- Email: [email protected]
Strain hardening in FCC metals and alloys
Materials Science and Engineering A309–310 (2001) 328–330
Strain hardening in FCC metals and alloys C.S. Pande∗ Naval Research Laboratory, Washington, DC 20375-5000, USA
Abstract Strain hardening (workhardening) in single crystal of ductile face centered cubic metals in stages I, II and III are briefly reviewed. A preliminary description of a new model of strain hardening based on stochastic and statistical considerations is given. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Strain hardening; Dislocation configuration; Models
1. Introduction One of the most challenging problems in dislocation theory has been the understanding of workhardening phenomena in single crystals in terms of a dislocation mechanism. Early work in this field is covered in great detail in the excellent review of Nabarro et al. [1] published in 1964. Their review starts with a quotation of Orowan made in 1951 [2] “Since the publication of Taylor’s theory there has been no significant progress in the understanding of strain hardening.” Taylor [3] in his celebrated work assumed the dislocations to be arranged in a regular lattice of positive and negative dislocations and calculated the flow stress of such an arrangement. Many models that followed have some elements of Taylor’s model. Taylor gave no mechanism for the formation of his dislocation lattice. Much information has since been obtained by transmission electron microscopy (TEM) [1] of deformed single crystals. We refer to a brief but rather complete review of such work by Hirsch [4]. See also a paper by Brown where he provides a model of his own [5]. Inspite of the wealth of information now available and claims and counterclaims in literature on workhardening, Orowan’s statement can still be made with some justification.
2. Dislocation configuration TEM observations were made in single crystals of copper alloys oriented for single slip. In stage I the dislocations were found confined to their slip plane and formed multipoles on two close parallel planes. Such multipoles were ∗ Tel.: +1-202-767-2144; fax: +1-202-767-2623. E-mail address: [email protected] (C.S. Pande).
more “regular” and easy to visualize if the stacking fault energy were lowered by making a solid solution [6]. It is found that stage I is surprisingly dependent on the nucleation of secondary dislocations and on their role as obstacles for ‘anchoring’ dislocation multipoles. In turn dislocations multipoles are capable of producing secondary slip dislocations on suitably oriented slip planes [7]. In stage II, both secondary and primary dislocations are active and many dislocation barriers exist. In stage III, cell formation takes place. [8]. Our observations are consistent with the model of Hirsch and Mitchell [9] and of Kuhlmann–Wilsdorf [10] especially for stage III. As mentioned before since our work numerous other studies have confirmed these results. The challenge is to use these results to build a consistent model of strain hardening for all the three stages. (We ignore for the time being stage IV, etc.).
3. Models Modeling of workhardening can be done at several levels of complication. Most satisfying, but also most difficult would be a model that predicts a variety of features such as stress strain curve, dislocation density versus stress relation, and slip line properties and nature of dislocations (primary or secondary dislocations with their Burgers vectors) from some dislocation configuration. It should also be able to predict how such a configuration develops. TEM gives such detail information in most of the cases about dislocations. However, we believe such a detailed model is possible only in the simplest of cases, such as stage I in FCC single crystal oriented for single slip. A second type of model will be a crude but simple attempt to explain stress strain curve and dislocation density vs strain
0921-5093/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 7 0 7 - X
C.S. Pande / Materials Science and Engineering A309–310 (2001) 328–330
without going into the detailed dislocation configuration. It is surprising that even at this level of simplification there is no universally accepted model. Attention is however, drawn to an excellent model which attempts to do this with some success [11] and which till now has been ignored. Finally there are now new attempts involving more sophisticated mathematical techniques. Some of attempts were discussed at the “Dislocation 2000” conference. We present here preliminary description a new model based on statistical and stochastic considerations. Of necessity, it ignores detailed dislocation picture though they may be needed to calculate the value of the various parameters in the model.
