1.12.4.1 Inclusion-Like Obstacles

1.12.4.1.1 Temperature T=0K

Voids in bcc and fcc metals at T = 0 K and >0 K are probably the most widely simulated obstacles of this type. Most simulations were made with edge disloca — tions.10,25-34 A recent and detailed comparison of strengthening by voids in Fe and Cu is to be found in Osetsky and Bacon.34 Examples of stress-strain curves (t vs. e) when an edge dislocation encounters and overcomes voids in Fe and Cu at 0 K are pre­sented in Figures 2 and 3, respectively. The four distinct stages in t versus e for the process are described in Osetsky and Bacon10 and Bacon and Osetsky. The difference in behavior between the two metals is due to the difference in their dislocation core structure, that is, dissociation into Shockley partials in Cu but no splitting in Fe (for details see Osetsky and Bacon34).

Under static conditions, T = 0 K, voids are strong obstacles and at maximum stress, an edge dislocation in Fe bows out strongly between the obstacles, creat­ing parallel screw segments in the form of a dipole pinned at the void surface. A consequence of this is that the screw arms cross-slip in the final stage when the dislocation is released from the void surface and this results in dislocation climb (see Figure 4), thereby reducing the number of vacancies in the void and therefore its size. In contrast to this, a Shockley partial cannot cross-slip. Partials of the dissociated dislocation in Cu interact individually with small voids whose diameter, D, is less than the partial spacing (~2 nm), thereby reducing the obstacle strength. Stress drops are seen in the stress-strain curve in Figure 3. The first occurs when the leading partial breaks from the void; the step formed by this on the exit surface is a partial step 1/6(112) and the stress required is small. Breakaway of the trailing partial controls the critical stress tc. For voids with D larger than the partial spacing, the two partials leave the void together at the same stress. However, extended screw segments do not form and the dislo­cation does not climb in this process. Consequently, large voids in Cu are stronger obstacles than those of the same size in Fe, as can be seen in Figure 6 and the number of vacancies in the sheared void in Cu is unchanged.

Cu-precipitates in Fe have been studied exten­sively23,27-29 due to their importance in raising the yield stress of irradiated pressure vessels steels35 and

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Figure 2 Stress-strain dependence for dislocation-void interaction in Fe at 0 K with L = 41.4 nm. Values of D are indicated below the individual plots. From Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2010, 90, 945. With permission from Taylor and Francis Ltd. (http://www. informaworld. com).

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Figure 4 [111] projection of atoms in the dislocation core showing climb of a1/2(111){110} edge dislocation after breakaway from voids of different diameter in Fe at 0 K. Climb-up indicates absorption of vacancies. The dislocation slip plane intersects the voids along their equator. From Osetsky, Yu. N.; Bacon, D. J. J. Nucl. Mater. 2003, 323, 268. Copyright (2003) with permission from Elsevier.

the availability of suitable IAPs for the Fe-Cu sys­tem.36 These precipitates are coherent with the sur­rounding Fe when small, that is, they have the bcc structure rather than the equilibrium fcc structure of Cu. Thus, the mechanism of edge dislocation inter­action with small Cu precipitates is similar to that of voids in Fe. The elastic shear modulus, G, of bcc Cu is lower than that of the Fe matrix and the dislocation is attracted into the precipitate by a reduction in its strain energy. Stress is required to overcome the
attraction and to form a 1/2(111) step on the Fe-Cu interface. This is lower than tc for a void, how­ever, for which G is zero and the void surface energy relatively high. Thus, small precipitates (< 3 nm) are relatively weak obstacles and, though sheared, remain coherent with the bcc Fe matrix after dislocation breakaway. tc is insufficient to draw out screw seg­ments and the dislocation is released without climb.

The Cu in larger precipitates is unstable, however, and their structure is partially transformed toward

dislocation breakaway at 0 K. The figure on the right shows the dislocation line in [111] projection after breakaway; climb to the left/right indicates absorption of vacancies/atoms by the dislocation. From Bacon, D. J.; Osetsky, Yu. N. Philos. Mag. 2009, 89, 3333. With permission from Taylor and Francis Ltd. (http://www. informaworld. com).

Figure 6 Critical stress tc (in units Gb/L) versus the harmonic mean of D and L (unit b) for voids and Cu-precipitates in Fe and voids in Cu at 0 K.

