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Deformation mechanism maps and microstructural influences
Materials Science and Engineering A 410–411 (2005) 12–15
Deformation mechanism maps and microstructural influences G.W. Greenwood ∗ Department of Engineering Materials, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK Received in revised form 27 April 2005
Abstract Areas on these maps define ranges of homologous temperature and of applied stress divided by an elastic modulus over which specific mechanisms of deformation are predominant. Langdon has extended their use to illustrate important features. Lines separating the mechanisms of overall sliding from diffusional drift can be difficult to locate. Observations of microstructural features assist in this location and provide information on the way in which the line positions can be influenced. Notably, the flux patterns of diffusional drift are dependent upon stress system as well as on microstructure and it is shown that specimen geometry may also be important. © 2005 Elsevier B.V. All rights reserved. Keywords: Deformation maps; Creep; Bending; Microstructure; Diffusion
1. Introduction The mechanism by which a crystalline solid deforms is strongly influenced by the ratios of applied tensile stress σ divided by Young’s Modulus E (or equivalently by the ratio of shear stress divided by shear modulus) and by the homologous temperature, T/Tm . Ashby [1] first demonstrated that these aspects can be represented by “maps”, often showing remarkably similar features when different materials are compared. Lines of constant strain rate, drawn as contours on these maps, usually show some small but definitive alteration in direction in crossing lines that separate regions where specific deformation modes predominate. Mohamed and Langdon [2] extended the map concept by showing that maps could be drawn with other axes to reveal features dependent upon microstructure when either the stress or temperature was maintained constant. A notable parameter in this respect is the grain size d that can be normalised by dividing by the Burgers vector b, so that the ratio, d/b, is plotted on one of the axes. Representations of this kind are valuable in determining the extent to which grain size is influential and in the consequential implications of the deformation mechanism that is rate controlling.
∗
Tel.: +44 1142225517; fax: +44 1142225943. E-mail address: [email protected].
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.098
Ashby and Langdon have always been careful in their recognition of the limitations of map representation. Adequate experimental data are not always available. The maps rely on unchanging microstructure that can be difficult to establish at temperatures approaching the melting point. Even small changes in microstructure can have large effects and for a given material it is important to be fully aware of compositional and structural influences. The lines forming the boundaries of different regions of the maps are not always easy to locate for there may be only a relatively small variation in the parameters governing the predominance of different mechanisms. This is the situation, for example, when the deformation mode is more strongly influenced by grain boundary than by lattice diffusion or when dislocation climb becomes significant in supplementation or replacement of the glide process. Even in situations where clear distinctions have been pointed out [3], it has remained difficult to resolve the relative importance between Nabarro–Herring [4,5] and Harper–Dorn [6] creep. The continuing debate [7,8] will not be entered into here. More profound changes in deformation mechanism arise if the role of dislocation motion is overtaken and replaced solely by the drift of vacancies under stress. Here, grain boundaries must play a decisive role in operating as vacancy sources and sinks and also in their ability to slide to accommodate grain deformation. Underlying the importance of different regions in the maps is the way in which the external stressing of a material leads to an
G.W. Greenwood / Materials Science and Engineering A 410–411 (2005) 12–15
13
internal redistribution of stress. This relates to the significance of microstructure, of the superimposed stress system, and, in some instances, of specimen geometry. These features will next be considered. 2. Microstructural observations Creep test data overall have been only of limited assistance in providing evidence of the range of dominance of specific deformation mechanisms. There is often a near continuity of stress–strain rate relationships over large ranges of T/Tm and σ/E, even to the extent that single equations have been proposed [9] to be representative of behaviour over almost the whole range of conditions. To throw light on these uncertainties, distinctive microstructural changes have been noted [10] that are indicative of definitive deformation modes. These include grain shape changes, grain boundary sliding, dislocation rearrangements and precipitate denuded regions in two-phase materials. This latter aspect, however, has given rise to much speculation. When precipitate denuded zones were first observed [11] in materials subjected to creep under low stresses, their occurrence was linked