- Email: [email protected]
A mechanism for the negative strain-rate sensitivity of dilute solid solutions
Acta Materialia 52 (2004) 3447–3458 www.actamat-journals.com
A mechanism for the negative strain-rate sensitivity of dilute solid solutions R.C. Picu
*
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA Received 19 November 2003; received in revised form 16 January 2004; accepted 26 March 2004 Available online 24 April 2004
Abstract A new mechanism is proposed for dynamic strain ageing and the negative strain-rate sensitivity (SRS) exhibited by dilute solid solutions containing mobile solute atoms. The mechanism is based on the strength variation of dislocation junctions due to the presence of solute clusters on forest dislocations. The strength of a Lomer–Cottrell lock in which the mobile dislocation is free of solute, while the forest dislocation is clustered, is studied by using an orientation-dependent line tension model. It is shown that the junction strength increases with the size of the cluster on the forest dislocation (binding energy of the forest dislocation to its cluster). The cluster forms by lattice diffusion and its size depends on the time lapsed from the formation of the respective dislocation segment. Therefore, the average size of clusters on new forest dislocations is smaller the larger the imposed strain rate. Consequently, the average strength of junctions decreases (after a transient) upon an increase of the strain rate, which leads to negative SRS. A model including the results of the mesoscopic analysis is developed to capture this mechanism. The model reproduces qualitatively a number of key features observed experimentally at the macroscopic scale. The new mechanism does not require solute diffusion to take place sufficiently fast for clustering of mobile dislocations to happen during their arrest time at obstacles, as assumed in previous models of the phenomenon. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Strain-rate sensitivity; Portevin–LeChatelier effect; Dislocation junctions; Mesoscopic modeling
1. Introduction Many dilute solid solutions of technological importance, both interstitial and substitutional, exhibit the Portevin–LeChatelier effect (PLC). The phenomenon is observed in a certain range of temperature and strain rate and consists in the deformation being inhomogeneous, dominated by successive strain localization events, which leads to serrated stress–strain curves. The PLC effect is usually undesirable in industrial processes since it is associated with a reduction of ductility and rippled surfaces of the deforming part. Some of the most studied material systems that exhibit the phenomenon are Al–Mg (e.g., [1–5]), Cu–Mn and Cu–Al alloys [6,7]. The PLC effect is attributed to dynamic strain ageing (DSA), a generic term representing a host of small-scale *
Tel.: +1-518-276-2195; fax: +1-518-276-6025. E-mail address: [email protected] (R.C. Picu).
phenomena associated with the interaction of dislocations and solute atoms. Dynamic strain ageing leads to negative strain-rate sensitivity (SRS). In fact, the PLC effect is a direct manifestation of the negative SRS, which generates material instabilities. The SRS is quantified using the SRS parameter m defined as the ratio of the flow stress variation corresponding to an imposed variation of the strain rate Dr m¼ : ð1Þ D log e_ It is usual to distinguished between the instantaneous SRS, mi , corresponding to the immediate response of the flow stress to a jump in strain rate, and the steady-state SRS, m, which is measured after a transient. Hence, m ¼ mi þ mt . The instantaneous measure is always positive, while the transient is negative in the DSA regime. When the total SRS parameter, m, becomes negative, conditions exist for the macroscopic observation of the PLC effect [8,9].
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.03.042
3448
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
The literature on the PLC effect is extensive, including a number of reviews such as those by Neuhauser [10], Robinson [11] and Estrin and Kubin [12]. The characteristics of DSA are: (a) the phenomenon manifests itself in a certain range of temperature and strain rate, (b) in the range in which DSA is observed, the yield stress and the strain-hardening rate are less temperature sensitive than outside this range, (c) m decreases as the mobility of solute atoms increases, (d) the negative SRS and PLC are usually observed after an incubation strain ec . A ‘‘normal’’ PLC domain is observed at low temperatures, in which ec increases with increasing the strain rate, while at high temperatures, an ‘‘inverse’’ PLC domain exists in which the critical strain decreases upon increasing the deformation rate [3–5] and (e) multiple activation energies appear to be related with the phenomenon [6]. The microscopic mechanism of DSA leading to negative SRS is still a matter of debate. The first model was developed by Cottrell and Bilby [13] who considered that solute atoms have sufficient mobility to follow dislocations as they move through the lattice. This leads to an increased friction force for mobile dislocations. As the deformation rate increases, the dislocations may lose their clouds and move freely. It was later realized that solute atoms do not have sufficient mobility to follow moving dislocations and ageing should take place while dislocations are arrested at obstacles such as at forest dislocations [14]. van den Beukel [15] developed this idea into a quantitative model of DSA. The basic mechanism envisioned is clustering of solute to the arrested mobile dislocations leading to an apparent increase of the lock strength. The process should occur by lattice diffusion of solute from the region nearby the defect. The amount of solute gathered to the core depends on the arrest time, tw , which, in turn, is a function of the strain rate and the average dislocation density. Mulford and Kocks [16] developed a model of DSA based on the concept that solute clustering affects the strain-hardening component of the flow stress rather than the friction stress. This is supported by experimental observations that strain hardening and the ultimate tensile strength are considerably more sensitive to DSA than the yield stress [2]. The mechanism underlying the model is inspired by an earlier suggestion by Sleeswyk [17] and requires that, at a junction of an unclustered mobile dislocation with a clustered forest dislocation, solute diffuses from the forest dislocation, along the core of the mobile dislocation (pipe diffusion). The lock strength increases with the arrest time as the solute pins a longer segment of the mobile dislocation. The Mulford–Kocks model received support from numerous experiments and it is now generally accepted that DSA perturbs the way mobile and forest dislocations interact. For example, PLC is observed in single crystals oriented for multiple slip and in polycrystals in
conditions in which it is not observed in single crystals oriented for single slip [18]. It is also believed that the critical strain ec is associated with a critical density of forest dislocations [2], ec being different in single slip compared with the multislip and polycrystal cases. The pipe diffusion idea promoted by this model is also considered a solution to the long-standing debate on the role of vacancies in solute diffusion to arrested dislocations. The idea that strain-induced vacancies may enhance bulk diffusion was proposed early in the development of the field [19], as it was observed that without extrinsic vacancies the diffusion coefficient in the bulk is too small to permit fitting the lattice friction model [15] to the experimental data. However, experiments by Cuddy and Leslie [20] and other authors (e.g., [21]) clearly indicated that vacancies are not the decisive ingredient in DSA. Hence, it was conjectured that pipe diffusion, which presumably may take place without being assisted by vacancies, is the leading mechanism responsible for DSA. The pipe diffusion mechanism was recently questioned based on an atomistic analysis of Mg diffusion along the core of dislocations in Al [22]. It was shown that solute diffusion in the absence of vacancies is as difficult along the pipe, as it is in the bulk (in the temperature range in which DSA is observed). The activation energies for vacancy-assisted solute diffusion in the core are lower than those in the bulk by about 25%, but the region where this condition exists is rather small (‘‘narrow pipe’’). Therefore, it appears that, if one discards the idea that vacancies are important in DSA, pipe diffusion is too slow to lead to reasonable strengthening of junctions during the arrest time of mobile dislocations (about 0.01–10 s, depending on the strain rate). A mechanism that avoids this discussion is proposed here. It is envisioned that solute atoms cluster at forest dislocations, the process being possible since forests are stationary for much longer time intervals than tw . The presence of such clusters affects the strength of dislocation junctions which, in turn, depends on the cluster size. The cluster size on forests is variable with the ageing time, ta , i.e., the time lapsed from the formation of the respective forest segment, or the time the forest is stationary. The ageing time of forests depends on the rate of forest production and the imposed strain rate. To quantify this mechanism, the strength of a Lomer–Cottrell lock is analyzed as a function of the cluster size and for various loading conditions ranging from those corresponding to single slip to those equivalent to multislip. This paper is organized as follows: the model used in the analysis of the dislocation junction is presented in Section 2, the effect of the presence of the cluster on junction configuration and strength is discussed in Section 3, the proposed mechanism is detailed in Section 4 and its macroscopic predictions are discussed against experimental observations in Section 5.
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
2. Anisotropic line tension model for the analysis of dislocation junctions In fcc crystals, the athermal component of the flow stress is largely attributed to the interaction of mobile and forest dislocations. In a random forest with uniformly distributed trees, about 20% of the trees form junctions. Many interactions are repulsive and, out of the attractive ones, only a fraction form stable junctions [23]. These junctions control to a large extent the hardening behavior and the flow stress. The various dislocation junctions have been classified in terms of their strength (and therefore contribution to the flow stress) [24–26]. It is currently considered that the Lomer–Cottrell lock has the highest strength, followed by the glissile junction and the Hirth lock [27]. It was recently suggested that the intersection of dislocations with collinear Burgers vectors (which do not form a stair rod) leads to a contribution to the flow stress higher than that of the Lomer–Cottrell lock [28]. The strength of these junctions was evaluated using a variety of models. The first attempts were made using an isotropic (direction independent) line tension model [24]. It was later pointed out that such models do not capture the difference between the Lomer–Cottrell lock and the glissile junction and cannot represent the Hirth lock. The anisotropic line tension model proved to be a good substitute and was extensively used to map the strength of junctions as a function of their type, line orientation and direction of loading [27,29–31]. Further confidence in this model was gained after it was shown that its predictions of the strength of the symmetric Lomer– Cottrell lock are in excellent agreement with the strength computed by nodal dislocation dynamics [32] and full 3D atomistics [33]. The success of this simplified model stems from the fact that core effects (e.g., dissociation into partials) and the close-range field-mediated dislocation–dislocation interactions are not essential in the energetics of junction formation and failure. In this work, the anisotropic line tension model is used to evaluate the strength of a symmetric Lomer– Cottrell lock for various loading modes and various
3449
degrees of clustering of the forest dislocation. The details of the model are similar to those described in [27,30] and are briefly reviewed here. Let us consider the configuration in Fig. 1. A mobile dislocation AB gliding in the horizontal plane comes in contact with a forest CD residing in the inclined plane. Both planes are of {1 1 1} type. The forest dislocation is clustered along its entire length CD (the cluster is not represented in the figure). The geometry is characterized by the angles made by the two dislocations with the line of intersection of the two glide planes, /f , and /m , and by the orientation of the Burgers vectors bf and bm with respect to the respective dislocation lines, af , am . The initial length of both dislocations is 2L. If attractive, the two dislocation reach in equilibrium the configuration in Fig. 1(b), in which a stair rod of length x1 þ x2 (EF) forms along the line of intersection of the two planes, MN. Since the forest dislocation is attracted to its cloud, there is an energy penalty for unbinding. Hence, the forest dislocation may partially remain trapped in the cluster (segments CF0 and DE0 ). The equilibrium configuration in Fig. 1(b), as well as the configuration the junction takes after imposing a resolved shear stress in the two planes, sm and sf , is determined by minimizing the total potential energy of the system over the space of all relevant parameters. The total potential energy reads p p p p Etotal ¼ EAE ðx1 ; sm ; bm Þ þ EBF ðx2 ; sm ; bm Þ þ EEF ðx1 ; x2 ; bj Þ p p p ðx ; d ; s ; bf Þ þ ECF þ EFF 0 ðx2 ; d2 ; sf ; bf Þ þ E 0 ðQ; d2 Þ EE0 1 1 f p þ EDE 0 ðQ; d1 Þ;
ð2Þ
where bj ¼ bm þ bf is the Burgers vector of the stair rod and Q is the binding energy per unit length of the forest dislocation to its cluster. The minimization problem is formulated as p min fEtotal gjsm ;sf ;bm ;bf ;Q :
x1 ;x2 ;d1 ;d2
ð3Þ
The potential energy of a dislocation segment is computed as Ep ¼ E W , where E is the total energy of the segment and W is the work done by the applied stress in moving the respective dislocation. The energy
Fig. 1. Configuration of a generic unloaded dislocation junction: (a) before formation of the stair rod and (b) after relaxation.
