellar dislocation boundaries. The lamellar boundaries formed during rolling are oriented within 15 deg of the rolling plane (Fig. 5). From ~ / / -- 0.4 to 2.0 both macroscopic orientations of the planar geometrically necessary boundaries are present in rolled structures (Fig. 6). Flow Stress and Flow Stress Anisotrop~ Both cells and geometrically necessary boundaries contribute to the flow strength. The qualitative and the functional differences between these two types of boundaries suggest that they make separate strength contributions. These separate strength contributions are different functions of their unit microstructural size. For cells, the flow stress has been found empirically to be inversely related to the call diameter(4, 5). This empirical relation has been explained semiquantitatively based on the concept of low energy structures (6). Although the exponent on the microstructural size of -I in this inverse relation fits a wide range of observations, reported values of the exponent range from -0.50 to less than -1 (5). Note that this reported range of exponents has been the source of much debate concerning the "true" value of the exponent on microstructural size. There are equal numbers of advocates for either an inverse relationship with an exponent of -1 or an inverse squareroot relation with an exponent of -0.5 (7). An inverse squareroot relationship between microstructural size and flow stress is generally recognized however for grain boundary strengthening in the Hall-Perch relation and has also been suggested by Thompson to be appropriate for subgrain
8
Vol.
27, No.
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M I C R O S T R U C T U R E AND FLOW STRESS
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strengthening (8). Since subgrains and grain boundaries are geometrically necessary boundaries, Hansen has suggested that the dense dislocation walls and microbands also contribute to the flow stress in an inverse square root relationship with their spacing (DGNB) (9). Thus the additive contribution of cells and geometrically necessary boundaries would give: o" = o'o + K1Gb/D + K~ (G b/DaNB) °'s
(2)
The two different types of boundaries have different regions of influence which is reflected to a large degree in this additive equation. At the start of deformation there are no cells and the only geometrically necessary boundaries are grain boundaries. Thus, the equation reduces to the Hall-Petch equation. At small strains cell strengthening dominates with a small contribution from the geometrlcaly necessary boundaries; since the spacing of these GNBs including grain boundaries, dense dislocation walls and microbands is large.' At intermediate to very large strains both cells and geometrically necessary boundaries are present and contribute to the flow stress. Cells persist to very large strains at low homologous temperatures, albeit in decreasing proportions. Thus, only at the largest strains, %/I > 4, would geometrically necessary boundaries contribute solely. It is also at these largest strains that Zimmer et al. considered that grain boundary strengthening would reemerge as the dominant strength parameter, since the minimum spacing of grain boundaries decreases at a higher rate than the other geometrically necessary boundaries (subgrains) (10). However at very large strains it is difficult to distinguish the other geometrically necessary boundaries (e.g. the lamellar structure in rolling) from grain boundaries, since both conform to the imposed shape change, have large misorientations and have been influenced by the texture development. Thus it is better to discuss the strength contribution from all geometrically necessary boundaries rather than consider a reemergence of the importance of the original grain boundaries. Equation 2 presents a simplified approach to calculating the microstructural contribution to strength. While it captures the contribution of the two most significant factors, i.e. that strength varies with the spacing of two different types of boundaries, geometrically necessary and incidental, other potentially important factors are hidden in the "constants" K1 and K2. These other factors include the width, dislocation density, misorientation, and resistance to dislocation glide of the boundary. Significant changes to the values of these factors during deformation will change the value of K; K is no longer a constant. In aluminum with its high stacking fault energy, the dislocation boundaries approach their ideal low energy configuration very early in the deformation process. Thus the incidental and geometrically necessary boundaries maintain, on average, characteristic widths and misorientations over a large strain range(11). Note the very similar appearance of the boundaries formed in aluminum at 30% c.r. (Fig. 4) compared to 90% c.r. (Fig. 6). Consequently, the value of K is reasonably constant for aluminum within those strain ranges. In contrast, for nickel and copper, the arrangement of dislocations towards a low energy configuration within a dislocation boundary occurs less rapidly and with more difficulty because of the lower dislocation mobilities. Thus there is a significant change in the average boundary structure during deformation which changes