to the operation of a diffusional creep process in which deformation occurred only through the stress directed migration of vacancies. For many years, this viewpoint remained unchallenged but more recently further questions have been raised [12]. Such zones have been observed when the creep rate was proportional to applied stress raised to a power more than unity. Grain boundaries are not generally in the centre of the zones. The zones have been considered to lie at orientations more random than those expected through diffusional creep operation. Processes other than creep have been considered to produce comparable zones. Following suggestions that precipitate free zones may not result from diffusional creep, the question naturally arises of an alternative cause of their formation. One suggestion is that precipitates are taken into solution by moving grain boundaries and subsequently re-precipitated. There is, however, substantial micro-analytical evidence that there are no regions of enhanced solute concentration in the matrix as would be anticipated in this proposal. Another proposal for the mechanism of zone creation is that they arise through grain boundary movement to cause particle sweeping such that a largely precipitate free zone is left behind. Such sweeping processes have been observed in a number of metallurgical treatments and it is necessary to enquire whether the microstructures after creep at low stresses bear resemblance to these. From Fig. 1, a precipitate denuded zone is illustrated in a hydrided Mg–Zr alloy after creep at low stresses. Here, the grain boundary is seen meandering within the zone which is largely free of precipitates. It appears that the grain boundary has had freedom to migrate within the zone though apparently pinned over short regions of the boundary where it has aligned with the edges of the precipitated regions. Equally noteworthy in Fig. 1 is the horizontal boundary, lying nearly parallel to the applied stress on which precipitates have accumulated. A critical feature here is the absence of any adjacent region from
Fig. 1. Optical micrograph of a longitudinal section of a hydrided Mg–Zr (ZR55) alloy after 3.5% elongation at 743 K under a stress of 1.7 MPa in a horizontal direction. Precipitate denuded zones are seen nearly perpendicular to the stress. A grain boundary meanders within these zones with some short segments along its length pinned by precipitates at the zone edges. There is no indication of the sweeping of precipitates by the boundary. At the grain boundary nearly aligned with the stress, precipitates have accumulated without evidence of their depletion on either side of the boundary.
which precipitates have been removed. All the features present in this illustration are fully in accord with those that would occur through the operation of a diffusional creep process. Further observations support the inferences from Fig. 1. In Fig. 2, a grain boundary is seen with sufficient curvature for one end region to be more nearly perpendicular to the applied stress and the other end to be nearly parallel to this stress. It is noted that precipitates have accumulated in the latter region, without diminution of precipitate concentration on either side of the boundary, and in the region, nearly perpendicular to the stress, there is a precipitate free zone and a grain boundary lying within it. These features are again totally in accord with their formation by diffusional creep. These observations clearly indicate a strong orientation dependence on precipitate redistribution to combat doubts about whether an orientation relationship exists. There is also further supporting evidence from observations on cross sections of specimens in planes perpendicular to the stress, where precipitate
Fig. 2. A longitudinal section of the gauge length of a similar alloy after 1.9% creep strain under a stress of 1.7 MPa at 743 K. The tensile stress axis is horizontal. It is noted that the regions of the grain boundaries most nearly perpendicular to the stress are associated with precipitate denuded zones though these zones contain some larger precipitates that have coarsened. In contrast, precipitates have accumulated in the curved grain boundary region aligned more closely in the direction of the stress.
14
G.W. Greenwood / Materials Science and Engineering A 410–411 (2005) 12–15
in creep is given by kTd 2 (2) BDEΩ When applied to the bending of a beam, it is noted that equation (2) is only applicable when the grain size is small compared with the beam thickness (that is d h) since the formula relates to the tensile and compressive stresses that act respectively on opposite sides of the neutral axis. The rate of change of curvature of the beam is then given by d/dt (1/R) = 12M/Ebh3 te and so from Eqs. (1) and (2), d 1 12MBDΩ = (3) dt R bh3 kTd 2
te =
Fig. 3. A section of a similar alloy, cut perpendicularly to the tensile stress, after a creep strain of 2% under a stress of 1.7 MPa at 743 K. A triangular section of a grain largely denuded of precipitates is observed. This is formed from precipitate denuded zones at three intersecting grain boundaries, all with orientations at large angles to the stress direction indicating the dependence of stress orientation on the location of the zones.
