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Stage IV work-hardening related to disorientations in dislocation structures
Materials Science and Engineering A 387–389 (2004) 257–261
Stage IV work-hardening related to disorientations in dislocation structures W. Pantleon∗ Center for Fundamental Research: Metal Structures in Four Dimensions, Materials Research Department, Risoe National Laboratory, Frederiksborgvej 399, 4000 Roskilde, Denmark Received 9 September 2003; received in revised form 10 November 2003
Abstract The effect of deformation-induced disorientations on the work-hardening of metals is modelled based on dislocation dynamics. Essentially, Kocks’ dislocation model describing stage III hardening is extended to stage IV by incorporation of excess dislocations related to the disorientations. Disorientations evolving from purely statistical reasons — leading to a square root dependence of the average disorientation angle on strain — affect the initial work-hardening rate (and the saturation stress) of stage III only slightly. On the other hand, deterministic contributions to the development of disorientations, as differences in the activated slip systems across boundaries, cause a linear increase of the flow stress at large strains. Such a constant work-hardening rate is characteristic for stage IV. © 2004 Elsevier B.V. All rights reserved. Keywords: Work-hardening; Stage IV; Disorientation; Dislocation; Modelling; Aluminium
1. Introduction In the stress–strain curve of plastically deforming metals different stages of work-hardening are clearly distinguished. During stage III the work-hardening rate is decreasing linearly with stress. This decrease is not continued in stage IV and the work-hardening rate often remains constant (cf. [1]). The work-hardening behavior in stage III has been modelled successfully by Kocks [2] based on a single microstructural variable — the total dislocation density ρ. The main assumption is a flow stress (of a material with shear modulus µ and Burgers vector b) √ τ = αµb ρ (1) entirely controlled by the total dislocation density ρ and a strain rate- and temperature-dependent interaction parameter α = α(γ, ˙ T). In combination with an appropriate evolution law for the dislocation density (see Section 2.1), the model describes flow curves covering the essential features of stage III, despite the fact that details of the microstructure as formation of dislocation boundaries and different types of boundaries are ignored. There have been several attempts ∗
Tel.: +45 4677 5791; fax: +45 4677 5758. E-mail address: [email protected] (W. Pantleon).
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.11.088
(cf. [1]) to modify the model by distinguishing between different types of dislocations (e.g. as dislocations in boundaries and dislocations between the boundaries or edge and screw dislocations [3]). These models are able to cover features of stage IV, but the involved quantities are difficult to access experimentally. Moreover, the formation of orientation differences in dislocation structures, which occurs during deformation independent of the deformation mode (e.g. [4]), is ignored. Argon and Haasen [5] suggested a link between the occurrence of stage IV and the formation of disorientations in the dislocation structure. In their model, disorientations across boundaries cause internal stresses and the work-hardening rate increases linearly with strain — even for a stochastic accumulation of disorientations and in conflict with experimental observations, e.g., on aluminium [6]. The effect of forming disorientations on the dislocation density due to excess dislocations and on the flow stress is investigated here. It will be shown that concise book-keeping of dislocations of different signs of the Burgers vector allows attributing the associated disorientations to a constant work-hardening rate as observed in stage IV. It is emphasized that dislocations will be categorized only according to the sign of their Burgers vector and no other distinction between different types of dislocations is made, in particular, no assumption about the dislocation character is required.