the relative strength contribution and thus K1 and K2. This change is illustrated by the different appearance of the two types of boundaries at small and large strains (Fig. 2 compared to Fig. 5) in nickel. Although the changes in K1 are less for aluminum than for nickel and copper, the trend in K1 values is the same for all three materials. The K1 values obtained at small strains, e e l / < 0.2, (4,5,12,13) are consistently much larger than those values obtained at larger strains (7,10,14-17). The source of this changing K value or strengthening behavior is partially masked because no distinction is consistently made between ceils and geometrically necessary boundaries and only the minimum spacing is reported without respect to the three dimensional morphology of the boundaries at large strains. Regardless of these differences in measurement parameters, the collective set of d a t a indicate that more parameters than just microstructural size or a change in the exponents on microstructural size is required to capture the strengthening behavior over the entire strain range. We are thus currently limited to defining a specific flow stress equation which is valid for a finite strain range. For a broad strain range, however, cells and GNBs are present and the relative strength contribution is controlled in part by the microstructural dimension. For cells, which are generally equiaxed, the cell diameter provides the simplest dimension. Frequently however, the geometrically necessary boundaries do not enclose equiaxed volumes, rather as described above, they enclose platelike or rod shaped volumes. Therefore it
972
MICROSTRUCTURE
AND FLOW STRESS
Vol.
27, No.
is important to determine the DGNB based on their morphology and estimates of the slip distance. The morphology is especially important in calculating flow stress anisotropy. The importance of incorporating geometrically necessary boundaries into the flow stress law is shown most clearly in predicting flow stress anisotropy. Hansen and Juul-Jensen have proposed a conical slip model to estimate the DCNV and to calculate the flow stress from the additive law (equation 2) (18). Flow stress anisotropy in this model can be predicted because the parameter DCNB becomes an average slip distance which depends on the three dimensional orientation of the geometrically necessary boundaries to the deformation axis. They also remove the anisotropy due to the texture in their calculations by using a Taylor factor determined from the measured textures. The model was tested for the case of deformation microstructures developed during rolling of aluminum. Flow stress was measured in tension for tensile samples cut from the rolled sheet at various angles c~ to the rolling direction. This procedure provided samples with geometrically necessary boundaries at different angles to the tensile axis. An example of their predictions shown in Fig. 7, shows that the ratio of the spacings of geometrically necessary boundaries in the model provides the necessary correction for flow stress anisotropy. Concluding Remarks The general framework for microstructural evolution during deformation includes grain subdivision into volume elements at two levels. The smallest level is that of dislocation cells which are incidental boundaries. The larger level is that of geometrically necessary boundaries which includes dense dislocation walls and microbands. These two features provide separate but additive contributions to strength. The geometrically necessary boundaries which enclose volume elements with high aspect ratios contribute to the observed flow stress anisotropy. Acknowledgements This work supported by the U. S. Department of Energy under contract DE-AC04-76DP00789. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
D. KuhlmanmWilsdorf and N. Hansen, Scripts Metall. Mater., 25, 1557 (1991). B. Bay, N. Hansen, D. A. Hughes and D. Kuhlmann-Wilsdorf, Acts Metall. Mater., 40, 205 (1992). O. I. Taylor, J. Inst. Metals, 62, 307 (1938). M. R. Staker and D. L. Holt, Acts Metall., 20, 569 (1972). S. V. Raj and G. M. Pharr, Mater. Sci. Engr., 81,217 (1986). D. Kuhlmann-Wilsdorf, Mater. Sci. Engr., 81, Al13, 1 (1989). 3. Gil Sevillano, P. van Houtte and E. Aernoudt, Progr. Mater. Sci., 25, 69 (1980). A. W. Thompson, Work Hardening in Tension and Fatigue, p. 89, TMS AIME, New York, NY (1977). N. Hansen and D. Juul Jensen, Proc. 9th Int. Conf. on Strength of Metals and Alloys, ICSMA 9, Pergamon Press, Oxford (1991). W. H. Zimmer, S. S. Hecker, D. L. Rohr and L. E. Murr, Metal Sci., 17, 198 (1983). B. Bay, N. Hansen and D. Kuhlmann-Wilsdorf, Mater. Sci. Engr. in press. T. Tabata, S. Yamanaka and H. Fujita, Acts Metal1., 26, 405 (1978) Yi-Yin Chao and S. K. Varma, Scripts Metall. Mater., 24, 1665 (1990). S.S. Recker and M. Stout, in G. Krauss (ed.), Deformation Processing and Structure, p. 1, ASM, Metals Park, OH (1984). D. A. Hughes and W. D. Nix, Mater. Sci. Engr., A122, 153 (1989). S. Thiagarajan and S. K. Varma, Metall. Trans., 22A, (1991). N. Bunsen, Trans. Metall. Soc. AIME, 245, 2061 (1969). N. Bunsen and D. Juul Jensen, Acts Metall. Mater. in press. B. Bay, N. Bunsen and D. Kuhlmann-Wilsdorf, Mater. Sci. Engr. Al13, 385 (1989). D. Hughes and N. Hansen, Mater. Sci. Tech., 7, 544 (1091).