denuded zones can occupy a part, or even the whole, of a grain section as in Fig. 3. Another feature, only recently pointed out [13,14] as a likely consequence of diffusional creep, lies in the demonstration of small rotations between adjacent grains. The presence of stringers of precipitates, created in lines along the axis of extrusion in manufacture, have provided suitable markers with which to follow any re-alignment of grains during deformation. Small rotations have been noted by this means in specimens deforming under low stresses and their cause has been attributed to a diffusional creep mechanism. This is a significant observation, for it illustrates some previously unsuspected aspects of the diffusional creep process. These will next be considered. 3. Creep under a small bending moment The observation of grain rotation has been linked with bending moments [13,14]. This has important consequences, for it provides not only a connection with microstructural influences but also with macroscopic geometrical features of the material. These aspects are appropriately considered by the bending of a beam of width w and thickness h of material with Young’s modulus E. If the beam is subjected to a bending moment M that results in its curvature 1/R under elastic conditions [15], 1/R = M/EI, where the second moment of area of the beam I = bh3 /12. Hence, 1 12M = R Ebh3
(1)
When the bending moment remains small and at higher temperature, if diffusional creep occurs, we can deduce a time interval during which the increment of creep strain equals the initial elastic strain. From the well established formulae of Nabarro [4] and Herring [5], under a tensile stress σ at a temperature T, the diffusional creep strain after a time t is given by ε = BDσΩt/kTd2 where D is the diffusion coefficient, Ω the atomic volume and k is Boltzmann’s constant. d is grain size and B is a numerical constant having a value about 10. Since the elastic strain ε = σ/E, it follows that the time te to achieve a similar strain increment
It is noted in equation (3) that the rate of bending (change in curvature) is dependent on grain size as 1/d2 and on the beam dimensions, varying as 1/bh3 . It is apparent that the latter dependency is identical with the relationship in the elastic condition so that the permanent distortion created by diffusional creep may be expected to replicate, albeit in an exaggerated way, the pattern of deformation established elastically. This is relevant, for example, to cantilever deflection where there is a continuous change in bending moment along the length. The present analysis, however, does not cover all the conditions that may arise. In contrast with the previous situation, if the beam is sufficiently thin or the grain size large so that each grain occupies the entire beam thickness, Burton has shown [13,14] that the diffusional fluxes are centred around the throughthickness grain boundaries. Atoms are plated on regions of the grain boundaries subjected to tensile stresses with a corresponding depletion of atoms in different regions of the same boundaries crossing to the opposite side of the neutral axis under compressive stress. The simplest case here is to consider the effect on the rotation of adjacent grains with their common boundary parallel to the axis of the bending moment. An analysis of this case [14] shows that, where lattice diffusion predominates, the angular rate of rotation dθ/dt = 4750MDΩ/bh4 kT. If such grains are spaced a distance x apart along the length of the beam, with x h, then the rate of change of curvature of the beam is d 1 1 dθ 4750MDΩ = = (4) dt R x dt xbh4 kT Now comparing this formula with equation (3) where grains were equiaxed with d h we note the different variation with respect to both grain and beam geometry. With small equiaxed grains, the rate of change of beam curvature, from equation (3), varies with the grain size as 1/d2 and with beam dimensions as 1/bh3 . In contrast, where grains occupy the entire beam thickness and are separated by a distance x along the beam length with the grain boundaries parallel to the bending moment, then the rate of change of curvature, from equation (4), depends on grain geometry as 1/x and on the beam dimensions as 1/bh4 . There are, of course, situations intermediate between the two specific conditions dealt with here, but the present analysis illustrates the need to consider beam dimensions as well as microstructure in dealing with the deformation that can arise when diffusional
G.W. Greenwood / Materials Science and Engineering A 410–411 (2005) 12–15
creep occurs. There is no analogue to equation (4) in the application of elasticity theory as the rate of creep is no longer governed by the parameters that determine the elastic deformation. 4. Conclusions Deformation mechanism maps have an important role to play in their illustration of the ranges of stress and temperature over which specific forms of atomic movements are likely to occur. They indicate the range of applicability of formulae for evaluation. The boundaries separating these ranges may not be easy to determine since different processes may occur simultaneously, so blurring the transition between them. Microstructural observations can be of particular value. These cover many aspects: dislocation densities and arrangements, surface features such as slip steps, twinning, orientation dependent grain boundary grooving, grain boundary sliding, grain shape changes and the redistribution of precipitates. Precipitate redistribution may provide important clues but various interpretations have been suggested. The characteristics of precipitate denuded zones and of regions of precipitate accumulation have been examined in detail with the conclusion that they result from a diffusional creep mechanism in the cases considered here. Valid identification of conditions for the occurrence of diffusional creep is important for there is now in place a substantial theoretical framework [13,14], involving the description of vacancy flux patterns that permit analytical or numerical solutions of a wide variety of practical problems.