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W. Pantleon / Materials Science and Engineering A 387–389 (2004) 257–261
2. Modelling 2.1. Original model of Kocks The evolution of the total dislocation density ρ can be described in terms of dislocation accumulation and their dynamic recovery [2]: during plastic deformation, the dislocation density increases due to trapping of mobile dislocations by already existing dislocations. The mean free path √ λ¯ = β/ ρ of mobile dislocations is governed by the total dislocation density with a proportionality coefficient β accounting for the details of the dislocation structure. Dynamic recovery occurs when a moving dislocation passes a dislocation of opposite sign and both annihilate. Such a dislocation pair annihilation may trigger even further annihilation events (by relaxation), removing a total dislocation length lr . Here, a slightly different interpretation of the recovery length lr = 2y in terms of the cross-section for annihilation events [7] is preferred: dislocations of opposite sign passing in a distance less than the capturing length y annihilate each other. By considering an effective y, secondary annihilation events are accounted for. Both processes are combined in the evolution of the dislocation density with plastic strain γ: √ ρ 2y dρ 2y 1 − ρ. − ρ= = (2) ¯ βb dγ b b λb The resulting work-hardening rate dτ αµb dρ τ Θ≡ = √ = ΘIII,0 1 − dγ 2 ρ dγ τIII,∞
(3)
decreases linearly with stress from an initial work-hardening K K = αµ/2β until a saturation stress τIII,∞ = rate ΘIII,0 K αµb/2yβ = ΘIII,0 b/y is reached (cf. the full lines in Fig. 3). The linear decrease of the work-hardening rate with stress is characteristic for stage III and leads to the well-established Palm-Voce behavior (cf. [2]) of the flow stress with strain: ΘIII,0 τ = τIII,∞ − (τIII,∞ − τ0 )exp − γ . (4) τIII,∞ Describing storage and annihilation of dislocations, the proposed balance equation for the total dislocation density does not differentiate between dislocations of opposite sign (⊥ and ) and neglects the possible existence of an excess of dislocations of one sign. The necessary modifications for incorporating an excess dislocation density ρ = ρ⊥ − ρ (and the related disorientation angle θ across boundaries) are obtained in a straightforward manner by a thorough analysis of the underlying dislocation dynamics.
Fig. 1. Possible interactions of mobile dislocations with a boundary of dense immobilized dislocations. The dislocation (⊥) with positive sign moving to the right may be trapped or pass the boundary, whereas the negative dislocation ( ) moving to the left passes a positive dislocation within the annihilation distance and both dislocations annihilate leading to a decrease in the dislocation densities of both signs.
dislocation densities can be formulated for dislocations of opposite signs (cf. [8]): 1 dρ⊥ = − 2yρ j → − 2yρ⊥ j ← , (5) dt λ¯ 1 dρ = − 2yρ⊥ j ← − 2yρ j → . (6) dt λ¯ The density of immobile dislocations with positive Burgers vectors (⊥) is increased by trapping of positive mobile dislocations from the flux j → to the right (see Fig. 1). Two processes lead to a loss of positive dislocations: (i) part of the mobile dislocations are stopped by annihilation with an existing immobile negative dislocations ( ), if they pass them within the annihilation distance y and (ii) already immobilized positive dislocations can annihilate with negative mobile dislocations moving to the left. The balance equations for the dislocation densities of both signs can be reformulated by introducing the total dislocation flux j = j → + j ← = γ/b ˙ (related to the strain rate γ) ˙ and the bias of fluxes j = j → − j ← as balance equations for the total dislocation density ρ = ρ⊥ + ρ and the excess dislocation density ρ = ρ⊥ − ρ : dρ 1 − 2yρ j + 2yρj, (7) = dt λ¯
2.2. Incorporation of excess dislocations
dρ 1 = j. dt λ¯
Mobile dislocations carrying the plastic flow move along a certain free path λ before they are stopped. They are either trapped in a dislocation boundary or annihilate with previously stored dislocations. The corresponding changes in the
The first two terms of the balance equation (7) for the total dislocation density are identical to the corresponding ones in the balance equation (2) of Kocks’ model. The additional third term leads to an increase in the total dislocation density without requiring additional storage mechanisms. It is
(8)
W. Pantleon / Materials Science and Engineering A 387–389 (2004) 257–261