3450
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
E is computed as an integral over the whole length of the segment, C: Z deðsÞ; ð4Þ E ¼ C
where the energy per unit length de is given as A½1 m cos2 aðsÞ ds unclustered; deðsÞ ¼ fA½1 m cos2 aðsÞ þ Qg ds clustered: ð5Þ Gb2
Here A ¼ 4pð1mÞ log rR0 is the energy per unit length of an edge dislocation, G is the shear modulus and m is the Poisson’s ratio. It is assumed that the material is isotropic and that the presence of the cluster does not perturb the far field of the dislocation [34]. The quantity A is taken as the reference energy (per unit length of dislocation line) in the system. Under the action of a stress field r, the dislocation moves and the work performed may be evaluated as: Z W ¼ ½ðr bÞ tðsÞ uðsÞ ds; ð6Þ C
where tðsÞ is the vector tangent to the dislocation line at position s. The quantity in the parenthesis is the Peach– Koehler force acting on the dislocation and uðsÞ is the displacement vector of segment ds located at s. If one considers glide only (no work done by the climb component of the Peach–Koehler force), and sRSS is the resolved shear stress acting in the respective glide plane and in the direction of b, Eq. (6) reduces to Z sRSS b uðsÞ ds ¼ sRSS jbjt; ð7Þ W ¼ C
where t is the area swept by the contour C. The resolved shear stress was defined above for the two glide planes as sm and sf , respectively. To render the integration tractable, it is assumed that the dislocation segments deform into arcs of ellipses as suggested in [29]. The large semiaxis of the ellipse, a1 , is in the direction of the Burgers vector b (and that of the respective sRSS ), while the length of the short semiaxis, a2 , depends on the magnitude of the applied shear stress through the Frank–Read condition: 2a2 sRSS b ¼ 2Að1 mÞ. Therefore, a2 ¼ Að1mÞ sRSS b and a1 ¼ ra2 . In the present calculations, the symmetric Lomer– Cottrell lock is considered, with the two dislocations being of 60° type, am ¼ af ¼ 60°, um ¼ uf ¼ 60°, bm ¼ 12 ½ 1 1 0, bf ¼ 12 ½1 0 1. The ratio of the semiaxes of the ellipse is taken r ¼ 1:75 [29] and m ¼ 0:347, corresponding to Al [27,32]. The parameters of the problem are the binding energy per unit length, Q, and the two shear stresses sm and sf . All quantities are normalized, a superscript denoting the respective normalized variable. The normalization quantities are L, for lengths and A, for energies. The
p is normalized by AL, and the shear total energy Etotal stress is normalized as s ¼ s bL A. The variables of the problem become: x1 , x2 , d1 , d2 . Further, it is observed (as in [27]) that the equilibrium condition for nodes E0 and F 0 require the condition di =xi ¼ const (i ¼ 1; 2). The constant depends on Q, since the binding energy to the cluster effectively increases the line tension of the segment still tied to the cloud. The specific dependence is discussed in the next section. Here, it suffices to show that the problem has, in fact, only two degrees of freedom, x1 and x2 . The energy minimization is performed using a conjugate-gradient routine. The equilibrium configuration is that which minimizes the energy. The critical stress at which the junction breaks results as the maximum stress beyond which no solution to Eq. (3) is found. The junction fails in two modes: by unzipping, in which the length of the stair rod gradually decreases to zero (x1 þ x2 ! 0) at the critical stress, and by the instability of one of the mobile dislocation branches. The mobile dislocation instabilities are monitored by following the position of the center of the ellipse corresponding to each segment. The instability sets in when the ellipse center aligns with the two ends of the bowedout segment. Conversely, it may be stated that the dislocation becomes unstable when the stress is so large that no elliptical dislocation loop with a2 ¼ Að1mÞ sRSS b and a1 ¼ ra2 may be fitted through the end points of the segment. Closing the model description, it is necessary to underline the approximations made. It was assumed that the dislocations are not split into partials (core details are not important), that the elasticity of the material (here, Al) is isotropic, and that the interaction between neighboring segments through their stress field is negligible compared to the effect of line tension. Furthermore, the energy associated with the formation of a jog in each dislocation after dislocation crossing is considered negligible compared to the other energies involved and is not represented in the model [24,32].
3. Effect of clustering on junction configuration and strength 3.1. Junction configuration in the unloaded state The equilibrium configuration of the unloaded lock is studied first. If the forest dislocation is not clustered (Q=A ¼ 0), d1 ¼ d2 ¼ L and x1 ¼ x2 ¼ x. In this case, the configuration is fully defined by the parameter x ¼ x=L. This parameter is shown in Fig. 2(a) for the whole range of angles /m made by the mobile dislocation with the line of intersection of the two glide planes, MN, before junction formation. The forest dislocation makes an angle /f ¼ 60° with the MN in all cases and both dis-
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
3451
Fig. 2. (a) Variation of the stair rod length with the angle made by the mobile dislocation with the line of intersection of the two glide planes, /m , for the Lomer–Cottrell lock. (b) Variation of the stair rod length and degree of unbinding of the forest dislocation from its cluster as a function of the binding energy of the forest to the cluster, for the symmetric Lomer–Cottrell lock. (c) Limit binding energies for which the junction structure is not perturbed, ðQ=AÞ1 , and for which the junction is transformed into a crossed-state, ðQ=AÞ2 , as a function of /m . (d) Variation of the ratio unbinding length to stair rod length with the binding energy (Q=A) for the symmetric Lomer–Cottrell lock.
locations have their Burgers vector running at 60° with respect to MN. Therefore, the type of the mobile dislocation changes with /m . As expected, the stair rod length increases as /m decreases. The value of x for the symmetric case (/m ¼ 60°) is identical to that reported in [27,32,33] for the same configuration (x ¼ 0:32). The configuration of the symmetric lock is studied next as a function of the normalized binding energy Q=A (Fig. 2(b)). At small Q=A (Q=A < ðQ=AÞ1 ¼ 0:04), the configuration is not perturbed by the presence of the cluster (x ¼ 0:32). For larger binding energies, the presence of the cluster reduces the size of the stair rod EF up to full constriction (at Q=A ¼ ðQ=AÞ2 ¼ 0:13). For larger Q=A, the junction is reduced to a ‘‘crossedstate.’’ In principle, a value of the binding energy exists that would transform any junction into a crossed-state. Fig. 2(c) shows the threshold Q=A values corresponding to the lower (unperturbed junction) and upper (crossedstate) limits of the transition region in Fig. 2(b). The plot shows data for all configurations considered in Fig. 2(a). Further, it is interesting to determine how the length over which the forest unbinds from the cluster,
d ¼ d=L, depends on the magnitude of Q=A. To this end, the symmetric junction is considered and Q=A is varied from 0 to 0.4. The results are shown in Fig. 2(d). As mentioned in Section 2, for a given Q=A, the ratio of d and x must be a constant, a condition ensuing from the equilibrium of nodes E0 and F0 . This observation reduces the dimensionality of the minimization problem stated in the previous section. When node F moves to the left O under the action of a shear stress, the of point ratio d2 =x2 , with x2 being now negative, becomes 1.1 times the ratio in Fig. 2(d). It is noted that the relaxation of the structure from the configuration in Fig. 1(a) to that in Fig. 1(b) requires overcoming an energy barrier for the nucleation of the stair rod. This barrier is expected to scale with Q and may be estimated by assuming that when OE ¼ OF 10b, the segments EE0 and FF0 are not interacting significantly with the cluster in the region E0 F0 and the stair rod has nucleated. This corresponds to an energy required for unbinding the forest over E0 F0 of magnitude 20bðd=xÞQ. This figure should be an upper limit. Deriving the actual magnitude of the barrier requires atomistic simulations of the 3D geometry. Once this
3452
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
barrier is overcome, the energy decays continuously up to the fully formed junction configuration (Fig. 1(b)). 3.2. Junction strength The most interesting aspect of the analysis is related to the effect of the cluster on the strength of the junction. To investigate this issue, the symmetric Lomer–Cottrell lock is considered. The stress is first applied in the mobile dislocation plane only. The critical stress corresponding to Q=A ¼ 0 was evaluated to be sm ¼ 0:79, in good agreement with the value reported in [27]. The variation of the critical stress sm with Q=A is shown in Fig. 3. At small cluster sizes, the critical stress increases almost linearly with the binding energy and reaches a maximum of 1.43 for Q=A 0:3. The critical stress remains unchanged as Q=A increases further. To gain insight into the origins of this behavior, it is instructive to look at the failure mode and the junction configuration for various Q=A as the applied stress increases from zero to the respective critical value. Fig. 4 shows the configuration of the dislocation in the forest plane for several reduced stresses and three values of Q=A. The first row refers to the unclustered dislocation. As the stress increases, the two nodes defining the stair rod, E and F, initially located at x1 ¼ x2 ¼ x ¼ 0:32, move in the direction of the applied stress. When the stress is half of the critical value, node F did not cross the median axis (point O). Close to the critical stress, F moves to the left and x2 becomes negative. The dislocation fails by unzipping (stair rod length goes to zero) at sm ¼ 0:79. When Q=A ¼ 0:13, a different situation is observed. The equilibrium configuration in the unloaded state is that of a ‘‘crossed-state’’: there is no stair rod formation,
Fig. 3. The dependence of the symmetric Lomer–Cottrell lock strength on the binding energy of the forest dislocation to its cluster. The forest dislocation is not loaded (sf ¼ 0). At small Q=A, the junction fails by unzipping and the critical stress increases with Q=A. At large Q=A the junction is reduced to a crossed state and the failure is associated with the instability of the mobile dislocation (MDI).
Fig. 4. Schematics indicating the failure modes at various Q=A. Only the forest dislocation plane is shown. The stress is applied in the mobile dislocation plane. The notations are identical with those in Fig. 1. When Q=A ¼ 0, the junction fails by unzipping at a reduced critical stress of 0.79. At Q=A ¼ 0:13, the evolution of the structure up to the critical point (sm ¼ 1:08) is different than for Q=A ¼ 0. The stress activates the unbinding of the forest branch EE0 . The segment OD does not unbind fully at the critical stress. At the same time, the segment CF remains trapped in the cluster up to the instability. At larger Q=A (Q=A > 0:3), none of the forest segments unbinds from the cluster (line CD) and the failure occurs at sm ¼ 1:43 by mobile dislocation instability.
the forest remaining bonded to its cloud (along line CD). As the stress increases to about half of the critical stress, segment EE0 unbinds from the cluster and point E moves to the left. Segment CF remains tied to the cloud. As the stress increases further, DF continues to unbind, while CF remains trapped. When the applied stress reaches the critical value of sm ¼ 1:08, the upper forest branch escapes from the cloud and the stair rod disappears. The increase of the critical stress for Q=A ranging from 0 to 0.13 is associated with the additional energy required for forest unbinding. The failure mode is either unzipping or mobile dislocation instability: unzipping results when node F reaches node E, however, the instability of the mobile dislocation branch BF is expected to set in before this happens. Since the whole process is dynamic and inertia is expected to play a role, no definite conclusion can be made as to which failure mode dominates. This mechanism controls failure up to Q=A 0:3. At larger binding energies no unbinding of the forest dislocation from its cloud occurs for any stress level and the junction fails by the instability of the mobile dislocation branch BF (Fig. 1(b)), at sm ¼ 1:43. This value of the critical stress corresponds to the motion of the mobile dislocation through an array of fixed rigid obstacles spaced L apart. Obviously, increasing the strength of the obstacles (Q=A) leads to no further modification of the critical stress.