(D~1 + L“V, b

the more stable fcc structure when penetrated by a dislocation at T = 0 K. This is demonstrated in Figure 5 by the projection of atom positions in four {110} atomic planes parallel to the slip plane near the equator of a 4 nm precipitate after dislocation break­away. In the bcc structure, the {110} planes have a twofold stacking sequence, as can be seen by the upright and inverted triangle symbols near the out­side of the precipitate, but atoms represented by circles are in a different sequence. Atoms away from the Fe-Cu interface are seen to have adopted a threefold sequence characteristic of the {111} planes in the fcc structure. This transformation of Cu struc­ture, first found in MS simulation of a screw disloca­tion penetrating a precipitate,37,38 increases the obstacle strength and results in a critical line shape that is close to those for voids of the same size.34 Under these conditions, a screw dipole is created
and effects associated with this, such as climb of the edge dislocation on breakaway described above for voids in Fe, are observed.23,27

The results above were obtained at T = 0 K by MS, in which the potential energy of the system is minimized to find the equilibrium arrangement of the atoms. The advantage of this modeling is that the results can be compared directly with continuum modeling ofdislocations in which the minimum elastic energy gives the equilibrium dislocation arrangement. An early and relevant example of this is provided by the linear elastic continuum modeling of edge and screw dislocations interacting with impenetrable Oro — wan particles39 and voids.40 By computing the equilib­rium shape of a dislocation moving under increasing stress through the periodic row of obstacles, as in the equivalent MS atomistic modeling, it was shown that the maximum stress fits the relationship

f~*~L

+ L—^ W

where G is the elastic shear modulus and D is an empirical constant; A equals 1 if the initial dislocation is pure edge and (1 — n) if pure screw, where n is Poisson’s ratio. Equation [1] holds for anisotropic elas­ticity if G and n are chosen appropriately for the slip system in question, that is, if Gb2/4я and Gb2/4яА are set equal to the prelogarithmic energy factor of screw and edge dislocations, respectively.39,40 The value of G obtained in this way is 64 GPa for (111){110} slip in Fe and 43 GPa for (110) {111} slip in Cu.41

The explanation for the D — and L-dependence of tc is that voids and impenetrable particles are ‘strong’ obstacles in that the dislocation segments at the obsta­cle surface are pulled into parallel, dipole alignment at tc by self-interaction.39,40 (Note that this shape would not be achieved at this stress in the line-tension approximation where self-stress effects are ignored.) For every obstacle, the forward force, tcbL, on the dislocation has to match the dipole tension, that is, energy per unit length, which is proportional to ln(D) when D ^L and ln(L) when L ^ D.39 Thus, tcbL correlates with Gb2ln(D—1 + L—1)—1. The correlation between tc obtained by the atomic-scale simulations above and the harmonic mean of D and L, as in eqn [1], is presented in Figure 6. A fairly good agreement can be seen across the size range down to about D < 2 nm for voids in Fe and 3-4 nm for the other obstacles. The explanation for this lies in the fact that in the atomic simulation, as in the earlier contin­uum modeling, obstacles with D > 2-3 nm are strong at T = 0 K and result in a dipole alignment at tc.

Smaller obstacles in Fe, for example, voids with D < 2 nm and Cu precipitates with D < 3 nm, are too weak to be treated by eqn [1]. Thus, the descriptions above and the data in Figure 6 demonstrate that the atomic — scale mechanisms that operate for small and large obstacles depend on their nature and are not pre­dicted by simple continuum treatments, such as the line-tension and modulus-difference approximations that form the basis of the Russell-Brown model of Cu — precipitate strengthening of Fe,42 often used in pre­dictions and treatment of experimental observations.

The importance of atomic-scale effects in inter­actions between an edge dislocation and voids and Cu-precipitates in Fe was recently stressed in a series of simulations with a variable geometry.43 In this study, obstacles were placed with their center at different distances from the dislocation slip plane. An example of the results for the case of 2 nm void at T = 0 K is presented in Figure 7. The surprising result is that a void with its center below the disloca­tion slip plane is still a strong obstacle and may increase its size after the dislocation breaks away. This can be seen in Figure 7, where a dislocation line climbs down absorbing atoms from the void surface. More details on larger voids, precipitates, and finite temperature effects can be found in Grammatikopoulos et al4