a direct consequence of an excess of dislocations of one sign (or equivalently of an existing disorientation). The extra work-hardening is caused by an inhibited annihilation stemming from the fact that an existing excess of dislocations of one sign of the Burgers vector affects the number of possible annihilation events. For instance, a persisting increased flux of positive dislocations leads to an excess of positive dislocations, and the relative frequency of finding a negative dislocation as partner for an annihilation event is reduced. √ The link between the mean free path λ¯ = β/ ρ of a mobile dislocation and the total dislocation density can be established for different dislocation arrangements. For a dislo√ cation cell structure, the cell size d = K/ ρ follows a scaling relation [9] and mobile dislocations passing cell walls or dislocation boundaries are immobilized with a constant immobilization probability P = d/λ¯ resulting in β = K/P. The excess dislocation density is intimately related to the disorientation angle θ = bdρ across dislocation boundaries (in mutual distance d). Therefore, the disorientation angle becomes the second relevant microstructural variable. Its evolution equation dθ = Pbj dt
(9)
does not contain any information on the total dislocation density and is treated independently in Section 2.3. Combining Eqs. (7) and (9) a single evolution equation for the total dislocation density ρ(γ) is obtained: √ 2y ρ dθ 2 dρ 2y P √ ρ− ρ+ = . (10) dγ Kb b 2PKb2 dγ 2.3. Disorientations The development of disorientation angles during plastic deformation has been modelled in detail [10,11] for different types of dislocation boundaries based on Eq. (9). In general, two contributions to the bias of the dislocation fluxes must be distinguished: dθ = Pbjfluc + bΓj. dt
259
cation boundaries (IDBs) are assumed to arise from mutual trapping of dislocations and excess dislocations accumulate only in a stochastic manner, whereas geometrically necessary boundaries (GNBs) are per definitionem associated with an activation imbalance between both sides of the boundary causing an additional deterministic contribution. In both cases, a Gaussian distribution of the disorientation angles results with a vanishing mean value θ = 0 and Pb 2 γ 2. (12) θ 2 = ∗ γ + σimb d A correlation length d ∗ of the disorientation angle along a dislocation boundary is introduced allowing for changes of the disorientation along a single boundary. It replaces the boundary spacing in earlier versions of the model and is assumed to remain constant. Disorientation angles across dislocation boundaries are in general caused by two dislocation sets in the boundary (cf. [13]) leading to a Rayleigh distribution of the disorientation √ angles θ with an average angle given by θ¯ = π/2 θ 2 . This has to be taken into account for any comparison of the predictions of Eq. (12) with experimental data. The measured average disorientation angles of cold-rolled aluminium [4] are shown separately for IDBs and GNBs in Fig. 2. The disorientation angles of both types of boundaries are well described by Eq. (12). (For simplicity the slip γ in a single slip system is equated to the true strain assuming effectively two slip systems with Schmid factor 1/2.) For IDBs, for which σimb vanishes per definitionem, a square of the average disorientation angle √ rootdependence √ θ¯ = π/2 Pb/d ∗ γ on strain is obtained. With Pb/d ∗ = 6 × 10−4 a good description of the experimental data is accomplished. A good correspondence is achieved for GNBs as well, using Pb/d ∗ = 6 × 10−3 and σimb = 6 × 10−2 . In this case, the initial square root behavior changes with
(11)
(i) A stochastic part due to fluctuations in the statistical mutual trapping of dislocations and (ii) a deterministic part from differences in the selection of the activated slip systems across the boundary. The latter is characterized by an imbalance parameter Γ describing the difference between the activation of slip systems on both sides of a boundary. Γ is assumed to be (approximately) Gaussian distributed with vanishing mean and a standard deviation σimb [10]. For the stochastic part different contributions may be considered. Here, the number of mobile dislocations is assumed to be Gaussian distributed causing a Gaussian white noise in the bias of the dislocation fluxes jfluc [10,12]. These two different contributions correspond closely to two different types of boundaries [10,11]: incidental dislo-
Fig. 2. Evolution of the average disorientation angle across different types of boundaries (IDBs and GNBs) in aluminium during cold-rolling. The √ full lines are fits of θ¯ = π/2 (Pb/d ∗ ) + σ 2 2 to the experimental imb
data from [4] for aluminium of two different purities. The two fitting parameters Pb/d ∗ and σimb in case of GNBs are reduced to one (Pb/d ∗ ) in case of IDBs due to their excluded activation imbalance (σimb = 0).
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increasing strain to a linear relationship between average disorientation angle and strain. The difference between the fitting parameters Pb/d ∗ obtained for both types of boundaries cannot be explained by their different immobilization probabilities (PIDB = 1/3 and PGNB = 1) and indicates a strong dependence of the correlation length d ∗ of the disorientation angle on the boundary type.