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
It is noted that thermal activation has no influence on the behavior of junctions. The energies involved are large enough such that thermal fluctuations (at temperatures in the vicinity of room temperature, i.e., in the range in which PLC is observed in Al–Mg alloys) would activate failure only when the applied stress is very close to the critical value. If the stress is significantly lower than the critical stress, the energy provided by thermal fluctuations is sufficient to slightly change the position of node E. It was verified that, if the system were activated up the energy landscape along the path it would normally take under stress, and were loaded to failure from that point, the critical stress would be identical to that reported in Fig. 3. Finally, various loading paths with nonzero sf were investigated in order to determine the failure envelope of this junction in the stress space. The results are shown in Fig. 5 for several values of Q=A and in the quadrant defined by sf 2 ðsm ; sm Þ. The failure surfaces are symmetric with respect to the lines sf ¼ sm and sf ¼ sm due to the symmetry of the structure. The failure envelope corresponding to Q=A ¼ 0 is identical to that derived by Dupuy and Fivel [27]. The failure mode is unzipping. As Q=A increases, the envelope moves to larger critical stresses. The shift correlates with the change in the failure mode as discussed in connection with Figs. 3 and 4. The envelope corresponding to Q=A P 0:3 is a square cen-
Fig. 5. Failure envelopes for the symmetric Lomer–Cottrell lock in the sm –sf plane and for various values of the normalized binding energy Q=A. Only one of the quadrants is shown since the envelope is symmetric with respect to the lines sm ¼ sf and sm ¼ sf . The curve corresponding to Q=A ¼ 0 is identical to that reported in [27]. As Q=A increases, the envelope shifts to larger stresses, while for Q=A > 0:3, the envelope becomes a square of side 2 1:43. The symbols indicate what mechanism controls the strength under the respective conditions. The filled circle indicates uncertainty over the failure mechanism under conditions represented by the second row in Fig. 4.
3453
tered at zero and of side 2 1:43, the failure mode being mobile dislocation instability (MDI) occurring under crossed-state conditions (Fig. 4 last row), irrespective of the loading path.
4. The negative strain-rate sensitivity mechanism The observation that the junction strength varies with the cluster size leads to a new mechanism for dynamic strain ageing. This mechanism is based on the idea that clusters form on forest dislocations rather than on the mobile dislocations, as assumed in the existing theories. Forest dislocations are generated during plastic deformation along with the mobile dislocations. Since they reside in planes subjected to a lower resolved shear stress, their residence time is considerably longer than that of mobile dislocations. Hence, their ageing time is sufficiently long for clusters to form even at relatively low temperatures (the lower temperature end of the PLC domain). Specifically, the average waiting time of a mobile dislocation at a forest obstacle may be computed as pffiffiffiffiffi tw ¼ X=_e, where X ¼ qm b= qf . The parameter X was evaluated by Kubin and Estrin [35] to be on the order of 3 103 or smaller, for realistic values of the forest and mobile dislocation densities, qm and qf . Considering the forest dislocations to be stationary, allows for the evaluation of the ageing time ta of a forest produced at time t after the inception of plastic deformation from ta þ t ¼ e=_e. Hence, the average ageing time of a (newly produced) forest is on the order of emax =2_e, where emax is the maximum strain reached in the respective test. The ratio of tw and ta may be evaluated from the above as tw =ta ¼ 2X=emax , which, for emax ¼ 0:3 (approximately equal to the failure strain of Al–Mg alloys), results to be smaller than 0.02 for all strain rates. The ageing time of forest dislocations is hence, in average, more than 50 times longer than the arrest time of mobile dislocations. The longer the residence time of forest dislocations, the larger the cluster they acquire and therefore the larger Q=A. For small Q=A (smaller than 0.3), the junction strength increases with the binding energy and it is immediately obvious that this leads to negative strainrate sensitivity: deformations at small strain rates imply longer ageing times for forests and therefore larger junction strength, while the opposite holds for deformations performed at large strain rates. The conclusion preserves for Q=A > 0:3. In order to quantify the proposed mechanism, a function is developed to represent the critical stress values presented in Fig. 5, for all Q=A and loading paths. This function is defined numerically and has the generic form Q c sm ¼ S ;n ; ð8Þ A
3454
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
where n ¼ sm =sf represents the loading path and takes values between )1 and 1. The current value of the binding energy Q=A is a function of the cluster size. Here, it is assumed that clustering occurs by lattice diffusion and therefore Friedel’s model modified by Louat’s correction [36] is selected to represent the clustering kinetics. The cluster size reaches a saturation value ðQ=AÞs at long ageing times ta . The kinetics is controlled by an intrinsic timescale t0 , which is associated with the rate of the diffusion process and depends on temperature. " 2=3 !# Q Q ta : ðta ; c; T Þ ¼ ðc; T Þ 1 exp A A s t0 ðT Þ ð9Þ
The dependence on temperature of t0 may be expressed by an Arrhenius law of the form t0 ¼ t00 exp½Ea1 =kT , as suggested in the literature. The saturation value depends on temperature and the average solute concentration, c. The cluster is expected to dissolve at high temperatures and therefore the saturation value is assumed to have an Arrhenius form: ðQ=AÞs ¼ ðQ=AÞ0 exp ðEa2 =kT Þ. Here, Ea2 is the activation energy for the cluster dissolution process and is expected to be on the order of the binding energy of a solute atom to the core of the respective type of dislocation. The dependence on the average solute concentration is unknown, however, data obtained by static ageing experiments on Al–Mg alloys [37] suggest that the cluster size decreases with decreasing c. The dependence of the saturated cluster size on both temperature and the average solute concentration needs to be calibrated from atomistic simulations. This work is ongoing for the Al–Mg binary system. Let us consider now a population of junctions of the type studied in Section 3. The forests were produced at various times during plastic deformation and their clusters have different sizes (Q=A). It will be approximated, for the sake of discussion, that the flow stress above the yield point (the strain-hardening component, Dsc m ) is fully associated with these junctions. The average shear stress applied in the mobile dislocation plane required to move a mobile dislocation through this set of obstacles may be approximated using the expression: 2R 31=q dq t q Q f _ e S ; n dt a 0 de A c 6 7 tta Dsm ðt; c; T ; nÞ ¼ 4 5 : ð10Þ R t dqf e_ dta 0 de tta
The average is made over the whole population and the density of forests with a given strength, S, is computed as the density of forests produced during the time interval dt (dt ¼ dta ) at a moment in the past (with respect to the current time, t) defined by the ageing time, ta . Therefore, the average over the population becomes an
average over time from the inception of plastic deformation. Evaluating the critical stress required to push a mobile dislocation through a set of obstacles of different strength is still a controversial issue. Here, the approximation developed by Hanson and Morris [38] is adopted and the quantity of interest is computed by making q ¼ 2 in Eq. (10). If it is assumed that, in addition to the variability associated with Q=A, there is also variability in terms of the loading path, a second average must be performed over all possible loading modes. Here, it is assumed that all loading paths are equally probable and therefore Eq. (10) becomes: 2R R 31=q dqf t 1 q Q _ S ; n dn dt e a 0 1 A de c 6 7 tta Dsm ðt; c; T Þ ¼ 4 5 : R t dqf 2 0 de e_ dta tta
ð11Þ This assumption is not critical. For example, considering that the loading paths are grouped in the neighborhood of the horizontal axis in Fig. 5 (the resolved stress in the forest plane is smaller than that in the mobile plane, or single slip conditions) leads to results similar to those discussed in the next section. The evolution of the density of forest dislocations, qf , with strain is represented by a monotonically increasing function with saturation, as suggested by the numerical results presented in [35]:
p qf ðeÞ ¼ qsat 1 exp e=e0 : ð12Þ f Here p and e0 are parameters controlling the rate of increase of the forest density with strain. The parameter p is always larger than 1. Further, the time dependence in Eq. (11) is transformed into strain dependence by imposing e ¼ e_ t and all parameters are rendered non-dimensional. Eq. (11) becomes
Dsc m ðe;c;T Þ ¼
Z
Z
i
h
2=3 ;n S 2 ðQ=AÞs 1 exp ðge=e0 Þ 0 1
p ð1 gÞp exp ð1 gÞp e=e0 dndg 1
1
1=2 Z 1
p p p dg 2 ð1 gÞ exp ð1 gÞ e=e0 ; 0
ð13Þ where g ¼ t=ta . The parameters in Eq. (13) are p and e0 , and ðQ=AÞs and e0 . The first two parameters are selected p ¼ 1:5 and e0 ¼ 1 which corresponds to a continuously rising curve of qf ðeÞ for e < 0:3. e 0:3 corresponds to the failure strain of AA5182-O Al–Mg alloy [5,39]. It was verified that considering other values for these parameters in the
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
range p < 3, e0 < 4 leads to no significant changes in model predictions (Section 5). The parameter e0 ¼ e_ t0 lumps the effects of strain rate and temperature (through the characteristic time of the diffusion process). It increases linearly with strain rate and increases with decreasing temperature. This is consistent with suggestions made in the literature according to which the strain rate and the temperature may be combined into a single constitutive parameter of the form e_ exp½Q=kT [3], where Q is the activation energy for DSA. An additional temperature dependence is included in ðQ=AÞs as discussed in connection with Eq. (9). The SRS is evaluated with Eqs. (1) and (13). This leads to the transient component of m only. The instantaneous component mi is always positive and is observed experimentally to increase with strain [8].
5. Macroscopic predictions and discussion The model described in the previous section was used to predict the SRS and the critical strain variation with strain, strain rate and temperature, and to derive a map of the negative SRS domain in the rate-temperature plane. The (transient) SRS parameter, mt , is evaluated based on Eqs. (1) and (13) and using a strain rate differential of e_ 1 ¼ 5_e2 . The variation of mt with strain results directly from Eq. (13) and is shown in Fig. 6 for several values of e0 ¼ e_ t0 and for ðQ=AÞs ¼ 0:3. Since the DSA mechanism described in the previous Section is built into the formulation, mt must be negative for all strains and strain rates. At low strain rates, mt is close to zero, which corresponds physically to the situation in which all forests are clustered and the clusters reach saturation. In these conditions varying the strain rate does not affect
Fig. 6. Variation of the transient SRS parameter with strain for various values of e0 ¼ e_ t0 and for ðQ=AÞs ¼ 0:3. mt < 0 at all strains. As the strain rate increases, mt decreases to a minimum (negative) and then returns in the neighborhood of zero, while remaining at negative values. The total SRS parameter, m, at large rates is positive since the absolute value of mt is small.
3455
the overall SRS. As the strain rate increases and the time constant associated with the imposed deformation (ta ) becomes comparable with the time constant of the bulk diffusion process (t0 ), mt decreases. Further increasing the rate leads to a return of mt in the vicinity of zero, which corresponds to no significant clustering (junction strength variation) occurring during ta . Hence, the total SRS parameter, m, is positive at large and small strain rates since mi > 0. This behavior is similar to that observed experimentally [8,9,39–42]. In most cases, the total SRS parameter m decreases from a positive value at the beginning of the plastic deformation and either stabilizes at a negative value [39–41], or rebounds toward the positive range at larger strains [6,16,42]. The actual shape of the curve depends on temperature and the base strain rate, feature captured by the model proposed here (see, e.g., the curves corresponding to e0 ¼ 0:1 and 0.01). In most solid solutions, even those that do not exhibit the PLC effect (m > 0), m is observed to decrease with strain (e.g. [41]). The larger the diffusivity of the solute, the more pronounced the strain dependence. This corresponds in the present model to the large e0 limit, i.e., to slow diffusion. Fig. 7 shows the variation of mt with logðe0 Þ for ðQ=AÞs ¼ 0:3 and for two plastic strains. It is observed that the strain has a relatively weak effect on the shape of the curve, a feature also observed in experiments [39]. The position of the minimum is dictated by the value of the saturation binding energy ðQ=AÞs . It is interesting to analyze how the function in Fig. 7 depends on ðQ=AÞs . Several curves corresponding to ðQ=AÞs ¼ 0:1, 0.3 and 0.5 and e ¼ 10% are shown in Fig. 8. As the binding energy increases, the transient SRS parameter becomes more negative, as expected. The effect is essentially insensitive to increasing Q=A above 0.3. The domain in which mt 0 expands, the expansion being more pronounced in the low temperature/high strain rate range. When Q=A > 0:3, the
Fig. 7. Variation of mt with e0 . mt is evaluated at two different strains.