8
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STRESS
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00w \Microband
Fig. 1. Schematic drawing of grain subdivision showing the geometrically necessary boundaries: dense dislocation walls (DDWs) and microbands delineating cell blocks containing incidental dislocation boundaries.
Fig. 2. Dense dislocation walls ( D D W ) and microband ( M B ) outlining ceils in nickel deformed in torsion to a shear strain of 3'sz = 0.6 which is a yon Mises equivalent strain of 0.3. The double arrows indicate the direction of the applied shear
stress. (From reference 20.)
Fig. 3. A closer spacing of microbands and dense dislocation walls occurs at larger strains. Nickel cold rolled 50%. The rolling direction is marked RD in this longitudinal plane section.
974
MICROSTRUCTURE
Fig. 4. Cells and microbands in aluminum cold rolled 30%. The direction of one set of microbands (MB) is marked by arrow A. Another weaker microband is marked at B. TEM micrograph is viewed in the longitudinal plane section with the rolling direction marked rd. (from reference 19)
AND FLOW STRESS
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8
Fig. 5. LameUar dislocation boundaries oriented parallel to the rolling plane in nickel cold rolled 90%. Note the cells between the lamellar boundaries. The rolling direction is marked RD in this longitudinal plane section.
i
I
i
I
=
1 60
I 75
21un
100
~E 80, z A
60
o
40
20
i 15
I 30
L 45
90
ANGLE TO ROLLING DIRECTION (¢0
Fig. 6. Lamellar dislocation boundaries and microbands in aluminum cold rolled 90%. TEM micrograph is viewed in the longitudinal plane section with the rolling direction marked rd. (from reference 11)
Fig. 7. The experimental flow s t r e s s (~reff) corrected for the effect of crystallographic texture compared to the flow stress calculated using equation 2 for 99.5 % aluminum cold rolled 20% The calculations are for two minimum spacings of geometrically necessary boundaries, 2 and 4/~m. The flow stress was measured in the rolling plane at various angles to the rolling direction. (From reference 18).