4. A stochastic model of strain hardening Taylor model attempted to explain the stress strain curve in polycrystals. This curve is approximately parabolic. However, in case of single crystals a three stage curve is obtained. A look at the resolved shear stress strain curve shows that most of the curve is made up of stage II and the transition to stage II from stage I and transition of stage II to stage III. These transitions are very smooth. Therefore, in our opinion a general theory should concentrate on stage II and its transition stages. Stage II is approximately linear with a slope which is of the order of G/300. Stage II shows a remarkable property. No matter whether the material is bcc or FCC, no matter it is a solid solution alloy or single element stress–strain curve is always linear with roughly same slope. This is true also for any orientation, any strain rate and for any temperature of deformation. It means that something very simple and fundamental is involved which is independent of various details of deformation of material. In our view there is no single mechanism that operates in stage II. In literature a variety of mechanisms have been proposed. In stage II, a large number of these can be seen to be operating at the same time. Hirsch [4] lists many of them. We briefly summarize his list below by just mentioning the mechanism and for some cases the corresponding flow stress: 1. Passing stress of two dislocations of opposite sign: passing stress is K/L where K = Gb/4π. 2. Passing stress of multipoles on two adjacent planes = K/4L. 3. Passing stress dislocation pileups. 4. Stress to operate a Frank–Read source. 5. Stress to overcome forest dislocations. 6. Stress to pull out dipoles sessile jogs. 7. Stress necessary to form a jog at a forest dislocation. 8. Stress necessary for a jog to create a vacancy. Most, if not all, of these contributions to flow stress can be expressed as K/L∗ [4] where L∗ is a characteristic length (one for each mechanism) and is related to dislocation arrangements.
329
Following Hirsch [4] we assume that the eventual flow stress of the system will the sum of all these components. It should also be noted that each of these components will have a distribution. What then will be the sum of the distributions? We take advantage of a theorem of statistics via Central Limit theorem, which states that no matter what the individual distributions are, the distribution of the sum is always Gaussian, i.e. the flow stress of the system in stage II is normally distributed. The proof of this result is given by Feller [12]. To be sure this theorem is valid only under some well-known constraints. Each distribution should be independent of each other and there are restrictions on the width of the distribution. Assuming that this theorem is at least approximately obeyed, we obtain that the distribution of the sum will also be Gaussian, i.e. the flow stress will be Gaussian. This is not enough to pin down the Fokker–Planck equation of this distribution since there are many Fokker–Plank equations with this distribution. However, the physics of the problem provides us some further guidelines such as 1. the mean of the distribution is not zero but finite, 2. the results at a particular strain are independent of prior history, i.e. it is a Markov process, 3. after a certain deformation the slip line length and the shear stress value saturates. This is equivalent to assuming that there is a stationary state of the distribution. These constraints limit the choice of the distribution very much. The simplest one that fits our requirement is the Fokker–Planck distribution corresponding to Ornstein–Uhlenbeck process. According to this process the mean slip line length should decrease exponentially with strain ε. A detailed calculation gives the very approximate relation dL = k (L − L0 ) dε This is the equation used in [11] without any rigorous proof. If the initial L value is Li then L = L0 + (Li − L0 ) exp (−kε) and the flow stress in appropriate units is given by Gb L0 + (Li − L0 ) exp (−kε) This equation is plotted in Fig. 1. It is seen that the three stages of stress strain curve is clearly present. As surmised the stage II is dominant with the other two stages acting more or less as transition stages to stage II. It should also be noted that stage I is only present if Li is sufficiently large. These results can also be applied to polycrystalline hardening in an approximate way if Li is taken of the same order as the grain size. The parameter k is related to the rate with which L approaches its saturation value L0 and hence, it would depend on the ease of the activity of the secondary dislocations. Hence, k for polycrystals should be a little higher.
330
C.S. Pande / Materials Science and Engineering A309–310 (2001) 328–330
Fig. 1. Stress–strain curve predicted from theory (see text). The three stages of workhardening are clearly visible.