15
Analysis of this form of creep highlights the importance of microstructure and of the stress system to which the material is subjected. It is shown here that specimen geometry may also be influential and an example is given where this feature must be into account. Acknowledgements The author is grateful for useful discussion with Prof. H. Jones and Drs. K.R. McNee and V. Srivastava. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15]
M.F. Ashby, Acta Metall. 20 (1972) 887. F.A. Mohamed, T.G. Langdon, Met. Trans. 5 (1974) 2339. T.G. Langdon, Key Eng. Mater. 171 (2000) 205. F.R.N. Nabarro, Report on a Conference on the Strength of Solids, The Physical Society, London, 1948, p. 75. C. Herring, J. Appl. Phys. 21 (1950) 437. J. Harper, J.E. Dorn, Acta Metall. 5 (1957) 654. W. Blum, W. Maier, Phys. Status Solidi (a) 171 (1999) 467. F.R.N. Nabarro, Met. Mater. Trans. 33A (2002) 213. B. Wilshire, C.J. Palmer, Scr. Mater. 436 (2002) 483. T.G. Langdon, Mater. Sci. Eng. A 283 (2000) 266. R.L. Squires, R.T. Weiner, M. Philips, J. Nucl. Mater. 8 (1963) 77. J. Wadsworth, O.A. Ruano, O.D. Sherby, in: A.K. Mukherjee, R.L. Murty (Eds.), Creep Behaviour of Advanced Materials for the 21st Century, TMS, Warrendale, PA, 1999, p. 425. B. Burton, Philos. Mag. 82A (2002) 51. B. Burton, Philos. Mag. 83 (2003) 2715. A.H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964, p. 125.
Deformation mechanism maps and microstructural influences G.W. Greenwood ∗ Department of Engineering Materials, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK Received in revised form 27 April 2005
Abstract Areas on these maps define ranges of homologous temperature and of applied stress divided by an elastic modulus over which specific mechanisms of deformation are predominant. Langdon has extended their use to illustrate important features. Lines separating the mechanisms of overall sliding from diffusional drift can be difficult to locate. Observations of microstructural features assist in this location and provide information on the way in which the line positions can be influenced. Notably, the flux patterns of diffusional drift are dependent upon stress system as well as on microstructure and it is shown that specimen geometry may also be important. © 2005 Elsevier B.V. All rights reserved. Keywords: Deformation maps; Creep; Bending; Microstructure; Diffusion
1. Introduction The mechanism by which a crystalline solid deforms is strongly influenced by the ratios of applied tensile stress σ divided by Young’s Modulus E (or equivalently by the ratio of shear stress divided by shear modulus) and by the homologous temperature, T/Tm . Ashby [1] first demonstrated that these aspects can be represented by “maps”, often showing remarkably similar features when different materials are compared. Lines of constant strain rate, drawn as contours on these maps, usually show some small but definitive alteration in direction in crossing lines that separate regions where specific deformation modes predominate. Mohamed and Langdon [2] extended the map concept by showing that maps could be drawn with other axes to reveal features dependent upon microstructure when either the stress or temperature was maintained constant. A notable parameter in this respect is the grain size d that can be normalised by dividing by the Burgers vector b, so that the ratio, d/b, is plotted on one of the axes. Representations of this kind are valuable in determining the extent to which grain size is influential and in the consequential implications of the deformation mechanism that is rate controlling.
∗
Tel.: +44 1142225517; fax: +44 1142225943. E-mail address: [email protected].