3. Results and discussion 3.1. Work-hardening rate From the evolution equation (10) of the total dislocation density in combination with the information on the formation of the disorientation angles from Eq. (12) or the corresponding strain derivative dθ 2 /dγ, finally a stress and strain dependent work-hardening rate τ − ΘIV γ Θ = ΘIII,0 1 − (13) τIII,∞ is obtained. An initial work-hardening rate ΘIII,0 = K (1 + y/Pd∗ ) and a saturation stress τ ΘIII,0 III,∞ = K ∗ τIII,∞ (1 + y/Pd ) are defined for stage III, and a constant work-hardening rate of stage IV is derived (this nomenclature will become obvious in the next section): ΘIV =
2 αµσimb . KP
(14)
3.2. Flow stress Eq. (13) can be integrated by elementary means resulting in a strain dependent flow stress
ΘIII,0 τ = τIII,∞ (1 − f) − (τIII,∞ (1 − f) − τ0 )exp − γ τIII,∞ + fΘIII,0 γ, (15) where the ratio f = ΘIV /ΘIII,0 between the work-hardening rates in the different stages is introduced. This behavior is illustrated in Fig. 3 for different ratios f . If only statistical accumulation of disorientations is considered (i.e. σimb = 0) as in the case of IDBs, ΘIV vanishes and a stage III behavior is observed with a work-hardening rate decreasing linearly with stress and a saturation of the flow stress. However, due to the statistically accumulating excess dislocations slightly different expressions (compared to the Kocks’ model) for the initial work-hardening rate ΘIII,0 of stage III as well as for the saturation stress τIII,∞ must be taken into account. The presence of a deterministic contribution to the accumulation of disorientations as in the case of GNBs (and for simplicity only the larger disorientations across GNBs
Fig. 3. Modelled work-hardening behavior for different ratios f = ΘIV /ΘIII,0 between the work-hardening rates in stage III and IV: (a) normalized stress strain curves according to Eq. (15) and (b) normalized work-hardening rate vs. normalized flow stress (‘Kocks–Mecking plot’) corresponding to Eq. (13). A vanishing initial flow stress is assumed (τ0 = τ(γ = 0) = 0). The full lines correspond to conventional stage III hardening (f = 0).
are considered now) causes at large stresses (and strains) a deviation from stage III. A pronounced stage IV with a flow stress increasing linearly with strain and a constant work-hardening rate ΘIV becomes obvious from Fig. 3. Both stages III and IV are clearly distinguishable with a smooth transition between both. With the data from the experimentally observed disorientation angles (and α = 0.5, K = 16, P = 1 for GNBs) a work-hardening rate ΘIV ≈ 1 × 10−4 µ is obtained. This is the right order of magnitude and — taking into account all simplifications on the geometry of the slip systems and the boundary structure — rather close to the value ΘIV = 2 × 10−4 µ reported from torsion tests on aluminium [6].
4. Conclusion An extension of the work-hardening model of Kocks is presented incorporating excess dislocations and disorienta-
W. Pantleon / Materials Science and Engineering A 387–389 (2004) 257–261
tions in the evolution of the dislocation density. A stochastic formation of disorientations causes stage III behavior, whereas a deterministic contribution to the formation of disorientations as a difference in the selection of slip systems on both sides of a boundary leads to stage IV behavior at large strains. The predicted constant work-hardening rate is in agreement with experimental data. Acknowledgements The author gratefully acknowledges the Danish National Research Foundation for supporting the Center of Fundamental Research: Metal Structures in Four Dimensions, within which this work was performed.