Apart from these differences polycrystal behavior should be similar. Next we show that the dislocation density versus flow stress can also be derived from this formulation. From dislocation theory, dρ 1 = dε Lb or dρ 1 dσ = dε Lb dε Using Gb/L = σ , σ dσ dρ = dσ 2 dε Gb In stage I or II, (dσ/dε)/G = constant = θ , so that σ − σ0 = αGbρ 1/2 where α is a constant of the order of 0.1 in stage I and about 0.5 in stage II [4] σ 0 is constant. It is seen that although the standard relation between dislocation density and flow stress is obtained for both stages I and II, the constant ␣ is different for the two stages. This is borne by experiments [4]. Finally, we can similarly show that in both stages I and II, L is inversely proportional to strain, again consistent with experiments [4].
5. Summary and conclusions In this short paper we have proposed that deformation behavior of metals and alloys can be treated as a stochastic process and specifically there are reasons to believe that it can be further specified as a Ornstein–Uhlenbeck process. Many features of the deformation process follow naturally from this formulation. A consistent theory of strain hardening can be formulated on these lines. References [1] F.R.N. Nabarro, Z.S. Basinski, D.B. Holt, Adv. in Phys 13, (1964). [2] E. Orowan, quoted in reference [1]. [3] G.I. Taylor, Proc. Roy. Soc. A 145 (1934) 362. [4] P.B. Hirsch (Ed.), Physics of Metals,Vol. 2, Cambridge University Press, Cambridge, 1975, p. 189. [5] L.M. Brown, Met. Trans. 22A (1991) 1693. [6] J.W. Steeds, P.M. Hazzledine, Dis. Faraday Soc. 38 (1964) 103. [7] C.S. Pande, P.M. Hazzledine, Philos. Mag. 24 (1971) 1093. [8] C.S. Pande, P.M. Hazzledine, Philos. Mag. 24 (1971) 1393. [9] P.B. Hirsch, T.E. Mitchell, Can. J. Phys. 45 (1967) 663. [10] D. Kuhlmann-Wilsdorf, Script. Mat. 36 (1997) 173. [11] W. Roberts, Y. Bergstrom, Acta. Met. 21 (1973) 457. [12] W. Feller, An Introduction to Probaility Theory and Applications, Vol. 2, 1971, Wiley, New York.
Strain hardening in FCC metals and alloys C.S. Pande∗ Naval Research Laboratory, Washington, DC 20375-5000, USA
Abstract Strain hardening (workhardening) in single crystal of ductile face centered cubic metals in stages I, II and III are briefly reviewed. A preliminary description of a new model of strain hardening based on stochastic and statistical considerations is given. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Strain hardening; Dislocation configuration; Models
1. Introduction One of the most challenging problems in dislocation theory has been the understanding of workhardening phenomena in single crystals in terms of a dislocation mechanism. Early work in this field is covered in great detail in the excellent review of Nabarro et al. [1] published in 1964. Their review starts with a quotation of Orowan made in 1951 [2] “Since the publication of Taylor’s theory there has been no significant progress in the understanding of strain hardening.” Taylor [3] in his celebrated work assumed the dislocations to be arranged in a regular lattice of positive and negative dislocations and calculated the flow stress of such an arrangement. Many models that followed have some elements of Taylor’s model. Taylor gave no mechanism for the formation of his dislocation lattice. Much information has since been obtained by transmission electron microscopy (TEM) [1] of deformed single crystals. We refer to a brief but rather complete review of such work by Hirsch [4]. See also a paper by Brown where he provides a model of his own [5]. Inspite of the wealth of information now available and claims and counterclaims in literature on workhardening, Orowan’s statement can still be made with some justification.
2. Dislocation configuration TEM observations were made in single crystals of copper alloys oriented for single slip. In stage I the dislocations were found confined to their slip plane and formed multipoles on two close parallel planes. Such multipoles were ∗ Tel.: +1-202-767-2144; fax: +1-202-767-2623. E-mail address: [email protected] (C.S. Pande).
more “regular” and easy to visualize if the stacking fault energy were lowered by making a solid solution [6]. It is found that stage I is surprisingly dependent on the nucleation of secondary dislocations and on their role as obstacles for ‘anchoring’ dislocation multipoles. In turn dislocations multipoles are capable of producing secondary slip dislocations on suitably oriented slip planes [7]. In stage II, both secondary and primary dislocations are active and many dislocation barriers exist. In stage III, cell formation takes place. [8]. Our observations are consistent with the model of Hirsch and Mitchell [9] and of Kuhlmann–Wilsdorf [10] especially for stage III. As mentioned before since our work numerous other studies have confirmed these results. The challenge is to use these results to build a consistent model of strain hardening for all the three stages. (We ignore for the time being stage IV, etc.).