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.098
Ashby and Langdon have always been careful in their recognition of the limitations of map representation. Adequate experimental data are not always available. The maps rely on unchanging microstructure that can be difficult to establish at temperatures approaching the melting point. Even small changes in microstructure can have large effects and for a given material it is important to be fully aware of compositional and structural influences. The lines forming the boundaries of different regions of the maps are not always easy to locate for there may be only a relatively small variation in the parameters governing the predominance of different mechanisms. This is the situation, for example, when the deformation mode is more strongly influenced by grain boundary than by lattice diffusion or when dislocation climb becomes significant in supplementation or replacement of the glide process. Even in situations where clear distinctions have been pointed out [3], it has remained difficult to resolve the relative importance between Nabarro–Herring [4,5] and Harper–Dorn [6] creep. The continuing debate [7,8] will not be entered into here. More profound changes in deformation mechanism arise if the role of dislocation motion is overtaken and replaced solely by the drift of vacancies under stress. Here, grain boundaries must play a decisive role in operating as vacancy sources and sinks and also in their ability to slide to accommodate grain deformation. Underlying the importance of different regions in the maps is the way in which the external stressing of a material leads to an
G.W. Greenwood / Materials Science and Engineering A 410–411 (2005) 12–15
13
internal redistribution of stress. This relates to the significance of microstructure, of the superimposed stress system, and, in some instances, of specimen geometry. These features will next be considered. 2. Microstructural observations Creep test data overall have been only of limited assistance in providing evidence of the range of dominance of specific deformation mechanisms. There is often a near continuity of stress–strain rate relationships over large ranges of T/Tm and σ/E, even to the extent that single equations have been proposed [9] to be representative of behaviour over almost the whole range of conditions. To throw light on these uncertainties, distinctive microstructural changes have been noted [10] that are indicative of definitive deformation modes. These include grain shape changes, grain boundary sliding, dislocation rearrangements and precipitate denuded regions in two-phase materials. This latter aspect, however, has given rise to much speculation. When precipitate denuded zones were first observed [11] in materials subjected to creep under low stresses, their occurrence was linked to the operation of a diffusional creep process in which deformation occurred only through the stress directed migration of vacancies. For many years, this viewpoint remained unchallenged but more recently further questions have been raised [12]. Such zones have been observed when the creep rate was proportional to applied stress raised to a power more than unity. Grain boundaries are not generally in the centre of the zones. The zones have been considered to lie at orientations more random than those expected through diffusional creep operation. Processes other than creep have been considered to produce comparable zones. Following suggestions that precipitate free zones may not result from diffusional creep, the question naturally arises of an alternative cause of their formation. One suggestion is that precipitates are taken into solution by moving grain boundaries and subsequently re-precipitated. There is, however, substantial micro-analytical evidence that there are no regions of enhanced solute concentration in the matrix as would be anticipated in this proposal. Another proposal for the mechanism of zone creation is that they arise through grain boundary movement to cause particle sweeping such that a largely precipitate free zone is left behind. Such sweeping processes have been observed in a number of metallurgical treatments and it is necessary to enquire whether the microstructures after creep at low stresses bear resemblance to these. From Fig. 1, a precipitate denuded zone is illustrated in a hydrided Mg–Zr alloy after creep at low stresses. Here, the grain boundary is seen meandering within the zone which is largely free of precipitates. It appears that the grain boundary has had freedom to migrate within the zone though apparently pinned over short regions of the boundary where it has aligned with the edges of the precipitated regions. Equally noteworthy in Fig. 1 is the horizontal boundary, lying nearly parallel to the applied stress on which precipitates have accumulated. A critical feature here is the absence of any adjacent region from
Fig. 1. Optical micrograph of a longitudinal section of a hydrided Mg–Zr (ZR55) alloy after 3.5% elongation at 743 K under a stress of 1.7 MPa in a horizontal direction. Precipitate denuded zones are seen nearly perpendicular to the stress. A grain boundary meanders within these zones with some short segments along its length pinned by precipitates at the zone edges. There is no indication of the sweeping of precipitates by the boundary. At the grain boundary nearly aligned with the stress, precipitates have accumulated without evidence of their depletion on either side of the boundary.