3456
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
Fig. 8. Dependence of the mt vs. e0 curve on the saturation value of Q=A. As ðQ=AÞs decreases, mt decreases in magnitude and the range of e0 corresponding to the pronounced effect (mt 0) shifts to lower e0 values.
domain ceases to expand and shifts to higher e0 values. This is in general agreement with experimental observations on the effect of the average solute concentration on the domain of negative SRS and PLC. The dependence of the critical strain on e0 (strain rate and temperature) is shown in Fig. 9. The critical strains are evaluated from the data in Fig. 6 by considering two strain-independent instantaneous rate sensitivity parameters mi ¼ 0:02 and 0.1, respectively. The two curves corresponding to each mi show the lower and the upper critical strains and therefore bound the domain of negative rate sensitivity. In this log–log plot, the boundaries are straight lines of similar slope. Both critical strains increase with increasing the strain rate (decreasing temperature), i.e., the behavior corresponds to the ‘‘normal PLC.’’ The result is in agreement with the experimental data presented for Al–Mg in [43]. It is noted that using a different value for the instantaneous SRS parameter has no qualitative influence on this result,
Fig. 9. Variation of the critical strain ec with e0 for two values of the instantaneous SRS parameter, mi . The lines represent the boundaries of the negative SRS domain. The model reproduces the ‘‘normal’’ PLC behavior only.
however, increasing the positive instantaneous SRS parameter reduces the m < 0 domain, as expected. Here, the lower critical strain is associated with a critical forest dislocation density, rather than to a critical concentration of vacancies, as proposed in earlier models. This is supported by the observation that in heavily cold rolled materials the lower critical strain shifts to lower values. In some cases the serrated flow is observed immediately after yielding [2,39]. The model does not capture the ‘‘inverse PLC’’ behavior, in which the critical strain decreases with increasing strain rate [1,5]. As discussed in [5,44], this type of response appears to be rooted in larger scale phenomena associated with the evolution of large populations of defects. Finally, the activation energy for appearance/disappearance of the negative SRS is investigated. To this end, the influence of temperature on the boundaries of the m < 0 domain needs to be determined. As discussed above, the temperature affects the time constant for diffusion, t0 , and the saturation value of Q=A, ðQ=AÞs . Therefore, the negative SRS domain has to be plotted in the space of these two variables, ðQ=AÞs and e0 . The curves in Fig. 8 are used for this purpose and the upper and lower limits of the domain m < 0, logðe0 Þc , are determined for several ðQ=AÞs values and for the instantaneous SRS mi ¼ 0:02 and 0.1. The result is shown in Fig. 10 for two values of mi . The domain is upperbounded by a straight line described by the equation (for mi ¼ 0:02): e0c1 ¼ 36:4ðQ=AÞ1:6 s :
ð14Þ
Fig. 10. Map of the negative SRS range in the e0 –ðQ=AÞs space for two values of the instantaneous SRS parameter, mi . The two variables are Arrhenius dependent on temperature with different activation energies. The activation energy for t0 (controlling e0 ) is associated with the clustering process of forest dislocations, i.e., the bulk diffusion of solute, while the activation energy for ðQ=AÞs is associated with the dissolution of the cluster (binding energy of a solute atom to the forest). These two energies translate into multiple activation energies for the negative SRS phenomenon (Eq. (16)).
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
The lower bound is a curve schematically represented here by two straight lines with equations (for mi ¼ 0:02): 1:9
e0 c1 ¼ 7 105 ðQ=AÞs ¼ 1:7 10
2
ðQ=AÞ2:5 s
at low ðQ=AÞs ; e0 c1 at high ðQ=AÞs :
ð15Þ
The curves corresponding to mi ¼ 0:1 are described by functions with similar exponents. This result may be rearranged in terms of the strain rate and temperature by considering e0 ¼ e_ t0 and including the explicit, Arrhenius dependence on temperature of t0 and ðQ=AÞs (Section 4). This procedure leads to the boundaries of the negative SRS parameter m in the log e_ 1=T plane as: Ea1 þ 1:6Ea2 e_ c1 / exp ; kT Ea1 þ 2:5Ea2 ð16Þ ; e_ c2 / exp kT Ea1 1:9Ea2 : e_ c2 / exp kT The domain is shown schematically in Fig. 11 in which the slopes of the three lines correspond to Ea1 þ 1:6Ea2 , Ea1 þ 2:5Ea2 and Ea1 1:9Ea2 . Experimental studies on various material systems indicate that the domain of occurrence of the PLC effect is bounded in the log e_ 1=T diagram by approximately parallel straight lines [1,16], or by lines that intersect at some angle [5,45], being limited on the high temperature side. Both experimental observations may be explained based on Eq. (16). Obviously, if Ea2 is much smaller than Ea1 , the boundaries appear in this representation as parallel straight lines. Furthermore, the model suggests that multiple activation energies are associated with the boundaries of the PLC (m < 0) domain, which is in agreement with experimental observations. The activation energy Ea1 is associated with the bulk diffusion of solute and is expected to be larger than 0.5 eV in most cases. For vacancy-assisted bulk diffusion in Al–Mg, it is 1.2 and 0.51 eV in the absence and presence of an excess of vacancies. The second figure
3457
here corresponds to the migration of a solute (Mg) in an existing vacancy, while the first is the sum of the vacancy formation and solute migration energies [34]. Ea2 is associated with the dissolution of the cluster and is expected to be on the order of the binding energy of a solute to the core of the respective forest dislocation. The binding energies of Mg to the core of edge, 60° and screw dislocations in Al were estimated by atomistic simulations to be 0.126, 0.092 and 0.043 eV, respectively [22,34]. Ling and McCormick [8] report that the dependence on temperature of the critical strain in an Al–Mg–Si alloy is characterized by an activation energy (for inception of the PLC effect) of 0.59 eV, while Kubin et al. [46] infer an activation energy equivalent with Ea1 of 0.5 eV by fitting their model to experimental data for Al–5%Mg. In closure is should be mentioned that the relationship developed in Section 4 between the dislocation junction-scale processes and the macroscopic behavior should be regarded as a proof of concept. Considering the rather dramatic simplifications made in the procedure of averaging over the population of junctions, and considering that the fundamental data regarding the behavior of junctions is restricted to only one junction type, the agreement with the macroscopic observations cannot be quantitative. However, the model correctly reproduces several basic features of the macroscopic behavior.
6. Conclusions The new mechanism leading to negative strain-rate sensitivity in dilute solid solutions proposed here is based on the effect of solute clusters formed on forest dislocations on the strength of dislocation junctions. It is shown that the presence of a cluster on a forest dislocation modifies the strength of junctions formed by that dislocation with mobile unclustered dislocations. Since the relevant clustering process is that at forest dislocations, the size of the cluster depends on the residence time of the forest. Faster deformation allows for less clustering to occur which, in turn, is associated with lower junction strength. This mechanism is related to the macroscopic behavior through a simplified model. The model reproduces qualitatively a number of features observed experimentally.
Acknowledgements
Fig. 11. Schematic map of the negative SRS range in the strain rate inverse temperature space.
This work was supported, in part, by Alcoa Inc. and by the NSF through Grant CMS-0084987. The author thanks Ms. M.A. Soare for help with some of the calculations leading to the results presented in Fig. 5.
3458
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Pink E, Grinberg A. Acta Metall 1982;30:2153. Balik J, Lukac P. Acta Metall Mater 1993;41:1447. Ling CP, McCormick PG. Acta Metall Mater 1993;41:3127. Kumar S. Scripta Metall Mater 1995;33:81. Abbadi M, Hahner P, Zeghloul A. Mater Sci Eng A 2002;337:194. Nortmann A, Schwink Ch. Acta Mater 1997;45:2043. Nortmann A, Schwink Ch. Acta Mater 1997;45:2051. Ling CP, McCormick PG. Acta Metall Mater 1990;38:2631. Kalk A, Nortmann A, Schwink Ch. Philos Mag A 1995;72:1239. Neuhauser H. Plastic instabilities and the deformation of metals. In: Walgref D, Ghoniem NM, editors. Patterns, defects and material instabilities. Kluwer; 1990. p. 241. Robinson JM. Int Mater Rev 1994;39:217. Estrin Y, Kubin LP. Spatial coupling and propagative plastic instabilities. In: Muhlhaus HB, editor. Continuum models for materials with microstructure. New York: Wiley; 1995. p. 395. Cottrell AH, Bilby BA. Proc Phys Soc London A 1949;62:49. McCormick PG. Acta Metall 1972;20:351. van den Beukel A. Phys Status Solidi A 1975;30:197. Mulford RA, Kocks UF. Acta Metall 1979;27:1125. Sleeswyk AW. Acta Metall 1958;6:598. Tabata T, Fujita H, Ueda N. Mater Sci Eng A 1980;44:81. Cottrell AH. Philos Mag 1953;44:31. Cuddy LJ, Leslie WC. Acta Metall 1972;20:1157. Schwaz RB, Funk LL. Acta Metall 1985;33:295. Picu RC, Zhang D. Acta Mater 2004;52:161. Hirth JP. J Appl Phys 1961;32:700.
[24] Saada G. Acta Metall 1960;8:841. [25] Wu TY, Bassani JL, Laird C. Proc Roy Soc London A 1991;435:1. [26] Franciosi P, Zaoui A. Acta Metall 1982;30:1627. [27] Dupuy L, Fivel MC. Acta Mater 2002;50:4873. [28] Madec R, Devincre B, Kubin L, Hoc T, Rodney D. Science 2003;301:1879. [29] De Wit G, Koehler JS. Phys Rev 1959;116:1113. [30] Schoeck G, Frydman R. Phys Status Solidi B 1972;53:661. [31] Wickham LK, Schwartz KW, Stolken JS. Phys Rev Lett 1999;83:4574. [32] Shenoy VB, Kukta RV, Phillips R. Phys Rev Lett 2000;84:1491. [33] Rodney D, Phillips R. Phys Rev Lett 1999;82:1704. [34] Zhang D, Picu RC. Model Simul Mater Sci Eng 2004;12:121. [35] Kubin LP, Estrin Y. Acta Metall Mater 1990;38:697. [36] Louat N. Scripta Metall 1981;15:1167. [37] Wycliffe P, Kocks UF, Embury JD. Scripta Metall 1980;14: 1349. [38] Hanson K, Morris JW. J Appl Phys 1975;46:2378. [39] Picu RC, Vincze G, Ozturk F, Gracio JJ, Barlat F, Maniatty AM. Mater Sci Eng. 2003 [submitted]. [40] van den Brink SH, van den Beukel A, McCormick PG. Phys Status Solidi A 1975;30:469. [41] McCormick PG. Scripta Metall 1978;12:197. [42] Schwink Ch, Nortmann A. Mater Sci Eng A 1997;234:1. [43] Guillot J, Grilhe T. Acta Metall 1972;20:291. [44] Hahner P, Rizzi E. Acta Mater 2003;51:3385. [45] Zeghloul A, Mliha-Touati M, Bakir S. J Phys III 1996;6:1467. [46] Kubin LP, Chihab K, Estrin Y. Acta Metall 1988;36:2707.