261
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
U.F. Kocks, H. Mecking, Prog. Mater. Sci. 48 (2003) 171. U.F. Kocks, J. Eng. Mater. Technol. 98 (1976) 76. M. Zehetbauer, Acta Metall. Mater. 41 (1993) 589. N. Hansen, Metall. Mater. Trans. A 32 (2001) 2917. A.S. Argon, P. Haasen, Acta Metall. Mater. 41 (1993) 3289. A.D. Rollett, Strain hardening at large strains in aluminum alloys, Ph.D. Thesis, Drexel University, Philadelphia, 1988. U. Essmann, H. Mughrabi, Philos. Mag. A 40 (1979) 731. D. Francke, W. Pantleon, P. Klimanek, Comput. Mater. Sci. 5 (1996) 111. D.L. Holt, J. Appl. Phys. 41 (1970) 3197. W. Pantleon, Mater. Sci. Eng. A 319 (2001) 211. W. Pantleon, J. Mater. Res. 17 (2002) 2433. W. Pantleon, Acta Mater. 46 (1998) 451. W. Pantleon, N. Hansen, Acta Mater. 49 (2001) 1479.
Stage IV work-hardening related to disorientations in dislocation structures W. Pantleon∗ Center for Fundamental Research: Metal Structures in Four Dimensions, Materials Research Department, Risoe National Laboratory, Frederiksborgvej 399, 4000 Roskilde, Denmark Received 9 September 2003; received in revised form 10 November 2003
Abstract The effect of deformation-induced disorientations on the work-hardening of metals is modelled based on dislocation dynamics. Essentially, Kocks’ dislocation model describing stage III hardening is extended to stage IV by incorporation of excess dislocations related to the disorientations. Disorientations evolving from purely statistical reasons — leading to a square root dependence of the average disorientation angle on strain — affect the initial work-hardening rate (and the saturation stress) of stage III only slightly. On the other hand, deterministic contributions to the development of disorientations, as differences in the activated slip systems across boundaries, cause a linear increase of the flow stress at large strains. Such a constant work-hardening rate is characteristic for stage IV. © 2004 Elsevier B.V. All rights reserved. Keywords: Work-hardening; Stage IV; Disorientation; Dislocation; Modelling; Aluminium
1. Introduction In the stress–strain curve of plastically deforming metals different stages of work-hardening are clearly distinguished. During stage III the work-hardening rate is decreasing linearly with stress. This decrease is not continued in stage IV and the work-hardening rate often remains constant (cf. [1]). The work-hardening behavior in stage III has been modelled successfully by Kocks [2] based on a single microstructural variable — the total dislocation density ρ. The main assumption is a flow stress (of a material with shear modulus µ and Burgers vector b) √ τ = αµb ρ (1) entirely controlled by the total dislocation density ρ and a strain rate- and temperature-dependent interaction parameter α = α(γ, ˙ T). In combination with an appropriate evolution law for the dislocation density (see Section 2.1), the model describes flow curves covering the essential features of stage III, despite the fact that details of the microstructure as formation of dislocation boundaries and different types of boundaries are ignored. There have been several attempts ∗
Tel.: +45 4677 5791; fax: +45 4677 5758. E-mail address: [email protected] (W. Pantleon).
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.11.088
(cf. [1]) to modify the model by distinguishing between different types of dislocations (e.g. as dislocations in boundaries and dislocations between the boundaries or edge and screw dislocations [3]). These models are able to cover features of stage IV, but the involved quantities are difficult to access experimentally. Moreover, the formation of orientation differences in dislocation structures, which occurs during deformation independent of the deformation mode (e.g. [4]), is ignored. Argon and Haasen [5] suggested a link between the occurrence of stage IV and the formation of disorientations in the dislocation structure. In their model, disorientations across boundaries cause internal stresses and the work-hardening rate increases linearly with strain — even for a stochastic accumulation of disorientations and in conflict with experimental observations, e.g., on aluminium [6]. The effect of forming disorientations on the dislocation density due to excess dislocations and on the flow stress is investigated here. It will be shown that concise book-keeping of dislocations of different signs of the Burgers vector allows attributing the associated disorientations to a constant work-hardening rate as observed in stage IV. It is emphasized that dislocations will be categorized only according to the sign of their Burgers vector and no other distinction between different types of dislocations is made, in particular, no assumption about the dislocation character is required.