3. Models Modeling of workhardening can be done at several levels of complication. Most satisfying, but also most difficult would be a model that predicts a variety of features such as stress strain curve, dislocation density versus stress relation, and slip line properties and nature of dislocations (primary or secondary dislocations with their Burgers vectors) from some dislocation configuration. It should also be able to predict how such a configuration develops. TEM gives such detail information in most of the cases about dislocations. However, we believe such a detailed model is possible only in the simplest of cases, such as stage I in FCC single crystal oriented for single slip. A second type of model will be a crude but simple attempt to explain stress strain curve and dislocation density vs strain
0921-5093/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 7 0 7 - X
C.S. Pande / Materials Science and Engineering A309–310 (2001) 328–330
without going into the detailed dislocation configuration. It is surprising that even at this level of simplification there is no universally accepted model. Attention is however, drawn to an excellent model which attempts to do this with some success [11] and which till now has been ignored. Finally there are now new attempts involving more sophisticated mathematical techniques. Some of attempts were discussed at the “Dislocation 2000” conference. We present here preliminary description a new model based on statistical and stochastic considerations. Of necessity, it ignores detailed dislocation picture though they may be needed to calculate the value of the various parameters in the model.
4. A stochastic model of strain hardening Taylor model attempted to explain the stress strain curve in polycrystals. This curve is approximately parabolic. However, in case of single crystals a three stage curve is obtained. A look at the resolved shear stress strain curve shows that most of the curve is made up of stage II and the transition to stage II from stage I and transition of stage II to stage III. These transitions are very smooth. Therefore, in our opinion a general theory should concentrate on stage II and its transition stages. Stage II is approximately linear with a slope which is of the order of G/300. Stage II shows a remarkable property. No matter whether the material is bcc or FCC, no matter it is a solid solution alloy or single element stress–strain curve is always linear with roughly same slope. This is true also for any orientation, any strain rate and for any temperature of deformation. It means that something very simple and fundamental is involved which is independent of various details of deformation of material. In our view there is no single mechanism that operates in stage II. In literature a variety of mechanisms have been proposed. In stage II, a large number of these can be seen to be operating at the same time. Hirsch [4] lists many of them. We briefly summarize his list below by just mentioning the mechanism and for some cases the corresponding flow stress: 1. Passing stress of two dislocations of opposite sign: passing stress is K/L where K = Gb/4π. 2. Passing stress of multipoles on two adjacent planes = K/4L. 3. Passing stress dislocation pileups. 4. Stress to operate a Frank–Read source. 5. Stress to overcome forest dislocations. 6. Stress to pull out dipoles sessile jogs. 7. Stress necessary to form a jog at a forest dislocation. 8. Stress necessary for a jog to create a vacancy. Most, if not all, of these contributions to flow stress can be expressed as K/L∗ [4] where L∗ is a characteristic length (one for each mechanism) and is related to dislocation arrangements.