A mechanism for the negative strain-rate sensitivity of dilute solid solutions R.C. Picu
*
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA Received 19 November 2003; received in revised form 16 January 2004; accepted 26 March 2004 Available online 24 April 2004
Abstract A new mechanism is proposed for dynamic strain ageing and the negative strain-rate sensitivity (SRS) exhibited by dilute solid solutions containing mobile solute atoms. The mechanism is based on the strength variation of dislocation junctions due to the presence of solute clusters on forest dislocations. The strength of a Lomer–Cottrell lock in which the mobile dislocation is free of solute, while the forest dislocation is clustered, is studied by using an orientation-dependent line tension model. It is shown that the junction strength increases with the size of the cluster on the forest dislocation (binding energy of the forest dislocation to its cluster). The cluster forms by lattice diffusion and its size depends on the time lapsed from the formation of the respective dislocation segment. Therefore, the average size of clusters on new forest dislocations is smaller the larger the imposed strain rate. Consequently, the average strength of junctions decreases (after a transient) upon an increase of the strain rate, which leads to negative SRS. A model including the results of the mesoscopic analysis is developed to capture this mechanism. The model reproduces qualitatively a number of key features observed experimentally at the macroscopic scale. The new mechanism does not require solute diffusion to take place sufficiently fast for clustering of mobile dislocations to happen during their arrest time at obstacles, as assumed in previous models of the phenomenon. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Strain-rate sensitivity; Portevin–LeChatelier effect; Dislocation junctions; Mesoscopic modeling
1. Introduction Many dilute solid solutions of technological importance, both interstitial and substitutional, exhibit the Portevin–LeChatelier effect (PLC). The phenomenon is observed in a certain range of temperature and strain rate and consists in the deformation being inhomogeneous, dominated by successive strain localization events, which leads to serrated stress–strain curves. The PLC effect is usually undesirable in industrial processes since it is associated with a reduction of ductility and rippled surfaces of the deforming part. Some of the most studied material systems that exhibit the phenomenon are Al–Mg (e.g., [1–5]), Cu–Mn and Cu–Al alloys [6,7]. The PLC effect is attributed to dynamic strain ageing (DSA), a generic term representing a host of small-scale *
Tel.: +1-518-276-2195; fax: +1-518-276-6025. E-mail address: [email protected] (R.C. Picu).
phenomena associated with the interaction of dislocations and solute atoms. Dynamic strain ageing leads to negative strain-rate sensitivity (SRS). In fact, the PLC effect is a direct manifestation of the negative SRS, which generates material instabilities. The SRS is quantified using the SRS parameter m defined as the ratio of the flow stress variation corresponding to an imposed variation of the strain rate Dr m¼ : ð1Þ D log e_ It is usual to distinguished between the instantaneous SRS, mi , corresponding to the immediate response of the flow stress to a jump in strain rate, and the steady-state SRS, m, which is measured after a transient. Hence, m ¼ mi þ mt . The instantaneous measure is always positive, while the transient is negative in the DSA regime. When the total SRS parameter, m, becomes negative, conditions exist for the macroscopic observation of the PLC effect [8,9].
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.03.042
3448
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
The literature on the PLC effect is extensive, including a number of reviews such as those by Neuhauser [10], Robinson [11] and Estrin and Kubin [12]. The characteristics of DSA are: (a) the phenomenon manifests itself in a certain range of temperature and strain rate, (b) in the range in which DSA is observed, the yield stress and the strain-hardening rate are less temperature sensitive than outside this range, (c) m decreases as the mobility of solute atoms increases, (d) the negative SRS and PLC are usually observed after an incubation strain ec . A ‘‘normal’’ PLC domain is observed at low temperatures, in which ec increases with increasing the strain rate, while at high temperatures, an ‘‘inverse’’ PLC domain exists in which the critical strain decreases upon increasing the deformation rate [3–5] and (e) multiple activation energies appear to be related with the phenomenon [6]. The microscopic mechanism of DSA leading to negative SRS is still a matter of debate. The first model was developed by Cottrell and Bilby [13] who considered that solute atoms have sufficient mobility to follow dislocations as they move through the lattice. This leads to an increased friction force for mobile dislocations. As the deformation rate increases, the dislocations may lose their clouds and move freely. It was later realized that solute atoms do not have sufficient mobility to follow moving dislocations and ageing should take place while dislocations are arrested at obstacles such as at forest dislocations [14]. van den Beukel [15] developed this idea into a quantitative model of DSA. The basic mechanism envisioned is clustering of solute to the arrested mobile dislocations leading to an apparent increase of the lock strength. The process should occur by lattice diffusion of solute from the region nearby the defect. The amount of solute gathered to the core depends on the arrest time, tw , which, in turn, is a function of the strain rate and the average dislocation density. Mulford and Kocks [16] developed a model of DSA based on the concept that solute clustering affects the strain-hardening component of the flow stress rather than the friction stress. This is supported by experimental observations that strain hardening and the ultimate tensile strength are considerably more sensitive to DSA than the yield stress [2]. The mechanism underlying the model is inspired by an earlier suggestion by Sleeswyk [17] and requires that, at a junction of an unclustered mobile dislocation with a clustered forest dislocation, solute diffuses from the forest dislocation, along the core of the mobile dislocation (pipe diffusion). The lock strength increases with the arrest time as the solute pins a longer segment of the mobile dislocation. The Mulford–Kocks model received support from numerous experiments and it is now generally accepted that DSA perturbs the way mobile and forest dislocations interact. For example, PLC is observed in single crystals oriented for multiple slip and in polycrystals in
conditions in which it is not observed in single crystals oriented for single slip [18]. It is also believed that the critical strain ec is associated with a critical density of forest dislocations [2], ec being different in single slip compared with the multislip and polycrystal cases. The pipe diffusion idea promoted by this model is also considered a solution to the long-standing debate on the role of vacancies in solute diffusion to arrested dislocations. The idea that strain-induced vacancies may enhance bulk diffusion was proposed early in the development of the field [19], as it was observed that without extrinsic vacancies the diffusion coefficient in the bulk is too small to permit fitting the lattice friction model [15] to the experimental data. However, experiments by Cuddy and Leslie [20] and other authors (e.g., [21]) clearly indicated that vacancies are not the decisive ingredient in DSA. Hence, it was conjectured that pipe diffusion, which presumably may take place without being assisted by vacancies, is the leading mechanism responsible for DSA. The pipe diffusion mechanism was recently questioned based on an atomistic analysis of Mg diffusion along the core of dislocations in Al [22]. It was shown that solute diffusion in the absence of vacancies is as difficult along the pipe, as it is in the bulk (in the temperature range in which DSA is observed). The activation energies for vacancy-assisted solute diffusion in the core are lower than those in the bulk by about 25%, but the region where this condition exists is rather small (‘‘narrow pipe’’). Therefore, it appears that, if one discards the idea that vacancies are important in DSA, pipe diffusion is too slow to lead to reasonable strengthening of junctions during the arrest time of mobile dislocations (about 0.01–10 s, depending on the strain rate). A mechanism that avoids this discussion is proposed here. It is envisioned that solute atoms cluster at forest dislocations, the process being possible since forests are stationary for much longer time intervals than tw . The presence of such clusters affects the strength of dislocation junctions which, in turn, depends on the cluster size. The cluster size on forests is variable with the ageing time, ta , i.e., the time lapsed from the formation of the respective forest segment, or the time the forest is stationary. The ageing time of forests depends on the rate of forest production and the imposed strain rate. To quantify this mechanism, the strength of a Lomer–Cottrell lock is analyzed as a function of the cluster size and for various loading conditions ranging from those corresponding to single slip to those equivalent to multislip. This paper is organized as follows: the model used in the analysis of the dislocation junction is presented in Section 2, the effect of the presence of the cluster on junction configuration and strength is discussed in Section 3, the proposed mechanism is detailed in Section 4 and its macroscopic predictions are discussed against experimental observations in Section 5.
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
2. Anisotropic line tension model for the analysis of dislocation junctions In fcc crystals, the athermal component of the flow stress is largely attributed to the interaction of mobile and forest dislocations. In a random forest with uniformly distributed trees, about 20% of the trees form junctions. Many interactions are repulsive and, out of the attractive ones, only a fraction form stable junctions [23]. These junctions control to a large extent the hardening behavior and the flow stress. The various dislocation junctions have been classified in terms of their strength (and therefore contribution to the flow stress) [24–26]. It is currently considered that the Lomer–Cottrell lock has the highest strength, followed by the glissile junction and the Hirth lock [27]. It was recently suggested that the intersection of dislocations with collinear Burgers vectors (which do not form a stair rod) leads to a contribution to the flow stress higher than that of the Lomer–Cottrell lock [28]. The strength of these junctions was evaluated using a variety of models. The first attempts were made using an isotropic (direction independent) line tension model [24]. It was later pointed out that such models do not capture the difference between the Lomer–Cottrell lock and the glissile junction and cannot represent the Hirth lock. The anisotropic line tension model proved to be a good substitute and was extensively used to map the strength of junctions as a function of their type, line orientation and direction of loading [27,29–31]. Further confidence in this model was gained after it was shown that its predictions of the strength of the symmetric Lomer– Cottrell lock are in excellent agreement with the strength computed by nodal dislocation dynamics [32] and full 3D atomistics [33]. The success of this simplified model stems from the fact that core effects (e.g., dissociation into partials) and the close-range field-mediated dislocation–dislocation interactions are not essential in the energetics of junction formation and failure. In this work, the anisotropic line tension model is used to evaluate the strength of a symmetric Lomer– Cottrell lock for various loading modes and various
3449
degrees of clustering of the forest dislocation. The details of the model are similar to those described in [27,30] and are briefly reviewed here. Let us consider the configuration in Fig. 1. A mobile dislocation AB gliding in the horizontal plane comes in contact with a forest CD residing in the inclined plane. Both planes are of {1 1 1} type. The forest dislocation is clustered along its entire length CD (the cluster is not represented in the figure). The geometry is characterized by the angles made by the two dislocations with the line of intersection of the two glide planes, /f , and /m , and by the orientation of the Burgers vectors bf and bm with respect to the respective dislocation lines, af , am . The initial length of both dislocations is 2L. If attractive, the two dislocation reach in equilibrium the configuration in Fig. 1(b), in which a stair rod of length x1 þ x2 (EF) forms along the line of intersection of the two planes, MN. Since the forest dislocation is attracted to its cloud, there is an energy penalty for unbinding. Hence, the forest dislocation may partially remain trapped in the cluster (segments CF0 and DE0 ). The equilibrium configuration in Fig. 1(b), as well as the configuration the junction takes after imposing a resolved shear stress in the two planes, sm and sf , is determined by minimizing the total potential energy of the system over the space of all relevant parameters. The total potential energy reads p p p p Etotal ¼ EAE ðx1 ; sm ; bm Þ þ EBF ðx2 ; sm ; bm Þ þ EEF ðx1 ; x2 ; bj Þ p p p ðx ; d ; s ; bf Þ þ ECF þ EFF 0 ðx2 ; d2 ; sf ; bf Þ þ E 0 ðQ; d2 Þ EE0 1 1 f p þ EDE 0 ðQ; d1 Þ;
ð2Þ
where bj ¼ bm þ bf is the Burgers vector of the stair rod and Q is the binding energy per unit length of the forest dislocation to its cluster. The minimization problem is formulated as p min fEtotal gjsm ;sf ;bm ;bf ;Q :
x1 ;x2 ;d1 ;d2
ð3Þ
The potential energy of a dislocation segment is computed as Ep ¼ E W , where E is the total energy of the segment and W is the work done by the applied stress in moving the respective dislocation. The energy
Fig. 1. Configuration of a generic unloaded dislocation junction: (a) before formation of the stair rod and (b) after relaxation.