Vol. 27, pp. 969-974, 1992 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
VIEWPOINT
SET No. 20
MICROSTRUCTURE AND FLOW STRESS OF DEFORMED POLYCRYSTALLINE METALS D. A. Hughes Center for Materials and Applied Mechanics Sandia National Laboratories Livermore, CA 94550, USA (Received July 27, 1992)
Introduction The end tom of many microstructural studies is to relate the microstructure to flow stress behavior. This relation is not always straightforward, since most deformation mlcrostructures are quite complex and rich in detail, encompassing many more structures than the simple cell structure. Thus it is first necessary to have a simplifying framework to identify and classify the important rnicrostructural components and their functions. A key example of this classification is the idea of geometrically necessary boundaries and incidental boundaries (1). These two types of boundaries have separate roles in deformation strengthening and in accommodating the imposed shape change during deformation. Secondly, it is necessary to consider each microstructural component's relative strength contribution. The relative strength contribution will depend on such parameters as the distribution of that structure, volume fractions, morphology (i.e. equiaxed or planar), macroscopic orientation to the deformation axis, the misorientations and dislocation densities of boundaries. The morphology and macroscopic orientation of deformation structures have further implications in calculating flow stress anisotropy. The following sections will first describe a general framework for deformation microstructures based on several experimental observations of grain subdivision. These observations apply mainly to microstructures which develop at low homologous temperatures, below 0.35 of the melting temperature. However, the evolutionary framework can also be applied to warm and hot working processes occurring at high strain rates. Secondly it will discuss the role of these microstructures during deformation and present some reasons for their formation. The effects of deformation mode and strain level on the observed structures will also be noted. Lastly, a method of incorporating complex microstructures into flow strength calculations is reviewed and its limitations discussed. This method considers that geometrically necessary boundaries and cells provide separate and additive strength contributions. Cell strength is based on the classical empirical inverse relationship between flow stress and cell size and the strength of geometrically necessary boundaries is estimated by using a Hall-Petch type equation. These equations are of the general form: o- = tr o + K G ( b / D )
rt
(1)
~r is the flow stress; ~0 is the friction stress; K is a constant; G is the shear modulus; b is the Burgers vector; D is the microstructural size, i.e cell diameter. Framework for Microstructural Evolution During deformation grains are subdivided into rotated volume elements at two levels for all deformation modes (Figure 1) (1,2). The smallest subdivision includes incidental boundaries, principally dislocation cells, which arise from the statistical trapping of glide dislocations together with forest dislocations. Cell walls contain small lattice misorientations. At a larger subdivision scale, geometrically necessary boundaries, principally dense dislocation wails and microbands form. These geometrically necessary boundaries incorporate the lattice rotations which arise from the geometrical requirements of strain accommodation. Consequently, there is a larger misorientation change across their boundaries than across cell walls. More familiar examples of geometrically necessary boundaries would include subgrains, twin boundaries and grain boundaries. 969 0 9 S 6 - 7 1 6 X / 9 2 $ 5 . 0 0 + .00 C o p y r i g h t ( c ) 1992 P e r g a m o n P r e s s
Ltd.
970
MICROSTRUCTURE
AND FLOW STRESS
Vol.
27, No.
The dense dislocation walls and microbands are large planar boundaries which delineate misoriented regions of cell blocks in which different slip systems operate (Fig. 1) (2). The development of these dislocation structures (boundaries), is caused by the conditions of polyslip and the driving force to minimize the energy of the structure per unit of dislocation line. Slip which occurs on a finite number of crystallographic slip planes and directions must occur in a combination which also meets the requirements of the imposed conditions on strain and/or stress. A minimum of five slip systems is needed to make any arbitrary shape change and achieve strain accommodation, the Taylor criterion (3). However activation of so many slip systems raises the flow stress, so that the crystal would prefer to deform on less than five systems. To achieve strain accommodation on fewer systems, grain subdivision occurs creating regions in which different slip systems operate. The boundaries between these regions are geometrically necessary boundaries (Figure 1). Together these regions act to achieve strain accommodation. Dense dislocation