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Geometrically necessary twins and their associated size effects
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Scripta Materialia 59 (2008) 135–138 www.elsevier.com/locate/scriptamat
Geometrically necessary twins and their associated size effects Javier Gil Sevillano CEIT and TECNUN (University of Navarra), M. de Lardiza´bal, 15, 20018 San Sebastia´n, Spain Received 8 October 2007; revised 12 February 2008; accepted 19 February 2008 Available online 14 March 2008
In metallic alloys or mineral crystals where twinning is a prominent deformation micromechanism, geometrically necessary twins (GNTs) have to be stored in order to accommodate plastic strain gradients, complementing or substituting the geometrically necessary dislocations (GNDs) that fulfil this function in materials deforming exclusively by crystallographic slip. The presence of GNTs will contribute to strain gradient hardening, just as GNDs contribute to it. Consequently, GNTs will be responsible for size effects. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Mechanical twinning; Plastic strain gradients; Size effects
The subdivision of strain-induced dislocation density into statistically stored dislocation (SSD) and geometrically necessary dislocation (GND) densities has gained widespread acceptance [1–3]. The convention for this distinction is to ascribe to the GND density the dislocations stored in order to accommodate the macroscopic or mesoscopic plastic strain gradients either externally imposed (as in torsion or indentation tests) or arising from the heterogeneity of the microstructure (as in polycrystals or in two-phase materials). The SSD density is the density of dislocations that would be stored during plastic deformation of the same material in the absence of those gradients. Of course, the distinction is rather artificial in some cases: any dislocation is geometrically necessary in its place; however, in most cases of interest, the distinction is clear and very practical. In low stacking-fault energy (SFE) metallic alloys or mineral materials under deformation conditions of relatively high strain rate and low temperature, deformation twinning is a prominent plastic micromechanism that competes with and complements dislocation-mediated slip [4–7]. In such materials, strain gradients will mainly be accommodated by the storage of an extra density of deformation twins instead of mainly by GNDs. We can thus speak of geometrically necessary twins (GNTs) in the same way that we speak of GNDs. On this basis, we can now consider the side effects of the presence of such GNTs on strength. Indeed, we can expect size effects associated with GNTs because of the strengthening effect of their presence. E-mail: [email protected]
Twin boundaries are special high-angle boundaries that represent strong barriers to the propagation of other non-coplanar deformation twins or gliding dislocations and, like other high-angle boundaries, they contribute to the Hall–Petch effect in polycrystals. In addition to this effect due to the confinement of plasticity, twin propagation and growth often involves an associated density of dislocations contributing to the hardening of the crystal matrix. The first effect seems to be the most important one, explaining the stage of extraordinary work-hardening rate (named stage B by Asgari et al. [8]), 0.02–0.03G, where G is the shear elastic modulus, exhibited up to medium strains (e 6 1) by materials that deform mainly by twinning, like twinning-induced plasticity (TWIP) steels and other low SFE face-centred cubic (fcc) or hexagonal close-packed (hcp) alloys [9–14]. The contribution of deformation twinning to the macroscopic strain e is very different from the contribution of crystallographic slip. For the latter dC ; ð1Þ M is an average orientation factor for the active where M slip systems and C is the total amount of crystallographic slip. The slip amount is, in principle, assuming the availability of mobile dislocations, unlimited. For twinning de ¼
de ¼
CT dfv ; MT
ð2Þ
T is where CT is the finite shear undergone by the twin, M an average orientation factor for the active twinning
1359-6462/$ - see front matter Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2008.02.052
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J. Gil Sevillano / Scripta Materialia 59 (2008) 135–138