3450
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
E is computed as an integral over the whole length of the segment, C: Z deðsÞ; ð4Þ E ¼ C
where the energy per unit length de is given as A½1 m cos2 aðsÞ ds unclustered; deðsÞ ¼ fA½1 m cos2 aðsÞ þ Qg ds clustered: ð5Þ Gb2
Here A ¼ 4pð1mÞ log rR0 is the energy per unit length of an edge dislocation, G is the shear modulus and m is the Poisson’s ratio. It is assumed that the material is isotropic and that the presence of the cluster does not perturb the far field of the dislocation [34]. The quantity A is taken as the reference energy (per unit length of dislocation line) in the system. Under the action of a stress field r, the dislocation moves and the work performed may be evaluated as: Z W ¼ ½ðr bÞ tðsÞ uðsÞ ds; ð6Þ C
where tðsÞ is the vector tangent to the dislocation line at position s. The quantity in the parenthesis is the Peach– Koehler force acting on the dislocation and uðsÞ is the displacement vector of segment ds located at s. If one considers glide only (no work done by the climb component of the Peach–Koehler force), and sRSS is the resolved shear stress acting in the respective glide plane and in the direction of b, Eq. (6) reduces to Z sRSS b uðsÞ ds ¼ sRSS jbjt; ð7Þ W ¼ C
where t is the area swept by the contour C. The resolved shear stress was defined above for the two glide planes as sm and sf , respectively. To render the integration tractable, it is assumed that the dislocation segments deform into arcs of ellipses as suggested in [29]. The large semiaxis of the ellipse, a1 , is in the direction of the Burgers vector b (and that of the respective sRSS ), while the length of the short semiaxis, a2 , depends on the magnitude of the applied shear stress through the Frank–Read condition: 2a2 sRSS b ¼ 2Að1 mÞ. Therefore, a2 ¼ Að1mÞ sRSS b and a1 ¼ ra2 . In the present calculations, the symmetric Lomer– Cottrell lock is considered, with the two dislocations being of 60° type, am ¼ af ¼ 60°, um ¼ uf ¼ 60°, bm ¼ 12 ½ 1 1 0, bf ¼ 12 ½1 0 1. The ratio of the semiaxes of the ellipse is taken r ¼ 1:75 [29] and m ¼ 0:347, corresponding to Al [27,32]. The parameters of the problem are the binding energy per unit length, Q, and the two shear stresses sm and sf . All quantities are normalized, a superscript denoting the respective normalized variable. The normalization quantities are L, for lengths and A, for energies. The
p is normalized by AL, and the shear total energy Etotal stress is normalized as s ¼ s bL A. The variables of the problem become: x1 , x2 , d1 , d2 . Further, it is observed (as in [27]) that the equilibrium condition for nodes E0 and F 0 require the condition di =xi ¼ const (i ¼ 1; 2). The constant depends on Q, since the binding energy to the cluster effectively increases the line tension of the segment still tied to the cloud. The specific dependence is discussed in the next section. Here, it suffices to show that the problem has, in fact, only two degrees of freedom, x1 and x2 . The energy minimization is performed using a conjugate-gradient routine. The equilibrium configuration is that which minimizes the energy. The critical stress at which the junction breaks results as the maximum stress beyond which no solution to Eq. (3) is found. The junction fails in two modes: by unzipping, in which the length of the stair rod gradually decreases to zero (x1 þ x2 ! 0) at the critical stress, and by the instability of one of the mobile dislocation branches. The mobile dislocation instabilities are monitored by following the position of the center of the ellipse corresponding to each segment. The instability sets in when the ellipse center aligns with the two ends of the bowedout segment. Conversely, it may be stated that the dislocation becomes unstable when the stress is so large that no elliptical dislocation loop with a2 ¼ Að1mÞ sRSS b and a1 ¼ ra2 may be fitted through the end points of the segment. Closing the model description, it is necessary to underline the approximations made. It was assumed that the dislocations are not split into partials (core details are not important), that the elasticity of the material (here, Al) is isotropic, and that the interaction between neighboring segments through their stress field is negligible compared to the effect of line tension. Furthermore, the energy associated with the formation of a jog in each dislocation after dislocation crossing is considered negligible compared to the other energies involved and is not represented in the model [24,32].
3. Effect of clustering on junction configuration and strength 3.1. Junction configuration in the unloaded state The equilibrium configuration of the unloaded lock is studied first. If the forest dislocation is not clustered (Q=A ¼ 0), d1 ¼ d2 ¼ L and x1 ¼ x2 ¼ x. In this case, the configuration is fully defined by the parameter x ¼ x=L. This parameter is shown in Fig. 2(a) for the whole range of angles /m made by the mobile dislocation with the line of intersection of the two glide planes, MN, before junction formation. The forest dislocation makes an angle /f ¼ 60° with the MN in all cases and both dis-
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
3451
Fig. 2. (a) Variation of the stair rod length with the angle made by the mobile dislocation with the line of intersection of the two glide planes, /m , for the Lomer–Cottrell lock. (b) Variation of the stair rod length and degree of unbinding of the forest dislocation from its cluster as a function of the binding energy of the forest to the cluster, for the symmetric Lomer–Cottrell lock. (c) Limit binding energies for which the junction structure is not perturbed, ðQ=AÞ1 , and for which the junction is transformed into a crossed-state, ðQ=AÞ2 , as a function of /m . (d) Variation of the ratio unbinding length to stair rod length with the binding energy (Q=A) for the symmetric Lomer–Cottrell lock.
locations have their Burgers vector running at 60° with respect to MN. Therefore, the type of the mobile dislocation changes with /m . As expected, the stair rod length increases as /m decreases. The value of x for the symmetric case (/m ¼ 60°) is identical to that reported in [27,32,33] for the same configuration (x ¼ 0:32). The configuration of the symmetric lock is studied next as a function of the normalized binding energy Q=A (Fig. 2(b)). At small Q=A (Q=A < ðQ=AÞ1 ¼ 0:04), the configuration is not perturbed by the presence of the cluster (x ¼ 0:32). For larger binding energies, the presence of the cluster reduces the size of the stair rod EF up to full constriction (at Q=A ¼ ðQ=AÞ2 ¼ 0:13). For larger Q=A, the junction is reduced to a ‘‘crossedstate.’’ In principle, a value of the binding energy exists that would transform any junction into a crossed-state. Fig. 2(c) shows the threshold Q=A values corresponding to the lower (unperturbed junction) and upper (crossedstate) limits of the transition region in Fig. 2(b). The plot shows data for all configurations considered in Fig. 2(a). Further, it is interesting to determine how the length over which the forest unbinds from the cluster,
d ¼ d=L, depends on the magnitude of Q=A. To this end, the symmetric junction is considered and Q=A is varied from 0 to 0.4. The results are shown in Fig. 2(d). As mentioned in Section 2, for a given Q=A, the ratio of d and x must be a constant, a condition ensuing from the equilibrium of nodes E0 and F0 . This observation reduces the dimensionality of the minimization problem stated in the previous section. When node F moves to the left O under the action of a shear stress, the of point ratio d2 =x2 , with x2 being now negative, becomes 1.1 times the ratio in Fig. 2(d). It is noted that the relaxation of the structure from the configuration in Fig. 1(a) to that in Fig. 1(b) requires overcoming an energy barrier for the nucleation of the stair rod. This barrier is expected to scale with Q and may be estimated by assuming that when OE ¼ OF 10b, the segments EE0 and FF0 are not interacting significantly with the cluster in the region E0 F0 and the stair rod has nucleated. This corresponds to an energy required for unbinding the forest over E0 F0 of magnitude 20bðd=xÞQ. This figure should be an upper limit. Deriving the actual magnitude of the barrier requires atomistic simulations of the 3D geometry. Once this
3452
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
barrier is overcome, the energy decays continuously up to the fully formed junction configuration (Fig. 1(b)). 3.2. Junction strength The most interesting aspect of the analysis is related to the effect of the cluster on the strength of the junction. To investigate this issue, the symmetric Lomer–Cottrell lock is considered. The stress is first applied in the mobile dislocation plane only. The critical stress corresponding to Q=A ¼ 0 was evaluated to be sm ¼ 0:79, in good agreement with the value reported in [27]. The variation of the critical stress sm with Q=A is shown in Fig. 3. At small cluster sizes, the critical stress increases almost linearly with the binding energy and reaches a maximum of 1.43 for Q=A 0:3. The critical stress remains unchanged as Q=A increases further. To gain insight into the origins of this behavior, it is instructive to look at the failure mode and the junction configuration for various Q=A as the applied stress increases from zero to the respective critical value. Fig. 4 shows the configuration of the dislocation in the forest plane for several reduced stresses and three values of Q=A. The first row refers to the unclustered dislocation. As the stress increases, the two nodes defining the stair rod, E and F, initially located at x1 ¼ x2 ¼ x ¼ 0:32, move in the direction of the applied stress. When the stress is half of the critical value, node F did not cross the median axis (point O). Close to the critical stress, F moves to the left and x2 becomes negative. The dislocation fails by unzipping (stair rod length goes to zero) at sm ¼ 0:79. When Q=A ¼ 0:13, a different situation is observed. The equilibrium configuration in the unloaded state is that of a ‘‘crossed-state’’: there is no stair rod formation,
Fig. 3. The dependence of the symmetric Lomer–Cottrell lock strength on the binding energy of the forest dislocation to its cluster. The forest dislocation is not loaded (sf ¼ 0). At small Q=A, the junction fails by unzipping and the critical stress increases with Q=A. At large Q=A the junction is reduced to a crossed state and the failure is associated with the instability of the mobile dislocation (MDI).
Fig. 4. Schematics indicating the failure modes at various Q=A. Only the forest dislocation plane is shown. The stress is applied in the mobile dislocation plane. The notations are identical with those in Fig. 1. When Q=A ¼ 0, the junction fails by unzipping at a reduced critical stress of 0.79. At Q=A ¼ 0:13, the evolution of the structure up to the critical point (sm ¼ 1:08) is different than for Q=A ¼ 0. The stress activates the unbinding of the forest branch EE0 . The segment OD does not unbind fully at the critical stress. At the same time, the segment CF remains trapped in the cluster up to the instability. At larger Q=A (Q=A > 0:3), none of the forest segments unbinds from the cluster (line CD) and the failure occurs at sm ¼ 1:43 by mobile dislocation instability.
the forest remaining bonded to its cloud (along line CD). As the stress increases to about half of the critical stress, segment EE0 unbinds from the cluster and point E moves to the left. Segment CF remains tied to the cloud. As the stress increases further, DF continues to unbind, while CF remains trapped. When the applied stress reaches the critical value of sm ¼ 1:08, the upper forest branch escapes from the cloud and the stair rod disappears. The increase of the critical stress for Q=A ranging from 0 to 0.13 is associated with the additional energy required for forest unbinding. The failure mode is either unzipping or mobile dislocation instability: unzipping results when node F reaches node E, however, the instability of the mobile dislocation branch BF is expected to set in before this happens. Since the whole process is dynamic and inertia is expected to play a role, no definite conclusion can be made as to which failure mode dominates. This mechanism controls failure up to Q=A 0:3. At larger binding energies no unbinding of the forest dislocation from its cloud occurs for any stress level and the junction fails by the instability of the mobile dislocation branch BF (Fig. 1(b)), at sm ¼ 1:43. This value of the critical stress corresponds to the motion of the mobile dislocation through an array of fixed rigid obstacles spaced L apart. Obviously, increasing the strength of the obstacles (Q=A) leads to no further modification of the critical stress.