walls and microbands are distributed throughout the volume of a grain indicating that the strain accommodation occurs throughout a grain rather than in a narrow region around the grain boundary. Since strain compatibility will also be needed at the interface between the differently deforming regions, the morphology and macroscopic orientation of the geometrically necessary boundaries at the interface will evolve to maximize the strain accommodation between the regions. Since neighboring cell blocks slip on different slip systems, continued slip within each cell block will increase the misorientations across the geometrically necessary boundaries surrounding each cell block. These diverging orientations create the need for further strain accommodation with increasing strain. The additional strain accommodation is achieved by further grain subdivision through the creation of microbands at dense dislocation walls and new dense dislocation walls within cell blocks. Thus the grain subdivision refines in size scaie with increasing strain as illustrated by the widely spaced dense dislocation walls in Figure 2 at small strains compared to the closer spacing of dense dislocation walls and microbands in Figure 3. Misorientations will also accummulate at the level of individual cells with increasing deformation. These increased misorientations will cause the slip systems across individual cell walls within a single ceU block to diverge. This slip divergence will result in the evolution of an individual cell (incidental boundary) into a subgrain (geometrically necessary boundary). Since geometrically necessary boundaries are interfaces between misoriented and differently deforming regions, they take on a morphology and macroscopic orientation to the deformation axis which facilitates the imposed deformation. This macroscopic orientation is fairly constant for a given deformation mode and within a particular strain regime (2). For example, for rolling deformation at small to intermediate strains, ~ef/ -- 0.05 to 0.5, the majority of geometrically necessary boundaries are platelike and are on average oriented parallel to the transverse direction and +40 deg or - 40 deg to the rolling direction (Fig. 4). For torsion within this strain range, one set of planar geometrically necessary boundaries is within 10 deg of the shear plane, while the other set is 70 deg (Fig. 2). These macroscopic orientations are general observations for fcc polycrystals. At very large strains ~ / / ~ 2, the geometrically necessary boundaries include equiaxed subgrains and lamellar dislocation boundaries. The lamellar boundaries formed during rolling are oriented within 15 deg of the rolling plane (Fig. 5). From ~ / / -- 0.4 to 2.0 both macroscopic orientations of the planar geometrically necessary boundaries are present in rolled structures (Fig. 6). Flow Stress and Flow Stress Anisotrop~ Both cells and geometrically necessary boundaries contribute to the flow strength. The qualitative and the functional differences between these two types of boundaries suggest that they make separate strength contributions. These separate strength contributions are different functions of their unit microstructural size. For cells, the flow stress has been found empirically to be inversely related to the call diameter(4, 5). This empirical relation has been explained semiquantitatively based on the concept of low energy structures (6). Although the exponent on the microstructural size of -I in this inverse relation fits a wide range of observations, reported values of the exponent range from -0.50 to less than -1 (5). Note that this reported range of exponents has been the source of much debate concerning the "true" value of the exponent on microstructural size. There are equal numbers of advocates for either an inverse relationship with an exponent of -1 or an inverse squareroot relation with an exponent of -0.5 (7). An inverse squareroot relationship between microstructural size and flow stress is generally recognized however for grain boundary strengthening in the Hall-Perch relation and has also been suggested by Thompson to be appropriate for subgrain
8
Vol.
27, No.
8
M I C R O S T R U C T U R E AND FLOW STRESS
971
strengthening (8). Since subgrains and grain boundaries are geometrically necessary boundaries, Hansen has suggested that the dense dislocation walls and microbands also contribute to the flow stress in an inverse square root relationship with their spacing (DGNB) (9). Thus the additive contribution of cells and geometrically necessary boundaries would give: o" = o'o + K1Gb/D + K~ (G b/DaNB) °'s
(2)