systems and fv is the twinned volume fraction in the volume element considered. On account of the polar character of twinning, the total amount of strain achievable by twinning when a monotonic strain path is followed would be limited to CT =M T unless secondary twinning could be activated inside the twinned volumes. For instance,pffiffiffifor fcc metals and h1 1 2i{1 1 1} twinning, CT ¼ 2=2. It is generally accepted that deformation twinning obeys a critical resolved shear stress (CRSS) law similar to the Schmid law for crystallographic slip (see Ref. [5] for a review and Ref. [15] for a recent experimental study). Under a macroscopic stress r, twinning will be observed when its CRSS is reached for ð3Þ r ¼ M T sT : However, the experimental evidence for a fixed value of the twinning CRSS sT is not always as clear as in the case of crystallographic slip, possibly because twinning is controlled either by nucleation or growth in the same material depending on the structure or pre-strain (and, because of the heterogeneity of the real structures, depending on the spatial location inside the same sample, and of the internal stresses associated with other mechanical twins and dislocation groups). Often twinning is observed to occur simultaneously with crystallographic slip. Under conditions of constrained deformation (i.e. approximately for polycrystalline deformation) of fcc metals, twinning will coexist with slip when the ratio pffiffiffi twinning and pffiffiffiof the CRSS for slip is in the range 1= 3 6 sT =sc 6 2= 3 [16,17], obviously under the assumption of availability of twinning and slip sources. Constrained deformation exclusively pffiffiffi by twinning of fcc metals is possible for sT =sc < 1= 3. The CRSS for twinning increases for increasing dislocation density or for decreasing grain or mean free path size. As twinning occurs by the movement of twinning dislocations, any microstructural feature representing an obstacle to dislocation propagation makes the twinning CRSS increase. The twinning CRSS versus mean free path relationship is of the Hall–Petch type [13,14], although some authors favour an exponent other than 1/2 (e.g. Vo¨hringer [18] and El-Danaf et al. [12] have found exponents of near 1). It should be noted that the Hall–Petch slope for the twinning CRSS is consistently larger than the corresponding slope for the crystallographic slip CRSS [6]. For virgin polycrystals with large grain size the CRSS for twinning corresponds to twin nucleation, which in pure metals or solid solutions is often followed by unstable propagation (serrated stress–strain curves and audible clicks). For small enough grain size polycrystals or after sufficient strain-induced structural fragmentation and hardening deformation, twinning ceases. This apparently implies that a lower limit to the mean free path for the occurrence of twinning exists, i.e. a size-induced transition in deformation mechanism would take place below some threshold value of the mean free path. For free nanocrystals, however, and for nanocrystalline polycrystals, molecular dynamics simulations predict twinning activity that is confirmed by experimental observations [19–21]. Inside an undeformed grain in a polycrystal, the first deformation twins start from a grain boundary and
cross the whole grain section until they are arrested by impingement with surrounding grain boundaries or with other twin boundaries. In a pre-deformed dislocated structure, twins can be nucleated from dislocation junctions and can be arrested by dislocation arrays. Deformation twins are very thin lenticular (hcp) or plate-like (fcc) reoriented layers, often of nanometric thickness (‘‘microtwins”), bounded by near-parallel twin boundaries. They do not thicken very much as strain proceeds; growth of the twinned volume fraction of the crystal in order to comply with Eq. (2) mainly takes place by continuous nucleation of new deformation microtwins. Microtwin lamellae are not homogeneously distributed; they group in bundles where the twin bands are approximately periodically spaced, which suggests an autocatalytic effect of nucleation. The energetics of nucleation and growth of such twins and twin bands has been recently discussed by Fisher and co-workers [7,22,23]. The accommodation of a macroscopic strain gradient v in a volume element internally deforming by dislocation glide is resolved by a combination of slip gradients of the active crystallographic slip systems a in the volume and its surroundings, leading, ignoring any possible dynamic or static recovery reactions, to the storage of a GND density [24,25] ðqg ÞGND ¼
X a
ðqg Þa ¼
1X jna rCa j: b a
ð4Þ
where n is the unit vector normal to a slip plane. A platelike deformation microtwin of thickness t can be geometrically seen as a ‘‘pseudo-dislocation” holding an effective Burgers vector with the twinning shear direction and sense and with a modulus beff ¼ tCT :
ð5Þ
The pseudo-dislocation line is constituted by the twin edge lying on the twinning habit plane of unit normal n. Consequently, the accommodation of a macroscopic strain gradient v by storage of GNT requires the storage of a volume density of twinning edge lines, TE, in the volume element X CT X ðqTE Þg ¼ ð6Þ qTE ¼ ng rðfv Þg ; b eff g g where g are now the active twinning systems resolving the plastic strain in the considered element and its surroundings. At the twin edge of a GNT, the intensity of the stress fields would be unrealistically high (corresponding to a dislocation line of Burgers vector beff) if no relaxation by dislocation emission and multiplication takes place. Indeed, the presence of accommodation dislocation arrays has been always observed around the edges of mechanically induced twins [5]. Consequently, the presence of a density of twinning edge lines in a volume of a crystal implies the content of a density of GNDs linked to the geometrically necessary twinning given on average – ignoring a geometrical factor of the order of unity – by beff CT X ¼ ð7Þ ðqg ÞGNT ffi qTE ng rðfv Þg : b b g