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
It is noted that thermal activation has no influence on the behavior of junctions. The energies involved are large enough such that thermal fluctuations (at temperatures in the vicinity of room temperature, i.e., in the range in which PLC is observed in Al–Mg alloys) would activate failure only when the applied stress is very close to the critical value. If the stress is significantly lower than the critical stress, the energy provided by thermal fluctuations is sufficient to slightly change the position of node E. It was verified that, if the system were activated up the energy landscape along the path it would normally take under stress, and were loaded to failure from that point, the critical stress would be identical to that reported in Fig. 3. Finally, various loading paths with nonzero sf were investigated in order to determine the failure envelope of this junction in the stress space. The results are shown in Fig. 5 for several values of Q=A and in the quadrant defined by sf 2 ðsm ; sm Þ. The failure surfaces are symmetric with respect to the lines sf ¼ sm and sf ¼ sm due to the symmetry of the structure. The failure envelope corresponding to Q=A ¼ 0 is identical to that derived by Dupuy and Fivel [27]. The failure mode is unzipping. As Q=A increases, the envelope moves to larger critical stresses. The shift correlates with the change in the failure mode as discussed in connection with Figs. 3 and 4. The envelope corresponding to Q=A P 0:3 is a square cen-
Fig. 5. Failure envelopes for the symmetric Lomer–Cottrell lock in the sm –sf plane and for various values of the normalized binding energy Q=A. Only one of the quadrants is shown since the envelope is symmetric with respect to the lines sm ¼ sf and sm ¼ sf . The curve corresponding to Q=A ¼ 0 is identical to that reported in [27]. As Q=A increases, the envelope shifts to larger stresses, while for Q=A > 0:3, the envelope becomes a square of side 2 1:43. The symbols indicate what mechanism controls the strength under the respective conditions. The filled circle indicates uncertainty over the failure mechanism under conditions represented by the second row in Fig. 4.
3453
tered at zero and of side 2 1:43, the failure mode being mobile dislocation instability (MDI) occurring under crossed-state conditions (Fig. 4 last row), irrespective of the loading path.
4. The negative strain-rate sensitivity mechanism The observation that the junction strength varies with the cluster size leads to a new mechanism for dynamic strain ageing. This mechanism is based on the idea that clusters form on forest dislocations rather than on the mobile dislocations, as assumed in the existing theories. Forest dislocations are generated during plastic deformation along with the mobile dislocations. Since they reside in planes subjected to a lower resolved shear stress, their residence time is considerably longer than that of mobile dislocations. Hence, their ageing time is sufficiently long for clusters to form even at relatively low temperatures (the lower temperature end of the PLC domain). Specifically, the average waiting time of a mobile dislocation at a forest obstacle may be computed as pffiffiffiffiffi tw ¼ X=_e, where X ¼ qm b= qf . The parameter X was evaluated by Kubin and Estrin [35] to be on the order of 3 103 or smaller, for realistic values of the forest and mobile dislocation densities, qm and qf . Considering the forest dislocations to be stationary, allows for the evaluation of the ageing time ta of a forest produced at time t after the inception of plastic deformation from ta þ t ¼ e=_e. Hence, the average ageing time of a (newly produced) forest is on the order of emax =2_e, where emax is the maximum strain reached in the respective test. The ratio of tw and ta may be evaluated from the above as tw =ta ¼ 2X=emax , which, for emax ¼ 0:3 (approximately equal to the failure strain of Al–Mg alloys), results to be smaller than 0.02 for all strain rates. The ageing time of forest dislocations is hence, in average, more than 50 times longer than the arrest time of mobile dislocations. The longer the residence time of forest dislocations, the larger the cluster they acquire and therefore the larger Q=A. For small Q=A (smaller than 0.3), the junction strength increases with the binding energy and it is immediately obvious that this leads to negative strainrate sensitivity: deformations at small strain rates imply longer ageing times for forests and therefore larger junction strength, while the opposite holds for deformations performed at large strain rates. The conclusion preserves for Q=A > 0:3. In order to quantify the proposed mechanism, a function is developed to represent the critical stress values presented in Fig. 5, for all Q=A and loading paths. This function is defined numerically and has the generic form Q c sm ¼ S ;n ; ð8Þ A
3454
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
where n ¼ sm =sf represents the loading path and takes values between )1 and 1. The current value of the binding energy Q=A is a function of the cluster size. Here, it is assumed that clustering occurs by lattice diffusion and therefore Friedel’s model modified by Louat’s correction [36] is selected to represent the clustering kinetics. The cluster size reaches a saturation value ðQ=AÞs at long ageing times ta . The kinetics is controlled by an intrinsic timescale t0 , which is associated with the rate of the diffusion process and depends on temperature. " 2=3 !# Q Q ta : ðta ; c; T Þ ¼ ðc; T Þ 1 exp A A s t0 ðT Þ ð9Þ
The dependence on temperature of t0 may be expressed by an Arrhenius law of the form t0 ¼ t00 exp½Ea1 =kT , as suggested in the literature. The saturation value depends on temperature and the average solute concentration, c. The cluster is expected to dissolve at high temperatures and therefore the saturation value is assumed to have an Arrhenius form: ðQ=AÞs ¼ ðQ=AÞ0 exp ðEa2 =kT Þ. Here, Ea2 is the activation energy for the cluster dissolution process and is expected to be on the order of the binding energy of a solute atom to the core of the respective type of dislocation. The dependence on the average solute concentration is unknown, however, data obtained by static ageing experiments on Al–Mg alloys [37] suggest that the cluster size decreases with decreasing c. The dependence of the saturated cluster size on both temperature and the average solute concentration needs to be calibrated from atomistic simulations. This work is ongoing for the Al–Mg binary system. Let us consider now a population of junctions of the type studied in Section 3. The forests were produced at various times during plastic deformation and their clusters have different sizes (Q=A). It will be approximated, for the sake of discussion, that the flow stress above the yield point (the strain-hardening component, Dsc m ) is fully associated with these junctions. The average shear stress applied in the mobile dislocation plane required to move a mobile dislocation through this set of obstacles may be approximated using the expression: 2R 31=q dq t q Q f _ e S ; n dt a 0 de A c 6 7 tta Dsm ðt; c; T ; nÞ ¼ 4 5 : ð10Þ R t dqf e_ dta 0 de tta
The average is made over the whole population and the density of forests with a given strength, S, is computed as the density of forests produced during the time interval dt (dt ¼ dta ) at a moment in the past (with respect to the current time, t) defined by the ageing time, ta . Therefore, the average over the population becomes an
average over time from the inception of plastic deformation. Evaluating the critical stress required to push a mobile dislocation through a set of obstacles of different strength is still a controversial issue. Here, the approximation developed by Hanson and Morris [38] is adopted and the quantity of interest is computed by making q ¼ 2 in Eq. (10). If it is assumed that, in addition to the variability associated with Q=A, there is also variability in terms of the loading path, a second average must be performed over all possible loading modes. Here, it is assumed that all loading paths are equally probable and therefore Eq. (10) becomes: 2R R 31=q dqf t 1 q Q _ S ; n dn dt e a 0 1 A de c 6 7 tta Dsm ðt; c; T Þ ¼ 4 5 : R t dqf 2 0 de e_ dta tta
ð11Þ This assumption is not critical. For example, considering that the loading paths are grouped in the neighborhood of the horizontal axis in Fig. 5 (the resolved stress in the forest plane is smaller than that in the mobile plane, or single slip conditions) leads to results similar to those discussed in the next section. The evolution of the density of forest dislocations, qf , with strain is represented by a monotonically increasing function with saturation, as suggested by the numerical results presented in [35]:
p qf ðeÞ ¼ qsat 1 exp e=e0 : ð12Þ f Here p and e0 are parameters controlling the rate of increase of the forest density with strain. The parameter p is always larger than 1. Further, the time dependence in Eq. (11) is transformed into strain dependence by imposing e ¼ e_ t and all parameters are rendered non-dimensional. Eq. (11) becomes
Dsc m ðe;c;T Þ ¼
Z
Z
i
h
2=3 ;n S 2 ðQ=AÞs 1 exp ðge=e0 Þ 0 1
p ð1 gÞp exp ð1 gÞp e=e0 dndg 1
1
1=2 Z 1
p p p dg 2 ð1 gÞ exp ð1 gÞ e=e0 ; 0
ð13Þ where g ¼ t=ta . The parameters in Eq. (13) are p and e0 , and ðQ=AÞs and e0 . The first two parameters are selected p ¼ 1:5 and e0 ¼ 1 which corresponds to a continuously rising curve of qf ðeÞ for e < 0:3. e 0:3 corresponds to the failure strain of AA5182-O Al–Mg alloy [5,39]. It was verified that considering other values for these parameters in the
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
range p < 3, e0 < 4 leads to no significant changes in model predictions (Section 5). The parameter e0 ¼ e_ t0 lumps the effects of strain rate and temperature (through the characteristic time of the diffusion process). It increases linearly with strain rate and increases with decreasing temperature. This is consistent with suggestions made in the literature according to which the strain rate and the temperature may be combined into a single constitutive parameter of the form e_ exp½Q=kT [3], where Q is the activation energy for DSA. An additional temperature dependence is included in ðQ=AÞs as discussed in connection with Eq. (9). The SRS is evaluated with Eqs. (1) and (13). This leads to the transient component of m only. The instantaneous component mi is always positive and is observed experimentally to increase with strain [8].
5. Macroscopic predictions and discussion The model described in the previous section was used to predict the SRS and the critical strain variation with strain, strain rate and temperature, and to derive a map of the negative SRS domain in the rate-temperature plane. The (transient) SRS parameter, mt , is evaluated based on Eqs. (1) and (13) and using a strain rate differential of e_ 1 ¼ 5_e2 . The variation of mt with strain results directly from Eq. (13) and is shown in Fig. 6 for several values of e0 ¼ e_ t0 and for ðQ=AÞs ¼ 0:3. Since the DSA mechanism described in the previous Section is built into the formulation, mt must be negative for all strains and strain rates. At low strain rates, mt is close to zero, which corresponds physically to the situation in which all forests are clustered and the clusters reach saturation. In these conditions varying the strain rate does not affect
Fig. 6. Variation of the transient SRS parameter with strain for various values of e0 ¼ e_ t0 and for ðQ=AÞs ¼ 0:3. mt < 0 at all strains. As the strain rate increases, mt decreases to a minimum (negative) and then returns in the neighborhood of zero, while remaining at negative values. The total SRS parameter, m, at large rates is positive since the absolute value of mt is small.
3455
the overall SRS. As the strain rate increases and the time constant associated with the imposed deformation (ta ) becomes comparable with the time constant of the bulk diffusion process (t0 ), mt decreases. Further increasing the rate leads to a return of mt in the vicinity of zero, which corresponds to no significant clustering (junction strength variation) occurring during ta . Hence, the total SRS parameter, m, is positive at large and small strain rates since mi > 0. This behavior is similar to that observed experimentally [8,9,39–42]. In most cases, the total SRS parameter m decreases from a positive value at the beginning of the plastic deformation and either stabilizes at a negative value [39–41], or rebounds toward the positive range at larger strains [6,16,42]. The actual shape of the curve depends on temperature and the base strain rate, feature captured by the model proposed here (see, e.g., the curves corresponding to e0 ¼ 0:1 and 0.01). In most solid solutions, even those that do not exhibit the PLC effect (m > 0), m is observed to decrease with strain (e.g. [41]). The larger the diffusivity of the solute, the more pronounced the strain dependence. This corresponds in the present model to the large e0 limit, i.e., to slow diffusion. Fig. 7 shows the variation of mt with logðe0 Þ for ðQ=AÞs ¼ 0:3 and for two plastic strains. It is observed that the strain has a relatively weak effect on the shape of the curve, a feature also observed in experiments [39]. The position of the minimum is dictated by the value of the saturation binding energy ðQ=AÞs . It is interesting to analyze how the function in Fig. 7 depends on ðQ=AÞs . Several curves corresponding to ðQ=AÞs ¼ 0:1, 0.3 and 0.5 and e ¼ 10% are shown in Fig. 8. As the binding energy increases, the transient SRS parameter becomes more negative, as expected. The effect is essentially insensitive to increasing Q=A above 0.3. The domain in which mt 0 expands, the expansion being more pronounced in the low temperature/high strain rate range. When Q=A > 0:3, the
Fig. 7. Variation of mt with e0 . mt is evaluated at two different strains.