The two different types of boundaries have different regions of influence which is reflected to a large degree in this additive equation. At the start of deformation there are no cells and the only geometrically necessary boundaries are grain boundaries. Thus, the equation reduces to the Hall-Petch equation. At small strains cell strengthening dominates with a small contribution from the geometrlcaly necessary boundaries; since the spacing of these GNBs including grain boundaries, dense dislocation walls and microbands is large.' At intermediate to very large strains both cells and geometrically necessary boundaries are present and contribute to the flow stress. Cells persist to very large strains at low homologous temperatures, albeit in decreasing proportions. Thus, only at the largest strains, %/I > 4, would geometrically necessary boundaries contribute solely. It is also at these largest strains that Zimmer et al. considered that grain boundary strengthening would reemerge as the dominant strength parameter, since the minimum spacing of grain boundaries decreases at a higher rate than the other geometrically necessary boundaries (subgrains) (10). However at very large strains it is difficult to distinguish the other geometrically necessary boundaries (e.g. the lamellar structure in rolling) from grain boundaries, since both conform to the imposed shape change, have large misorientations and have been influenced by the texture development. Thus it is better to discuss the strength contribution from all geometrically necessary boundaries rather than consider a reemergence of the importance of the original grain boundaries. Equation 2 presents a simplified approach to calculating the microstructural contribution to strength. While it captures the contribution of the two most significant factors, i.e. that strength varies with the spacing of two different types of boundaries, geometrically necessary and incidental, other potentially important factors are hidden in the "constants" K1 and K2. These other factors include the width, dislocation density, misorientation, and resistance to dislocation glide of the boundary. Significant changes to the values of these factors during deformation will change the value of K; K is no longer a constant. In aluminum with its high stacking fault energy, the dislocation boundaries approach their ideal low energy configuration very early in the deformation process. Thus the incidental and geometrically necessary boundaries maintain, on average, characteristic widths and misorientations over a large strain range(11). Note the very similar appearance of the boundaries formed in aluminum at 30% c.r. (Fig. 4) compared to 90% c.r. (Fig. 6). Consequently, the value of K is reasonably constant for aluminum within those strain ranges. In contrast, for nickel and copper, the arrangement of dislocations towards a low energy configuration within a dislocation boundary occurs less rapidly and with more difficulty because of the lower dislocation mobilities. Thus there is a significant change in the average boundary structure during deformation which changes the relative strength contribution and thus K1 and K2. This change is illustrated by the different appearance of the two types of boundaries at small and large strains (Fig. 2 compared to Fig. 5) in nickel. Although the changes in K1 are less for aluminum than for nickel and copper, the trend in K1 values is the same for all three materials. The K1 values obtained at small strains, e e l / < 0.2, (4,5,12,13) are consistently much larger than those values obtained at larger strains (7,10,14-17). The source of this changing K value or strengthening behavior is partially masked because no distinction is consistently made between ceils and geometrically necessary boundaries and only the minimum spacing is reported without respect to the three dimensional morphology of the boundaries at large strains. Regardless of these differences in measurement parameters, the collective set of d a t a indicate that more parameters than just microstructural size or a change in the exponents on microstructural size is required to capture the strengthening behavior over the entire strain range. We are thus currently limited to defining a specific flow stress equation which is valid for a finite strain range. For a broad strain range, however, cells and GNBs are present and the relative strength contribution is controlled in part by the microstructural dimension. For cells, which are generally equiaxed, the cell diameter provides the simplest dimension. Frequently however, the geometrically necessary boundaries do not enclose equiaxed volumes, rather as described above, they enclose platelike or rod shaped volumes. Therefore it
972
MICROSTRUCTURE
AND FLOW STRESS
Vol.
27, No.