J. Gil Sevillano / Scripta Materialia 59 (2008) 135–138
In principle, stress relaxation at the GNT edge could also take place by secondary twinning or by any kinematically admissible combination of slip and twinning. As a first approximation we take Eq. (7) as an effective GND density for estimation of GNT strengthening effects whatever their origin (back-stresses or forest-type strengthening) and whatever the actual relaxed structure that grows around the twin edges. The approximate GNT strengthening contribution can then easily be incorporated into strain gradient-dependent crystal plasticity calculations. The total strengthening from dislocation density origin will be induced by q ¼ qs þ ðqg ÞGND þ ðqg ÞGNT :
ð8Þ
A comparison of Eqs. (4) and (7) leads to the conclusion that the size effects on plastic strength arising from strain gradients will be essentially similar for materials deforming either by slip or twinning or by slip and twinning combinations, at least for the strengthening effect on the CRSS for dislocation-mediated slip. The quantitative effect of a previous dislocation density on the CRSS for twinning is not well documented [5]. Probably both nucleation and growth are affected. Some quantitative experiments that introduced a well-controlled dislocation density, i.e. by deformation at high temperature, above the twinning activity region, followed by testing in the twinning region have been published [26,27], but not, to our knowledge, for metallic alloys. The aforementioned experiments, on sapphire, are compatible with a proportionality between the twinning CRSS and the square root of the dislocation density, i.e. a dislocation hardening relationship similar to that shown by the CRSS for dislocation slip, but with a smaller proportionality factor. The proportionality was explained in Refs. [26,27] by an argument based on the energy penalty imposed on the advancing twin plate by its engulfing of forest dislocation lines. An alternative argument based on the crossing of forest dislocations with the help of the effective line tension TTE of the twin edges leads to the same functionality and predicts a strengthening factor for the twin edges smaller than for the dislocation–dislocation interaction Gb2eff : ð9Þ 2 This upper bound corresponds to the action of the twin edge as an individual entity (a single pseudo-dislocation of Burgers vector beff). Obviously, a lower bound for TT would be given by the line tension of an individual emissary dislocation of the twin edge. Fragmentation of twins advancing across regions of high dislocation density has been observed [5]. Tentatively, we can thus assume, as a first approximation pffiffiffi ð10Þ sT ¼ aT Gb q: T TE 6
A formally similar relationship is obtained if the twin edges are modelled as disclination dipole loops [28,29]. Experiments are needed to confirm the validity of Eq. (10) and to obtain values of the factor aT. Constrained deformation experiments of Chin et al. [16] with fcc single crystals deforming by simultaneous slip and twinning suggest that the critical stresses for both mechanisms
137
undergo a comparable strain-hardening as a function of the total slip plus twinning-resolved shear strain. Several detailed physical models for evaluating the CRSS for twinning have been developed (e.g. [30]). Only a very simple approximation is developed here. As described above, the CRSS for twinning is composed of a term that depends both on the effective grain size (i.e. on the mean free path for twin propagation) and on the effective dislocation density plus a term dependent on the stacking-fault energy of the material. In the absence of any imposed macroscopic strain gradients (e.g. in a tensile tests), both will significantly change as deformation proceeds when twinning and slip coexist. When twinning predominates, the mean intertwining true spacing s can be easily related to the mean twinning plate thickness t through the stereological relationship 1 1 : ð11Þ s¼t fv Ignoring orientation changes, on account of Eq. (2) and assuming M T 3 (for a macroscopic tensile deformation) CT CT 1 ; e6 : ð12Þ st 3e 3 The subdivision of the original microstructure is so rapid that for fcc materials s t for e 0.1. On account of observed microtwinning thickness values, t 100 nm, this implies that the material would be fully nanostructured after such a small strain. Consequently, on account of the Hall–Petch relationships empirically observed, it is expected that the mean free path influencing the CRSS either for twinning or slip will be controlled by the inter-twinning spacing after very small twinning-resolved plastic strains even for very fine grain polycrystals. Such a dynamic Hall–Petch effect can explain by itself the extraordinary strain-hardening rate observed in materials where deformation twinning prevails (‘‘stage B” [8]). For small strains, e 1 1 3e : ð13Þ s e1 tCT The mere extrusion through the matrix intertwining channels of either the emitted dislocations for new twins or the slip dislocations in case of slip activity simultaneously with twinning would need a CRSS approximately given by the Orowan stress, i.e. the stress for bowing the dislocation lines to a circular shape of diameter s ðsc ÞOrowan ffi