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W. Pantleon / Materials Science and Engineering A 387–389 (2004) 257–261
2. Modelling 2.1. Original model of Kocks The evolution of the total dislocation density ρ can be described in terms of dislocation accumulation and their dynamic recovery [2]: during plastic deformation, the dislocation density increases due to trapping of mobile dislocations by already existing dislocations. The mean free path √ λ¯ = β/ ρ of mobile dislocations is governed by the total dislocation density with a proportionality coefficient β accounting for the details of the dislocation structure. Dynamic recovery occurs when a moving dislocation passes a dislocation of opposite sign and both annihilate. Such a dislocation pair annihilation may trigger even further annihilation events (by relaxation), removing a total dislocation length lr . Here, a slightly different interpretation of the recovery length lr = 2y in terms of the cross-section for annihilation events [7] is preferred: dislocations of opposite sign passing in a distance less than the capturing length y annihilate each other. By considering an effective y, secondary annihilation events are accounted for. Both processes are combined in the evolution of the dislocation density with plastic strain γ: √ ρ 2y dρ 2y 1 − ρ. − ρ= = (2) ¯ βb dγ b b λb The resulting work-hardening rate dτ αµb dρ τ Θ≡ = √ = ΘIII,0 1 − dγ 2 ρ dγ τIII,∞
(3)
decreases linearly with stress from an initial work-hardening K K = αµ/2β until a saturation stress τIII,∞ = rate ΘIII,0 K αµb/2yβ = ΘIII,0 b/y is reached (cf. the full lines in Fig. 3). The linear decrease of the work-hardening rate with stress is characteristic for stage III and leads to the well-established Palm-Voce behavior (cf. [2]) of the flow stress with strain: ΘIII,0 τ = τIII,∞ − (τIII,∞ − τ0 )exp − γ . (4) τIII,∞ Describing storage and annihilation of dislocations, the proposed balance equation for the total dislocation density does not differentiate between dislocations of opposite sign (⊥ and ) and neglects the possible existence of an excess of dislocations of one sign. The necessary modifications for incorporating an excess dislocation density ρ = ρ⊥ − ρ (and the related disorientation angle θ across boundaries) are obtained in a straightforward manner by a thorough analysis of the underlying dislocation dynamics.
Fig. 1. Possible interactions of mobile dislocations with a boundary of dense immobilized dislocations. The dislocation (⊥) with positive sign moving to the right may be trapped or pass the boundary, whereas the negative dislocation ( ) moving to the left passes a positive dislocation within the annihilation distance and both dislocations annihilate leading to a decrease in the dislocation densities of both signs.
dislocation densities can be formulated for dislocations of opposite signs (cf. [8]): 1 dρ⊥ = − 2yρ j → − 2yρ⊥ j ← , (5) dt λ¯ 1 dρ = − 2yρ⊥ j ← − 2yρ j → . (6) dt λ¯ The density of immobile dislocations with positive Burgers vectors (⊥) is increased by trapping of positive mobile dislocations from the flux j → to the right (see Fig. 1). Two processes lead to a loss of positive dislocations: (i) part of the mobile dislocations are stopped by annihilation with an existing immobile negative dislocations ( ), if they pass them within the annihilation distance y and (ii) already immobilized positive dislocations can annihilate with negative mobile dislocations moving to the left. The balance equations for the dislocation densities of both signs can be reformulated by introducing the total dislocation flux j = j → + j ← = γ/b ˙ (related to the strain rate γ) ˙ and the bias of fluxes j = j → − j ← as balance equations for the total dislocation density ρ = ρ⊥ + ρ and the excess dislocation density ρ = ρ⊥ − ρ : dρ 1 − 2yρ j + 2yρj, (7) = dt λ¯
2.2. Incorporation of excess dislocations
dρ 1 = j. dt λ¯
Mobile dislocations carrying the plastic flow move along a certain free path λ before they are stopped. They are either trapped in a dislocation boundary or annihilate with previously stored dislocations. The corresponding changes in the
The first two terms of the balance equation (7) for the total dislocation density are identical to the corresponding ones in the balance equation (2) of Kocks’ model. The additional third term leads to an increase in the total dislocation density without requiring additional storage mechanisms. It is
(8)
W. Pantleon / Materials Science and Engineering A 387–389 (2004) 257–261