is important to determine the DGNB based on their morphology and estimates of the slip distance. The morphology is especially important in calculating flow stress anisotropy. The importance of incorporating geometrically necessary boundaries into the flow stress law is shown most clearly in predicting flow stress anisotropy. Hansen and Juul-Jensen have proposed a conical slip model to estimate the DCNV and to calculate the flow stress from the additive law (equation 2) (18). Flow stress anisotropy in this model can be predicted because the parameter DCNB becomes an average slip distance which depends on the three dimensional orientation of the geometrically necessary boundaries to the deformation axis. They also remove the anisotropy due to the texture in their calculations by using a Taylor factor determined from the measured textures. The model was tested for the case of deformation microstructures developed during rolling of aluminum. Flow stress was measured in tension for tensile samples cut from the rolled sheet at various angles c~ to the rolling direction. This procedure provided samples with geometrically necessary boundaries at different angles to the tensile axis. An example of their predictions shown in Fig. 7, shows that the ratio of the spacings of geometrically necessary boundaries in the model provides the necessary correction for flow stress anisotropy. Concluding Remarks The general framework for microstructural evolution during deformation includes grain subdivision into volume elements at two levels. The smallest level is that of dislocation cells which are incidental boundaries. The larger level is that of geometrically necessary boundaries which includes dense dislocation walls and microbands. These two features provide separate but additive contributions to strength. The geometrically necessary boundaries which enclose volume elements with high aspect ratios contribute to the observed flow stress anisotropy. Acknowledgements This work supported by the U. S. Department of Energy under contract DE-AC04-76DP00789. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
D. KuhlmanmWilsdorf and N. Hansen, Scripts Metall. Mater., 25, 1557 (1991). B. Bay, N. Hansen, D. A. Hughes and D. Kuhlmann-Wilsdorf, Acts Metall. Mater., 40, 205 (1992). O. I. Taylor, J. Inst. Metals, 62, 307 (1938). M. R. Staker and D. L. Holt, Acts Metall., 20, 569 (1972). S. V. Raj and G. M. Pharr, Mater. Sci. Engr., 81,217 (1986). D. Kuhlmann-Wilsdorf, Mater. Sci. Engr., 81, Al13, 1 (1989). 3. Gil Sevillano, P. van Houtte and E. Aernoudt, Progr. Mater. Sci., 25, 69 (1980). A. W. Thompson, Work Hardening in Tension and Fatigue, p. 89, TMS AIME, New York, NY (1977). N. Hansen and D. Juul Jensen, Proc. 9th Int. Conf. on Strength of Metals and Alloys, ICSMA 9, Pergamon Press, Oxford (1991). W. H. Zimmer, S. S. Hecker, D. L. Rohr and L. E. Murr, Metal Sci., 17, 198 (1983). B. Bay, N. Hansen and D. Kuhlmann-Wilsdorf, Mater. Sci. Engr. in press. T. Tabata, S. Yamanaka and H. Fujita, Acts Metal1., 26, 405 (1978) Yi-Yin Chao and S. K. Varma, Scripts Metall. Mater., 24, 1665 (1990). S.S. Recker and M. Stout, in G. Krauss (ed.), Deformation Processing and Structure, p. 1, ASM, Metals Park, OH (1984). D. A. Hughes and W. D. Nix, Mater. Sci. Engr., A122, 153 (1989). S. Thiagarajan and S. K. Varma, Metall. Trans., 22A, (1991). N. Bunsen, Trans. Metall. Soc. AIME, 245, 2061 (1969). N. Bunsen and D. Juul Jensen, Acts Metall. Mater. in press. B. Bay, N. Bunsen and D. Kuhlmann-Wilsdorf, Mater. Sci. Engr. Al13, 385 (1989). D. Hughes and N. Hansen, Mater. Sci. Tech., 7, 544 (1091).
8
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MICROSTRUCTURE
AND FLOW
STRESS
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00w \Microband
Fig. 1. Schematic drawing of grain subdivision showing the geometrically necessary boundaries: dense dislocation walls (DDWs) and microbands delineating cell blocks containing incidental dislocation boundaries.
Fig. 2. Dense dislocation walls ( D D W ) and microband ( M B ) outlining ceils in nickel deformed in torsion to a shear strain of 3'sz = 0.6 which is a yon Mises equivalent strain of 0.3. The double arrows indicate the direction of the applied shear
stress. (From reference 20.)
Fig. 3. A closer spacing of microbands and dense dislocation walls occurs at larger strains. Nickel cold rolled 50%. The rolling direction is marked RD in this longitudinal plane section.
974
MICROSTRUCTURE
Fig. 4. Cells and microbands in aluminum cold rolled 30%. The direction of one set of microbands (MB) is marked by arrow A. Another weaker microband is marked at B. TEM micrograph is viewed in the longitudinal plane section with the rolling direction marked rd. (from reference 19)
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Fig. 5. LameUar dislocation boundaries oriented parallel to the rolling plane in nickel cold rolled 90%. Note the cells between the lamellar boundaries. The rolling direction is marked RD in this longitudinal plane section.
i
I
i
I
=
1 60
I 75
21un
100
~E 80, z A
60
o
40
20
i 15
I 30
L 45
90
ANGLE TO ROLLING DIRECTION (¢0
Fig. 6. Lamellar dislocation boundaries and microbands in aluminum cold rolled 90%. TEM micrograph is viewed in the longitudinal plane section with the rolling direction marked rd. (from reference 11)
Fig. 7. The experimental flow s t r e s s (~reff) corrected for the effect of crystallographic texture compared to the flow stress calculated using equation 2 for 99.5 % aluminum cold rolled 20% The calculations are for two minimum spacings of geometrically necessary boundaries, 2 and 4/~m. The flow stress was measured in the rolling plane at various angles to the rolling direction. (From reference 18).