329
Following Hirsch [4] we assume that the eventual flow stress of the system will the sum of all these components. It should also be noted that each of these components will have a distribution. What then will be the sum of the distributions? We take advantage of a theorem of statistics via Central Limit theorem, which states that no matter what the individual distributions are, the distribution of the sum is always Gaussian, i.e. the flow stress of the system in stage II is normally distributed. The proof of this result is given by Feller [12]. To be sure this theorem is valid only under some well-known constraints. Each distribution should be independent of each other and there are restrictions on the width of the distribution. Assuming that this theorem is at least approximately obeyed, we obtain that the distribution of the sum will also be Gaussian, i.e. the flow stress will be Gaussian. This is not enough to pin down the Fokker–Planck equation of this distribution since there are many Fokker–Plank equations with this distribution. However, the physics of the problem provides us some further guidelines such as 1. the mean of the distribution is not zero but finite, 2. the results at a particular strain are independent of prior history, i.e. it is a Markov process, 3. after a certain deformation the slip line length and the shear stress value saturates. This is equivalent to assuming that there is a stationary state of the distribution. These constraints limit the choice of the distribution very much. The simplest one that fits our requirement is the Fokker–Planck distribution corresponding to Ornstein–Uhlenbeck process. According to this process the mean slip line length should decrease exponentially with strain ε. A detailed calculation gives the very approximate relation dL = k (L − L0 ) dε This is the equation used in [11] without any rigorous proof. If the initial L value is Li then L = L0 + (Li − L0 ) exp (−kε) and the flow stress in appropriate units is given by Gb L0 + (Li − L0 ) exp (−kε) This equation is plotted in Fig. 1. It is seen that the three stages of stress strain curve is clearly present. As surmised the stage II is dominant with the other two stages acting more or less as transition stages to stage II. It should also be noted that stage I is only present if Li is sufficiently large. These results can also be applied to polycrystalline hardening in an approximate way if Li is taken of the same order as the grain size. The parameter k is related to the rate with which L approaches its saturation value L0 and hence, it would depend on the ease of the activity of the secondary dislocations. Hence, k for polycrystals should be a little higher.
330
C.S. Pande / Materials Science and Engineering A309–310 (2001) 328–330
Fig. 1. Stress–strain curve predicted from theory (see text). The three stages of workhardening are clearly visible.
Apart from these differences polycrystal behavior should be similar. Next we show that the dislocation density versus flow stress can also be derived from this formulation. From dislocation theory, dρ 1 = dε Lb or dρ 1 dσ = dε Lb dε Using Gb/L = σ , σ dσ dρ = dσ 2 dε Gb In stage I or II, (dσ/dε)/G = constant = θ , so that σ − σ0 = αGbρ 1/2 where α is a constant of the order of 0.1 in stage I and about 0.5 in stage II [4] σ 0 is constant. It is seen that although the standard relation between dislocation density and flow stress is obtained for both stages I and II, the constant ␣ is different for the two stages. This is borne by experiments [4]. Finally, we can similarly show that in both stages I and II, L is inversely proportional to strain, again consistent with experiments [4].
5. Summary and conclusions In this short paper we have proposed that deformation behavior of metals and alloys can be treated as a stochastic process and specifically there are reasons to believe that it can be further specified as a Ornstein–Uhlenbeck process. Many features of the deformation process follow naturally from this formulation. A consistent theory of strain hardening can be formulated on these lines. References [1] F.R.N. Nabarro, Z.S. Basinski, D.B. Holt, Adv. in Phys 13, (1964). [2] E. Orowan, quoted in reference [1]. [3] G.I. Taylor, Proc. Roy. Soc. A 145 (1934) 362. [4] P.B. Hirsch (Ed.), Physics of Metals,Vol. 2, Cambridge University Press, Cambridge, 1975, p. 189. [5] L.M. Brown, Met. Trans. 22A (1991) 1693. [6] J.W. Steeds, P.M. Hazzledine, Dis. Faraday Soc. 38 (1964) 103. [7] C.S. Pande, P.M. Hazzledine, Philos. Mag. 24 (1971) 1093. [8] C.S. Pande, P.M. Hazzledine, Philos. Mag. 24 (1971) 1393. [9] P.B. Hirsch, T.E. Mitchell, Can. J. Phys. 45 (1967) 663. [10] D. Kuhlmann-Wilsdorf, Script. Mat. 36 (1997) 173. [11] W. Roberts, Y. Bergstrom, Acta. Met. 21 (1973) 457. [12] W. Feller, An Introduction to Probaility Theory and Applications, Vol. 2, 1971, Wiley, New York.