Gb 3Gbe : s tCT
ð14Þ
Assuming again for tension an orientation factor r/ sc 3 and a microtwin thickness t 100 nm, a strainhardening rate in tension is predicted of the right order of magnitude hOrowan
9Gb ffi 0:03G: tCT
ð15Þ
The fragmentation of the original crystal volume implies the arrest of propagating microtwins by other twin boundaries, i.e. the presence of an increasing twin edge
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J. Gil Sevillano / Scripta Materialia 59 (2008) 135–138
density scaling with 1/s2. This presence contributes additionally to the flow stress either through a back-stress or through an additional dislocation density relaxing the twin edge stresses, as described above for the twin edges arising from the geometrical necessity of meso- or macro-strain gradient accommodation. We can estimate an upper bound for such a contribution as an additional effective GND term. For small strains, on account of Eq. (13) and assuming again an orientation factor of 3 ðqg ÞTEarrest 6
beff 1 tCT ð3eÞ2 ¼ ffi : b s2 bs2 btCT
ð16Þ
This term represents a linear strain-hardening of the same order of magnitude as observed in tensile testing 9aT Gb hTEarrest 6 pffiffiffiffiffiffiffiffiffiffi ffi 0:5aT G: btCT
ð17Þ
A comparison of Eq. (7) with Eqs. (14) and (16) shows that for the strengthening effect of GNT to dominate over the effect of fragmentation of the crystals by microtwinning, the resolved shear strain gradients need to be very high; GNT size effects originating from macro- or mesoscopic gradients will only be important for very small sizes (e.g. very small imprint sizes for the indentation size effect (ISE)) although this also happens with size effects induced by GND-resolved gradients in materials with normal initial contents of dislocation densities. Finally, note that the fragmentation rate and the twinning contributions to strain-hardening will be smaller than those predicted by Eqs. (15) and (17) if dislocation-mediated slip participates in the deformation together with twinning. On account of the similarity of Eqs. (4) and (7), the gradient-dependent size effects on strength coming from GNT or from GND will have the same functional form. Measurements of the ISE in TWIP steels have been performed that show the same proportionality between hardness to the square and the inverse of indentation depth found in crystals deforming exclusively by slip, also with the same deviation from linearity at low depths [31]. A particular example of GNT has been recently presented by Appel et al. [32] under the name of ‘‘precipitation twins” (twins nucleated and grown in a TiAl matrix for absorbing the misfit of precipitates). Finally, it is obvious that a geometrically necessary martensitic transformation (GNMT) will occur in materials where such strain-induced phase transformation contributes to the macroscopic strain. In conclusion: A first attempt to analyse the role of GNTs in accommodating macroscopic or mesoscopic strain gradients highlights the importance of the volume density of twin edges in the size effects expected from GNT. The effect of GNT can be incorporated into the analysis of gradient-dependent crystal plasticity in the same way as the effect of GNDs. However, before any quantitative application it is first necessary to experimentally determine the quantitative dependence of the CRSS for mechanical twinning on the dislocation and twin edge densities.
This work is part of the projects UET 120905 (Basque Government, SAIOTEK 2006), NANOTRON (Basque Government and Diputacio´n Foral de Gipuzkoa) and P6/24, Belgian State (Belgian Science Policy, Interuniversity Attraction Poles IAP Programme). [1] J.F. Nye, Acta Mater. 1 (1953) 153. [2] M.F. Ashby, Philos. Mag. 21 (1970) 399. [3] A. Needleman, J. Gil Sevillano, Scripta Mater. 48 (2003) 109. [4] N. Narita, J.I. Takamura, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, vol. 9, Elsevier, Amsterdam, 1992, pp. 135–189. [5] J.W. Christian, S. Mahajan, Prog. Mater. Sci. 39 (1995) 1. [6] M.A. Meyers, O. Vo¨ringer, V.A. Lubarda, Acta Mater. 49 (2001) 4025. [7] F.D. Fischer, T. Schaden, F. Appel, H. Clemens, Eur. J. Mech. A/Solids 22 (2003) 709. [8] S. Asgari, E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Metall. Mater. Trans. A 28A (1997) 1781. [9] O. Gra¨ssel, G. Frommeyer, Mater. Sci. Technol. 14 (1998) 1213. [10] O. Grassel, L. Kru¨ger, G. Frommeyer, L.W. Meyer, Int. J. Plasticity 16 (2000) 1391. [11] A. Rohatgi, K.S. Vecchio, G.T. Gray III, Metall. Trans. A 32A (2001) 135–145. [12] E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Metall. Mater. Trans. A 30A (1999) 1223. [13] M.R. Barnett, Mater. Sci. Eng. A 464 (2007) 1. [14] M.R. Barnett, Mater. Sci. Eng. A 464 (2007) 8. [15] M.S. Szczerba, T. Bajor, T. Tokarski, Philos. Mag. 21 (2004) 481. [16] G.Y. Chin, W.F. Hosford, D.R. Mendorf, Proc. Roy. Soc. A 309 (1969) 433. [17] Z. Chen, H. Cai, S. Li, X. Zhang, F. Wang, C. Tan, Mater. Sci. Eng. A 464 (2007) 101. [18] O. Vo¨hringer, in: Proceedings of the International Conference on the Strength of Metals and Alloys (ICSMA-2), ASM, Metals Park, OH, 1970, p. 294. [19] V. Yamakov, D. Wolf, S.R. Phillpot, A.K. Mukherjee, H. Gleiter, Nature Mater. 