a direct consequence of an excess of dislocations of one sign (or equivalently of an existing disorientation). The extra work-hardening is caused by an inhibited annihilation stemming from the fact that an existing excess of dislocations of one sign of the Burgers vector affects the number of possible annihilation events. For instance, a persisting increased flux of positive dislocations leads to an excess of positive dislocations, and the relative frequency of finding a negative dislocation as partner for an annihilation event is reduced. √ The link between the mean free path λ¯ = β/ ρ of a mobile dislocation and the total dislocation density can be established for different dislocation arrangements. For a dislo√ cation cell structure, the cell size d = K/ ρ follows a scaling relation [9] and mobile dislocations passing cell walls or dislocation boundaries are immobilized with a constant immobilization probability P = d/λ¯ resulting in β = K/P. The excess dislocation density is intimately related to the disorientation angle θ = bdρ across dislocation boundaries (in mutual distance d). Therefore, the disorientation angle becomes the second relevant microstructural variable. Its evolution equation dθ = Pbj dt
(9)
does not contain any information on the total dislocation density and is treated independently in Section 2.3. Combining Eqs. (7) and (9) a single evolution equation for the total dislocation density ρ(γ) is obtained: √ 2y ρ dθ 2 dρ 2y P √ ρ− ρ+ = . (10) dγ Kb b 2PKb2 dγ 2.3. Disorientations The development of disorientation angles during plastic deformation has been modelled in detail [10,11] for different types of dislocation boundaries based on Eq. (9). In general, two contributions to the bias of the dislocation fluxes must be distinguished: dθ = Pbjfluc + bΓj. dt
259
cation boundaries (IDBs) are assumed to arise from mutual trapping of dislocations and excess dislocations accumulate only in a stochastic manner, whereas geometrically necessary boundaries (GNBs) are per definitionem associated with an activation imbalance between both sides of the boundary causing an additional deterministic contribution. In both cases, a Gaussian distribution of the disorientation angles results with a vanishing mean value θ = 0 and Pb 2 γ 2. (12) θ 2 = ∗ γ + σimb d A correlation length d ∗ of the disorientation angle along a dislocation boundary is introduced allowing for changes of the disorientation along a single boundary. It replaces the boundary spacing in earlier versions of the model and is assumed to remain constant. Disorientation angles across dislocation boundaries are in general caused by two dislocation sets in the boundary (cf. [13]) leading to a Rayleigh distribution of the disorientation √ angles θ with an average angle given by θ¯ = π/2 θ 2 . This has to be taken into account for any comparison of the predictions of Eq. (12) with experimental data. The measured average disorientation angles of cold-rolled aluminium [4] are shown separately for IDBs and GNBs in Fig. 2. The disorientation angles of both types of boundaries are well described by Eq. (12). (For simplicity the slip γ in a single slip system is equated to the true strain assuming effectively two slip systems with Schmid factor 1/2.) For IDBs, for which σimb vanishes per definitionem, a square of the average disorientation angle √ rootdependence √ θ¯ = π/2 Pb/d ∗ γ on strain is obtained. With Pb/d ∗ = 6 × 10−4 a good description of the experimental data is accomplished. A good correspondence is achieved for GNBs as well, using Pb/d ∗ = 6 × 10−3 and σimb = 6 × 10−2 . In this case, the initial square root behavior changes with
(11)
(i) A stochastic part due to fluctuations in the statistical mutual trapping of dislocations and (ii) a deterministic part from differences in the selection of the activated slip systems across the boundary. The latter is characterized by an imbalance parameter Γ describing the difference between the activation of slip systems on both sides of a boundary. Γ is assumed to be (approximately) Gaussian distributed with vanishing mean and a standard deviation σimb [10]. For the stochastic part different contributions may be considered. Here, the number of mobile dislocations is assumed to be Gaussian distributed causing a Gaussian white noise in the bias of the dislocation fluxes jfluc [10,12]. These two different contributions correspond closely to two different types of boundaries [10,11]: incidental dislo-
Fig. 2. Evolution of the average disorientation angle across different types of boundaries (IDBs and GNBs) in aluminium during cold-rolling. The √ full lines are fits of θ¯ = π/2 (Pb/d ∗ ) + σ 2 2 to the experimental imb
data from [4] for aluminium of two different purities. The two fitting parameters Pb/d ∗ and σimb in case of GNBs are reduced to one (Pb/d ∗ ) in case of IDBs due to their excluded activation imbalance (σimb = 0).