which precipitates have been removed. All the features present in this illustration are fully in accord with those that would occur through the operation of a diffusional creep process. Further observations support the inferences from Fig. 1. In Fig. 2, a grain boundary is seen with sufficient curvature for one end region to be more nearly perpendicular to the applied stress and the other end to be nearly parallel to this stress. It is noted that precipitates have accumulated in the latter region, without diminution of precipitate concentration on either side of the boundary, and in the region, nearly perpendicular to the stress, there is a precipitate free zone and a grain boundary lying within it. These features are again totally in accord with their formation by diffusional creep. These observations clearly indicate a strong orientation dependence on precipitate redistribution to combat doubts about whether an orientation relationship exists. There is also further supporting evidence from observations on cross sections of specimens in planes perpendicular to the stress, where precipitate
Fig. 2. A longitudinal section of the gauge length of a similar alloy after 1.9% creep strain under a stress of 1.7 MPa at 743 K. The tensile stress axis is horizontal. It is noted that the regions of the grain boundaries most nearly perpendicular to the stress are associated with precipitate denuded zones though these zones contain some larger precipitates that have coarsened. In contrast, precipitates have accumulated in the curved grain boundary region aligned more closely in the direction of the stress.
14
G.W. Greenwood / Materials Science and Engineering A 410–411 (2005) 12–15
in creep is given by kTd 2 (2) BDEΩ When applied to the bending of a beam, it is noted that equation (2) is only applicable when the grain size is small compared with the beam thickness (that is d h) since the formula relates to the tensile and compressive stresses that act respectively on opposite sides of the neutral axis. The rate of change of curvature of the beam is then given by d/dt (1/R) = 12M/Ebh3 te and so from Eqs. (1) and (2), d 1 12MBDΩ = (3) dt R bh3 kTd 2
te =
Fig. 3. A section of a similar alloy, cut perpendicularly to the tensile stress, after a creep strain of 2% under a stress of 1.7 MPa at 743 K. A triangular section of a grain largely denuded of precipitates is observed. This is formed from precipitate denuded zones at three intersecting grain boundaries, all with orientations at large angles to the stress direction indicating the dependence of stress orientation on the location of the zones.
denuded zones can occupy a part, or even the whole, of a grain section as in Fig. 3. Another feature, only recently pointed out [13,14] as a likely consequence of diffusional creep, lies in the demonstration of small rotations between adjacent grains. The presence of stringers of precipitates, created in lines along the axis of extrusion in manufacture, have provided suitable markers with which to follow any re-alignment of grains during deformation. Small rotations have been noted by this means in specimens deforming under low stresses and their cause has been attributed to a diffusional creep mechanism. This is a significant observation, for it illustrates some previously unsuspected aspects of the diffusional creep process. These will next be considered. 3. Creep under a small bending moment The observation of grain rotation has been linked with bending moments [13,14]. This has important consequences, for it provides not only a connection with microstructural influences but also with macroscopic geometrical features of the material. These aspects are appropriately considered by the bending of a beam of width w and thickness h of material with Young’s modulus E. If the beam is subjected to a bending moment M that results in its curvature 1/R under elastic conditions [15], 1/R = M/EI, where the second moment of area of the beam I = bh3 /12. Hence, 1 12M = R Ebh3
(1)
When the bending moment remains small and at higher temperature, if diffusional creep occurs, we can deduce a time interval during which the increment of creep strain equals the initial elastic strain. From the well established formulae of Nabarro [4] and Herring [5], under a tensile stress σ at a temperature T, the diffusional creep strain after a time t is given by ε = BDσΩt/kTd2 where D is the diffusion coefficient, Ω the atomic volume and k is Boltzmann’s constant. d is grain size and B is a numerical constant having a value about 10. Since the elastic strain ε = σ/E, it follows that the time te to achieve a similar strain increment