1 (2002) 45. [20] Y.T. Zhu, X.Z. Liao, S.G. Srinivasan, Y.H. Zhao, M.I. Baskes, Appl. Phys. Lett. 85 (2004) 5049. [21] X. Wu, Y.T. Zhu, M.W. Chen, E. Ma, Scripta Mater. 54 (2006) 1685. [22] F.D. Fisher, F. Appel, H. Clemens, Acta Mater. 51 (2003) 1249. [23] H. Petryk, F.D.F.D. Fisher, W. Marketz, H. Clemens, F. Appel, Metall. Mater. Trans. A 34A (2003) 2827. [24] J.M. Martı´nez-Esnaola, M. Montagnat, P. Duval, J. Gil Sevillano, Scripta Mater. 50 (2004) 273. [25] J. Gil Sevillano, D. Gonza´lez, J.M. Martı´nez Esnaola, Mater. Sci. Forum 550 (2007) 75. [26] J. Castaing, A. Mun˜oz, A. Dominguez Rodriguez, Philos. Mag. A 82 (2002) 1419. [27] A. Mun˜oz, J. Castaing, A. Domı´nguez Rodriguez, Rev. Metal. Madrid 37 (2001) 285. [28] A.E. Romanov, V.I. Vladimirov, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, Vol. 9, Elsevier, Amsterdam, 1992, p. 191. [29] P. Mu¨llner, Z. Metallkde. 97 (2006) 205. [30] S. Allain, J.-P. Chateau, O. Bouaziz, Mater. Sci. Eng. A 387–389 (2004) 143. [31] J. Gil Sevillano, J. Molina, J. Alkorta, in preparation. [32] F. Appel, F.D. Fisher, H. Clemens, Acta Mater. 55 (2007) 4915.
Scripta Materialia 59 (2008) 135–138 www.elsevier.com/locate/scriptamat
Geometrically necessary twins and their associated size effects Javier Gil Sevillano CEIT and TECNUN (University of Navarra), M. de Lardiza´bal, 15, 20018 San Sebastia´n, Spain Received 8 October 2007; revised 12 February 2008; accepted 19 February 2008 Available online 14 March 2008
In metallic alloys or mineral crystals where twinning is a prominent deformation micromechanism, geometrically necessary twins (GNTs) have to be stored in order to accommodate plastic strain gradients, complementing or substituting the geometrically necessary dislocations (GNDs) that fulfil this function in materials deforming exclusively by crystallographic slip. The presence of GNTs will contribute to strain gradient hardening, just as GNDs contribute to it. Consequently, GNTs will be responsible for size effects. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Mechanical twinning; Plastic strain gradients; Size effects
The subdivision of strain-induced dislocation density into statistically stored dislocation (SSD) and geometrically necessary dislocation (GND) densities has gained widespread acceptance [1–3]. The convention for this distinction is to ascribe to the GND density the dislocations stored in order to accommodate the macroscopic or mesoscopic plastic strain gradients either externally imposed (as in torsion or indentation tests) or arising from the heterogeneity of the microstructure (as in polycrystals or in two-phase materials). The SSD density is the density of dislocations that would be stored during plastic deformation of the same material in the absence of those gradients. Of course, the distinction is rather artificial in some cases: any dislocation is geometrically necessary in its place; however, in most cases of interest, the distinction is clear and very practical. In low stacking-fault energy (SFE) metallic alloys or mineral materials under deformation conditions of relatively high strain rate and low temperature, deformation twinning is a prominent plastic micromechanism that competes with and complements dislocation-mediated slip [4–7]. In such materials, strain gradients will mainly be accommodated by the storage of an extra density of deformation twins instead of mainly by GNDs. We can thus speak of geometrically necessary twins (GNTs) in the same way that we speak of GNDs. On this basis, we can now consider the side effects of the presence of such GNTs on strength. Indeed, we can expect size effects associated with GNTs because of the strengthening effect of their presence. E-mail: [email protected]
Twin boundaries are special high-angle boundaries that represent strong barriers to the propagation of other non-coplanar deformation twins or gliding dislocations and, like other high-angle boundaries, they contribute to the Hall–Petch effect in polycrystals. In addition to this effect due to the confinement of plasticity, twin propagation and growth often involves an associated density of dislocations contributing to the hardening of the crystal matrix. The first effect seems to be the most important one, explaining the stage of extraordinary work-hardening rate (named stage B by Asgari et al. [8]), 0.02–0.03G, where G is the shear elastic modulus, exhibited up to medium strains (e 6 1) by materials that deform mainly by twinning, like twinning-induced plasticity (TWIP) steels and other low SFE face-centred cubic (fcc) or hexagonal close-packed (hcp) alloys [9–14]. The contribution of deformation twinning to the macroscopic strain e is very different from the contribution of crystallographic slip. For the latter dC ; ð1Þ M is an average orientation factor for the active where M slip systems and C is the total amount of crystallographic slip. The slip amount is, in principle, assuming the availability of mobile dislocations, unlimited. For twinning de ¼