3456
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
Fig. 8. Dependence of the mt vs. e0 curve on the saturation value of Q=A. As ðQ=AÞs decreases, mt decreases in magnitude and the range of e0 corresponding to the pronounced effect (mt 0) shifts to lower e0 values.
domain ceases to expand and shifts to higher e0 values. This is in general agreement with experimental observations on the effect of the average solute concentration on the domain of negative SRS and PLC. The dependence of the critical strain on e0 (strain rate and temperature) is shown in Fig. 9. The critical strains are evaluated from the data in Fig. 6 by considering two strain-independent instantaneous rate sensitivity parameters mi ¼ 0:02 and 0.1, respectively. The two curves corresponding to each mi show the lower and the upper critical strains and therefore bound the domain of negative rate sensitivity. In this log–log plot, the boundaries are straight lines of similar slope. Both critical strains increase with increasing the strain rate (decreasing temperature), i.e., the behavior corresponds to the ‘‘normal PLC.’’ The result is in agreement with the experimental data presented for Al–Mg in [43]. It is noted that using a different value for the instantaneous SRS parameter has no qualitative influence on this result,
Fig. 9. Variation of the critical strain ec with e0 for two values of the instantaneous SRS parameter, mi . The lines represent the boundaries of the negative SRS domain. The model reproduces the ‘‘normal’’ PLC behavior only.
however, increasing the positive instantaneous SRS parameter reduces the m < 0 domain, as expected. Here, the lower critical strain is associated with a critical forest dislocation density, rather than to a critical concentration of vacancies, as proposed in earlier models. This is supported by the observation that in heavily cold rolled materials the lower critical strain shifts to lower values. In some cases the serrated flow is observed immediately after yielding [2,39]. The model does not capture the ‘‘inverse PLC’’ behavior, in which the critical strain decreases with increasing strain rate [1,5]. As discussed in [5,44], this type of response appears to be rooted in larger scale phenomena associated with the evolution of large populations of defects. Finally, the activation energy for appearance/disappearance of the negative SRS is investigated. To this end, the influence of temperature on the boundaries of the m < 0 domain needs to be determined. As discussed above, the temperature affects the time constant for diffusion, t0 , and the saturation value of Q=A, ðQ=AÞs . Therefore, the negative SRS domain has to be plotted in the space of these two variables, ðQ=AÞs and e0 . The curves in Fig. 8 are used for this purpose and the upper and lower limits of the domain m < 0, logðe0 Þc , are determined for several ðQ=AÞs values and for the instantaneous SRS mi ¼ 0:02 and 0.1. The result is shown in Fig. 10 for two values of mi . The domain is upperbounded by a straight line described by the equation (for mi ¼ 0:02): e0c1 ¼ 36:4ðQ=AÞ1:6 s :
ð14Þ
Fig. 10. Map of the negative SRS range in the e0 –ðQ=AÞs space for two values of the instantaneous SRS parameter, mi . The two variables are Arrhenius dependent on temperature with different activation energies. The activation energy for t0 (controlling e0 ) is associated with the clustering process of forest dislocations, i.e., the bulk diffusion of solute, while the activation energy for ðQ=AÞs is associated with the dissolution of the cluster (binding energy of a solute atom to the forest). These two energies translate into multiple activation energies for the negative SRS phenomenon (Eq. (16)).
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
The lower bound is a curve schematically represented here by two straight lines with equations (for mi ¼ 0:02): 1:9
e0 c1 ¼ 7 105 ðQ=AÞs ¼ 1:7 10
2
ðQ=AÞ2:5 s
at low ðQ=AÞs ; e0 c1 at high ðQ=AÞs :
ð15Þ
The curves corresponding to mi ¼ 0:1 are described by functions with similar exponents. This result may be rearranged in terms of the strain rate and temperature by considering e0 ¼ e_ t0 and including the explicit, Arrhenius dependence on temperature of t0 and ðQ=AÞs (Section 4). This procedure leads to the boundaries of the negative SRS parameter m in the log e_ 1=T plane as: Ea1 þ 1:6Ea2 e_ c1 / exp ; kT Ea1 þ 2:5Ea2 ð16Þ ; e_ c2 / exp kT Ea1 1:9Ea2 : e_ c2 / exp kT The domain is shown schematically in Fig. 11 in which the slopes of the three lines correspond to Ea1 þ 1:6Ea2 , Ea1 þ 2:5Ea2 and Ea1 1:9Ea2 . Experimental studies on various material systems indicate that the domain of occurrence of the PLC effect is bounded in the log e_ 1=T diagram by approximately parallel straight lines [1,16], or by lines that intersect at some angle [5,45], being limited on the high temperature side. Both experimental observations may be explained based on Eq. (16). Obviously, if Ea2 is much smaller than Ea1 , the boundaries appear in this representation as parallel straight lines. Furthermore, the model suggests that multiple activation energies are associated with the boundaries of the PLC (m < 0) domain, which is in agreement with experimental observations. The activation energy Ea1 is associated with the bulk diffusion of solute and is expected to be larger than 0.5 eV in most cases. For vacancy-assisted bulk diffusion in Al–Mg, it is 1.2 and 0.51 eV in the absence and presence of an excess of vacancies. The second figure
3457
here corresponds to the migration of a solute (Mg) in an existing vacancy, while the first is the sum of the vacancy formation and solute migration energies [34]. Ea2 is associated with the dissolution of the cluster and is expected to be on the order of the binding energy of a solute to the core of the respective forest dislocation. The binding energies of Mg to the core of edge, 60° and screw dislocations in Al were estimated by atomistic simulations to be 0.126, 0.092 and 0.043 eV, respectively [22,34]. Ling and McCormick [8] report that the dependence on temperature of the critical strain in an Al–Mg–Si alloy is characterized by an activation energy (for inception of the PLC effect) of 0.59 eV, while Kubin et al. [46] infer an activation energy equivalent with Ea1 of 0.5 eV by fitting their model to experimental data for Al–5%Mg. In closure is should be mentioned that the relationship developed in Section 4 between the dislocation junction-scale processes and the macroscopic behavior should be regarded as a proof of concept. Considering the rather dramatic simplifications made in the procedure of averaging over the population of junctions, and considering that the fundamental data regarding the behavior of junctions is restricted to only one junction type, the agreement with the macroscopic observations cannot be quantitative. However, the model correctly reproduces several basic features of the macroscopic behavior.
6. Conclusions The new mechanism leading to negative strain-rate sensitivity in dilute solid solutions proposed here is based on the effect of solute clusters formed on forest dislocations on the strength of dislocation junctions. It is shown that the presence of a cluster on a forest dislocation modifies the strength of junctions formed by that dislocation with mobile unclustered dislocations. Since the relevant clustering process is that at forest dislocations, the size of the cluster depends on the residence time of the forest. Faster deformation allows for less clustering to occur which, in turn, is associated with lower junction strength. This mechanism is related to the macroscopic behavior through a simplified model. The model reproduces qualitatively a number of features observed experimentally.
Acknowledgements
Fig. 11. Schematic map of the negative SRS range in the strain rate inverse temperature space.
This work was supported, in part, by Alcoa Inc. and by the NSF through Grant CMS-0084987. The author thanks Ms. M.A. Soare for help with some of the calculations leading to the results presented in Fig. 5.
3458
R.C. Picu / Acta Materialia 52 (2004) 3447–3458
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Pink E, Grinberg A. Acta Metall 1982;30:2153. Balik J, Lukac P. Acta Metall Mater 1993;41:1447. Ling CP, McCormick PG. Acta Metall Mater 1993;41:3127. Kumar S. Scripta Metall Mater 1995;33:81. Abbadi M, Hahner P, Zeghloul A. Mater Sci Eng A 2002;337:194. Nortmann A, Schwink Ch. Acta Mater 1997;45:2043. Nortmann A, Schwink Ch. Acta Mater 1997;45:2051. Ling CP, McCormick PG. Acta Metall Mater 1990;38:2631. Kalk A, Nortmann A, Schwink Ch. Philos Mag A 1995;72:1239. Neuhauser H. Plastic instabilities and the deformation of metals. In: Walgref D, Ghoniem NM, editors. Patterns, defects and material instabilities. Kluwer; 1990. p. 241. Robinson JM. Int Mater Rev 1994;39:217. Estrin Y, Kubin LP. Spatial coupling and propagative plastic instabilities. In: Muhlhaus HB, editor. Continuum models for materials with microstructure. New York: Wiley; 1995. p. 395. Cottrell AH, Bilby BA. Proc Phys Soc London A 1949;62:49. McCormick PG. Acta Metall 1972;20:351. van den Beukel A. Phys Status Solidi A 1975;30:197. Mulford RA, Kocks UF. Acta Metall 1979;27:1125. Sleeswyk AW. Acta Metall 1958;6:598. Tabata T, Fujita H, Ueda N. Mater Sci Eng A 1980;44:81. Cottrell AH. Philos Mag 1953;44:31. Cuddy LJ, Leslie WC. Acta Metall 1972;20:1157. Schwaz RB, Funk LL. Acta Metall 1985;33:295. Picu RC, Zhang D. Acta Mater 2004;52:161. Hirth JP. J Appl Phys 1961;32:700.
[24] Saada G. Acta Metall 1960;8:841. [25] Wu TY, Bassani JL, Laird C. Proc Roy Soc London A 1991;435:1. [26] Franciosi P, Zaoui A. Acta Metall 1982;30:1627. [27] Dupuy L, Fivel MC. Acta Mater 2002;50:4873. [28] Madec R, Devincre B, Kubin L, Hoc T, Rodney D. Science 2003;301:1879. [29] De Wit G, Koehler JS. Phys Rev 1959;116:1113. [30] Schoeck G, Frydman R. Phys Status Solidi B 1972;53:661. [31] Wickham LK, Schwartz KW, Stolken JS. Phys Rev Lett 1999;83:4574. [32] Shenoy VB, Kukta RV, Phillips R. Phys Rev Lett 2000;84:1491. [33] Rodney D, Phillips R. Phys Rev Lett 1999;82:1704. [34] Zhang D, Picu RC. Model Simul Mater Sci Eng 2004;12:121. [35] Kubin LP, Estrin Y. Acta Metall Mater 1990;38:697. [36] Louat N. Scripta Metall 1981;15:1167. [37] Wycliffe P, Kocks UF, Embury JD. Scripta Metall 1980;14: 1349. [38] Hanson K, Morris JW. J Appl Phys 1975;46:2378. [39] Picu RC, Vincze G, Ozturk F, Gracio JJ, Barlat F, Maniatty AM. Mater Sci Eng. 2003 [submitted]. [40] van den Brink SH, van den Beukel A, McCormick PG. Phys Status Solidi A 1975;30:469. [41] McCormick PG. Scripta Metall 1978;12:197. [42] Schwink Ch, Nortmann A. Mater Sci Eng A 1997;234:1. [43] Guillot J, Grilhe T. Acta Metall 1972;20:291. [44] Hahner P, Rizzi E. Acta Mater 2003;51:3385. [45] Zeghloul A, Mliha-Touati M, Bakir S. J Phys III 1996;6:1467. [46] Kubin LP, Chihab K, Estrin Y. Acta Metall 1988;36:2707.