It is noted in equation (3) that the rate of bending (change in curvature) is dependent on grain size as 1/d2 and on the beam dimensions, varying as 1/bh3 . It is apparent that the latter dependency is identical with the relationship in the elastic condition so that the permanent distortion created by diffusional creep may be expected to replicate, albeit in an exaggerated way, the pattern of deformation established elastically. This is relevant, for example, to cantilever deflection where there is a continuous change in bending moment along the length. The present analysis, however, does not cover all the conditions that may arise. In contrast with the previous situation, if the beam is sufficiently thin or the grain size large so that each grain occupies the entire beam thickness, Burton has shown [13,14] that the diffusional fluxes are centred around the throughthickness grain boundaries. Atoms are plated on regions of the grain boundaries subjected to tensile stresses with a corresponding depletion of atoms in different regions of the same boundaries crossing to the opposite side of the neutral axis under compressive stress. The simplest case here is to consider the effect on the rotation of adjacent grains with their common boundary parallel to the axis of the bending moment. An analysis of this case [14] shows that, where lattice diffusion predominates, the angular rate of rotation dθ/dt = 4750MDΩ/bh4 kT. If such grains are spaced a distance x apart along the length of the beam, with x h, then the rate of change of curvature of the beam is d 1 1 dθ 4750MDΩ = = (4) dt R x dt xbh4 kT Now comparing this formula with equation (3) where grains were equiaxed with d h we note the different variation with respect to both grain and beam geometry. With small equiaxed grains, the rate of change of beam curvature, from equation (3), varies with the grain size as 1/d2 and with beam dimensions as 1/bh3 . In contrast, where grains occupy the entire beam thickness and are separated by a distance x along the beam length with the grain boundaries parallel to the bending moment, then the rate of change of curvature, from equation (4), depends on grain geometry as 1/x and on the beam dimensions as 1/bh4 . There are, of course, situations intermediate between the two specific conditions dealt with here, but the present analysis illustrates the need to consider beam dimensions as well as microstructure in dealing with the deformation that can arise when diffusional
G.W. Greenwood / Materials Science and Engineering A 410–411 (2005) 12–15
creep occurs. There is no analogue to equation (4) in the application of elasticity theory as the rate of creep is no longer governed by the parameters that determine the elastic deformation. 4. Conclusions Deformation mechanism maps have an important role to play in their illustration of the ranges of stress and temperature over which specific forms of atomic movements are likely to occur. They indicate the range of applicability of formulae for evaluation. The boundaries separating these ranges may not be easy to determine since different processes may occur simultaneously, so blurring the transition between them. Microstructural observations can be of particular value. These cover many aspects: dislocation densities and arrangements, surface features such as slip steps, twinning, orientation dependent grain boundary grooving, grain boundary sliding, grain shape changes and the redistribution of precipitates. Precipitate redistribution may provide important clues but various interpretations have been suggested. The characteristics of precipitate denuded zones and of regions of precipitate accumulation have been examined in detail with the conclusion that they result from a diffusional creep mechanism in the cases considered here. Valid identification of conditions for the occurrence of diffusional creep is important for there is now in place a substantial theoretical framework [13,14], involving the description of vacancy flux patterns that permit analytical or numerical solutions of a wide variety of practical problems.
15
Analysis of this form of creep highlights the importance of microstructure and of the stress system to which the material is subjected. It is shown here that specimen geometry may also be influential and an example is given where this feature must be into account. Acknowledgements The author is grateful for useful discussion with Prof. H. Jones and Drs. K.R. McNee and V. Srivastava. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15]
M.F. Ashby, Acta Metall. 20 (1972) 887. F.A. Mohamed, T.G. Langdon, Met. Trans. 5 (1974) 2339. T.G. Langdon, Key Eng. Mater. 171 (2000) 205. F.R.N. Nabarro, Report on a Conference on the Strength of Solids, The Physical Society, London, 1948, p. 75. C. Herring, J. Appl. Phys. 21 (1950) 437. J. Harper, J.E. Dorn, Acta Metall. 5 (1957) 654. W. Blum, W. Maier, Phys. Status Solidi (a) 171 (1999) 467. F.R.N. Nabarro, Met. Mater. Trans. 33A (2002) 213. B. Wilshire, C.J. Palmer, Scr. Mater. 436 (2002) 483. T.G. Langdon, Mater. Sci. Eng. A 283 (2000) 266. R.L. Squires, R.T. Weiner, M. Philips, J. Nucl. Mater. 8 (1963) 77. J. Wadsworth, O.A. Ruano, O.D. Sherby, in: A.K. Mukherjee, R.L. Murty (Eds.), Creep Behaviour of Advanced Materials for the 21st Century, TMS, Warrendale, PA, 1999, p. 425. B. Burton, Philos. Mag. 82A (2002) 51. B. Burton, Philos. Mag. 83 (2003) 2715. A.H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964, p. 125.