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W. Pantleon / Materials Science and Engineering A 387–389 (2004) 257–261
increasing strain to a linear relationship between average disorientation angle and strain. The difference between the fitting parameters Pb/d ∗ obtained for both types of boundaries cannot be explained by their different immobilization probabilities (PIDB = 1/3 and PGNB = 1) and indicates a strong dependence of the correlation length d ∗ of the disorientation angle on the boundary type.
3. Results and discussion 3.1. Work-hardening rate From the evolution equation (10) of the total dislocation density in combination with the information on the formation of the disorientation angles from Eq. (12) or the corresponding strain derivative dθ 2 /dγ, finally a stress and strain dependent work-hardening rate τ − ΘIV γ Θ = ΘIII,0 1 − (13) τIII,∞ is obtained. An initial work-hardening rate ΘIII,0 = K (1 + y/Pd∗ ) and a saturation stress τ ΘIII,0 III,∞ = K ∗ τIII,∞ (1 + y/Pd ) are defined for stage III, and a constant work-hardening rate of stage IV is derived (this nomenclature will become obvious in the next section): ΘIV =
2 αµσimb . KP
(14)
3.2. Flow stress Eq. (13) can be integrated by elementary means resulting in a strain dependent flow stress
ΘIII,0 τ = τIII,∞ (1 − f) − (τIII,∞ (1 − f) − τ0 )exp − γ τIII,∞ + fΘIII,0 γ, (15) where the ratio f = ΘIV /ΘIII,0 between the work-hardening rates in the different stages is introduced. This behavior is illustrated in Fig. 3 for different ratios f . If only statistical accumulation of disorientations is considered (i.e. σimb = 0) as in the case of IDBs, ΘIV vanishes and a stage III behavior is observed with a work-hardening rate decreasing linearly with stress and a saturation of the flow stress. However, due to the statistically accumulating excess dislocations slightly different expressions (compared to the Kocks’ model) for the initial work-hardening rate ΘIII,0 of stage III as well as for the saturation stress τIII,∞ must be taken into account. The presence of a deterministic contribution to the accumulation of disorientations as in the case of GNBs (and for simplicity only the larger disorientations across GNBs
Fig. 3. Modelled work-hardening behavior for different ratios f = ΘIV /ΘIII,0 between the work-hardening rates in stage III and IV: (a) normalized stress strain curves according to Eq. (15) and (b) normalized work-hardening rate vs. normalized flow stress (‘Kocks–Mecking plot’) corresponding to Eq. (13). A vanishing initial flow stress is assumed (τ0 = τ(γ = 0) = 0). The full lines correspond to conventional stage III hardening (f = 0).
are considered now) causes at large stresses (and strains) a deviation from stage III. A pronounced stage IV with a flow stress increasing linearly with strain and a constant work-hardening rate ΘIV becomes obvious from Fig. 3. Both stages III and IV are clearly distinguishable with a smooth transition between both. With the data from the experimentally observed disorientation angles (and α = 0.5, K = 16, P = 1 for GNBs) a work-hardening rate ΘIV ≈ 1 × 10−4 µ is obtained. This is the right order of magnitude and — taking into account all simplifications on the geometry of the slip systems and the boundary structure — rather close to the value ΘIV = 2 × 10−4 µ reported from torsion tests on aluminium [6].
4. Conclusion An extension of the work-hardening model of Kocks is presented incorporating excess dislocations and disorienta-
W. Pantleon / Materials Science and Engineering A 387–389 (2004) 257–261
tions in the evolution of the dislocation density. A stochastic formation of disorientations causes stage III behavior, whereas a deterministic contribution to the formation of disorientations as a difference in the selection of slip systems on both sides of a boundary leads to stage IV behavior at large strains. The predicted constant work-hardening rate is in agreement with experimental data. Acknowledgements The author gratefully acknowledges the Danish National Research Foundation for supporting the Center of Fundamental Research: Metal Structures in Four Dimensions, within which this work was performed.
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