de ¼
CT dfv ; MT
ð2Þ
T is where CT is the finite shear undergone by the twin, M an average orientation factor for the active twinning
1359-6462/$ - see front matter Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2008.02.052
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J. Gil Sevillano / Scripta Materialia 59 (2008) 135–138
systems and fv is the twinned volume fraction in the volume element considered. On account of the polar character of twinning, the total amount of strain achievable by twinning when a monotonic strain path is followed would be limited to CT =M T unless secondary twinning could be activated inside the twinned volumes. For instance,pffiffiffifor fcc metals and h1 1 2i{1 1 1} twinning, CT ¼ 2=2. It is generally accepted that deformation twinning obeys a critical resolved shear stress (CRSS) law similar to the Schmid law for crystallographic slip (see Ref. [5] for a review and Ref. [15] for a recent experimental study). Under a macroscopic stress r, twinning will be observed when its CRSS is reached for ð3Þ r ¼ M T sT : However, the experimental evidence for a fixed value of the twinning CRSS sT is not always as clear as in the case of crystallographic slip, possibly because twinning is controlled either by nucleation or growth in the same material depending on the structure or pre-strain (and, because of the heterogeneity of the real structures, depending on the spatial location inside the same sample, and of the internal stresses associated with other mechanical twins and dislocation groups). Often twinning is observed to occur simultaneously with crystallographic slip. Under conditions of constrained deformation (i.e. approximately for polycrystalline deformation) of fcc metals, twinning will coexist with slip when the ratio pffiffiffi twinning and pffiffiffiof the CRSS for slip is in the range 1= 3 6 sT =sc 6 2= 3 [16,17], obviously under the assumption of availability of twinning and slip sources. Constrained deformation exclusively pffiffiffi by twinning of fcc metals is possible for sT =sc < 1= 3. The CRSS for twinning increases for increasing dislocation density or for decreasing grain or mean free path size. As twinning occurs by the movement of twinning dislocations, any microstructural feature representing an obstacle to dislocation propagation makes the twinning CRSS increase. The twinning CRSS versus mean free path relationship is of the Hall–Petch type [13,14], although some authors favour an exponent other than 1/2 (e.g. Vo¨hringer [18] and El-Danaf et al. [12] have found exponents of near 1). It should be noted that the Hall–Petch slope for the twinning CRSS is consistently larger than the corresponding slope for the crystallographic slip CRSS [6]. For virgin polycrystals with large grain size the CRSS for twinning corresponds to twin nucleation, which in pure metals or solid solutions is often followed by unstable propagation (serrated stress–strain curves and audible clicks). For small enough grain size polycrystals or after sufficient strain-induced structural fragmentation and hardening deformation, twinning ceases. This apparently implies that a lower limit to the mean free path for the occurrence of twinning exists, i.e. a size-induced transition in deformation mechanism would take place below some threshold value of the mean free path. For free nanocrystals, however, and for nanocrystalline polycrystals, molecular dynamics simulations predict twinning activity that is confirmed by experimental observations [19–21]. Inside an undeformed grain in a polycrystal, the first deformation twins start from a grain boundary and
cross the whole grain section until they are arrested by impingement with surrounding grain boundaries or with other twin boundaries. In a pre-deformed dislocated structure, twins can be nucleated from dislocation junctions and can be arrested by dislocation arrays. Deformation twins are very thin lenticular (hcp) or plate-like (fcc) reoriented layers, often of nanometric thickness (‘‘microtwins”), bounded by near-parallel twin boundaries. They do not thicken very much as strain proceeds; growth of the twinned volume fraction of the crystal in order to comply with Eq. (2) mainly takes place by continuous nucleation of new deformation microtwins. Microtwin lamellae are not homogeneously distributed; they group in bundles where the twin bands are approximately periodically spaced, which suggests an autocatalytic effect of nucleation. The energetics of nucleation and growth of such twins and twin bands has been recently discussed by Fisher and co-workers [7,22,23]. The accommodation of a macroscopic strain gradient v in a volume element internally deforming by dislocation glide is resolved by a combination of slip gradients of the active crystallographic slip systems a in the volume and its surroundings, leading, ignoring any possible dynamic or static recovery reactions, to the storage of a GND density [24,25] ðqg ÞGND ¼
X a
ðqg Þa ¼
1X jna rCa j: b a
ð4Þ
where n is the unit vector normal to a slip plane. A platelike deformation microtwin of thickness t can be geometrically seen as a ‘‘pseudo-dislocation” holding an effective Burgers vector with the twinning shear direction and sense and with a modulus beff ¼ tCT :
ð5Þ
The pseudo-dislocation line is constituted by the twin edge lying on the twinning habit plane of unit normal n. Consequently, the accommodation of a macroscopic strain gradient v by storage of GNT requires the storage of a volume density of twinning edge lines, TE, in the volume element X CT X ðqTE Þg ¼ ð6Þ qTE ¼ ng rðfv Þg ; b eff g g where g are now the active twinning systems resolving the plastic strain in the considered element and its surroundings. At the twin edge of a GNT, the intensity of the stress fields would be unrealistically high (corresponding to a dislocation line of Burgers vector beff) if no relaxation by dislocation emission and multiplication takes place. Indeed, the presence of accommodation dislocation arrays has been always observed around the edges of mechanically induced twins [5]. Consequently, the presence of a density of twinning edge lines in a volume of a crystal implies the content of a density of GNDs linked to the geometrically necessary twinning given on average – ignoring a geometrical factor of the order of unity – by beff CT X ¼ ð7Þ ðqg ÞGNT ffi qTE ng rðfv Þg : b b g
J. Gil Sevillano / Scripta Materialia 59 (2008) 135–138
In principle, stress relaxation at the GNT edge could also take place by secondary twinning or by any kinematically admissible combination of slip and twinning. As a first approximation we take Eq. (7) as an effective GND density for estimation of GNT strengthening effects whatever their origin (back-stresses or forest-type strengthening) and whatever the actual relaxed structure that grows around the twin edges. The approximate GNT strengthening contribution can then easily be incorporated into strain gradient-dependent crystal plasticity calculations. The total strengthening from dislocation density origin will be induced by q ¼ qs þ ðqg ÞGND þ ðqg ÞGNT :
ð8Þ
A comparison of Eqs. (4) and (7) leads to the conclusion that the size effects on plastic strength arising from strain gradients will be essentially similar for materials deforming either by slip or twinning or by slip and twinning combinations, at least for the strengthening effect on the CRSS for dislocation-mediated slip. The quantitative effect of a previous dislocation density on the CRSS for twinning is not well documented [5]. Probably both nucleation and growth are affected. Some quantitative experiments that introduced a well-controlled dislocation density, i.e. by deformation at high temperature, above the twinning activity region, followed by testing in the twinning region have been published [26,27], but not, to our knowledge, for metallic alloys. The aforementioned experiments, on sapphire, are compatible with a proportionality between the twinning CRSS and the square root of the dislocation density, i.e. a dislocation hardening relationship similar to that shown by the CRSS for dislocation slip, but with a smaller proportionality factor. The proportionality was explained in Refs. [26,27] by an argument based on the energy penalty imposed on the advancing twin plate by its engulfing of forest dislocation lines. An alternative argument based on the crossing of forest dislocations with the help of the effective line tension TTE of the twin edges leads to the same functionality and predicts a strengthening factor for the twin edges smaller than for the dislocation–dislocation interaction Gb2eff : ð9Þ 2 This upper bound corresponds to the action of the twin edge as an individual entity (a single pseudo-dislocation of Burgers vector beff). Obviously, a lower bound for TT would be given by the line tension of an individual emissary dislocation of the twin edge. Fragmentation of twins advancing across regions of high dislocation density has been observed [5]. Tentatively, we can thus assume, as a first approximation pffiffiffi ð10Þ sT ¼ aT Gb q: T TE 6
A formally similar relationship is obtained if the twin edges are modelled as disclination dipole loops [28,29]. Experiments are needed to confirm the validity of Eq. (10) and to obtain values of the factor aT. Constrained deformation experiments of Chin et al. [16] with fcc single crystals deforming by simultaneous slip and twinning suggest that the critical stresses for both mechanisms
137
undergo a comparable strain-hardening as a function of the total slip plus twinning-resolved shear strain. Several detailed physical models for evaluating the CRSS for twinning have been developed (e.g. [30]). Only a very simple approximation is developed here. As described above, the CRSS for twinning is composed of a term that depends both on the effective grain size (i.e. on the mean free path for twin propagation) and on the effective dislocation density plus a term dependent on the stacking-fault energy of the material. In the absence of any imposed macroscopic strain gradients (e.g. in a tensile tests), both will significantly change as deformation proceeds when twinning and slip coexist. When twinning predominates, the mean intertwining true spacing s can be easily related to the mean twinning plate thickness t through the stereological relationship 1 1 : ð11Þ s¼t fv Ignoring orientation changes, on account of Eq. (2) and assuming M T 3 (for a macroscopic tensile deformation) CT CT 1 ; e6 : ð12Þ st 3e 3 The subdivision of the original microstructure is so rapid that for fcc materials s t for e 0.1. On account of observed microtwinning thickness values, t 100 nm, this implies that the material would be fully nanostructured after such a small strain. Consequently, on account of the Hall–Petch relationships empirically observed, it is expected that the mean free path influencing the CRSS either for twinning or slip will be controlled by the inter-twinning spacing after very small twinning-resolved plastic strains even for very fine grain polycrystals. Such a dynamic Hall–Petch effect can explain by itself the extraordinary strain-hardening rate observed in materials where deformation twinning prevails (‘‘stage B” [8]). For small strains, e 1 1 3e : ð13Þ s e1 tCT The mere extrusion through the matrix intertwining channels of either the emitted dislocations for new twins or the slip dislocations in case of slip activity simultaneously with twinning would need a CRSS approximately given by the Orowan stress, i.e. the stress for bowing the dislocation lines to a circular shape of diameter s ðsc ÞOrowan ffi
Gb 3Gbe : s tCT
ð14Þ
Assuming again for tension an orientation factor r/ sc 3 and a microtwin thickness t 100 nm, a strainhardening rate in tension is predicted of the right order of magnitude hOrowan
9Gb ffi 0:03G: tCT
ð15Þ
The fragmentation of the original crystal volume implies the arrest of propagating microtwins by other twin boundaries, i.e. the presence of an increasing twin edge
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density scaling with 1/s2. This presence contributes additionally to the flow stress either through a back-stress or through an additional dislocation density relaxing the twin edge stresses, as described above for the twin edges arising from the geometrical necessity of meso- or macro-strain gradient accommodation. We can estimate an upper bound for such a contribution as an additional effective GND term. For small strains, on account of Eq. (13) and assuming again an orientation factor of 3 ðqg ÞTEarrest 6
beff 1 tCT ð3eÞ2 ¼ ffi : b s2 bs2 btCT
ð16Þ
This term represents a linear strain-hardening of the same order of magnitude as observed in tensile testing 9aT Gb hTEarrest 6 pffiffiffiffiffiffiffiffiffiffi ffi 0:5aT G: btCT
ð17Þ
A comparison of Eq. (7) with Eqs. (14) and (16) shows that for the strengthening effect of GNT to dominate over the effect of fragmentation of the crystals by microtwinning, the resolved shear strain gradients need to be very high; GNT size effects originating from macro- or mesoscopic gradients will only be important for very small sizes (e.g. very small imprint sizes for the indentation size effect (ISE)) although this also happens with size effects induced by GND-resolved gradients in materials with normal initial contents of dislocation densities. Finally, note that the fragmentation rate and the twinning contributions to strain-hardening will be smaller than those predicted by Eqs. (15) and (17) if dislocation-mediated slip participates in the deformation together with twinning. On account of the similarity of Eqs. (4) and (7), the gradient-dependent size effects on strength coming from GNT or from GND will have the same functional form. Measurements of the ISE in TWIP steels have been performed that show the same proportionality between hardness to the square and the inverse of indentation depth found in crystals deforming exclusively by slip, also with the same deviation from linearity at low depths [31]. A particular example of GNT has been recently presented by Appel et al. [32] under the name of ‘‘precipitation twins” (twins nucleated and grown in a TiAl matrix for absorbing the misfit of precipitates). Finally, it is obvious that a geometrically necessary martensitic transformation (GNMT) will occur in materials where such strain-induced phase transformation contributes to the macroscopic strain. In conclusion: A first attempt to analyse the role of GNTs in accommodating macroscopic or mesoscopic strain gradients highlights the importance of the volume density of twin edges in the size effects expected from GNT. The effect of GNT can be incorporated into the analysis of gradient-dependent crystal plasticity in the same way as the effect of GNDs. However, before any quantitative application it is first necessary to experimentally determine the quantitative dependence of the CRSS for mechanical twinning on the dislocation and twin edge densities.
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