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Post-irradiation plastic deformation in bcc Fe grains investigated by means of 3D dislocation dynamics simulations
Accepted Manuscript Post-irradiation plastic deformation in bcc Fe grains investigated by means of 3D dislocation dynamics simulations K. Gururaj, C. Robertson, M. Fivel PII: DOI: Reference:
S0022-3115(15)00035-5 http://dx.doi.org/10.1016/j.jnucmat.2015.01.031 NUMA 48889
To appear in:
Journal of Nuclear Materials
Received Date: Accepted Date:
20 May 2014 13 January 2015
Please cite this article as: K. Gururaj, C. Robertson, M. Fivel, Post-irradiation plastic deformation in bcc Fe grains investigated by means of 3D dislocation dynamics simulations, Journal of Nuclear Materials (2015), doi: http:// dx.doi.org/10.1016/j.jnucmat.2015.01.031
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Post-irradiation plastic deformation in bcc Fe grains investigated by means of 3D dislocation dynamics simulations
K. Gururaj 1, C. Robertson2*, M. Fivel3
1
Materials Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, 603 102 Tamil Nadu, India 2
CEA, DEN, SRMA, Building-455, F91191 Gif-sur-Yvette, France
3
SIMaP-GPM2, Grenoble INP, CNRS/UJF, 101 Rue de la Physique, BP 46, 38402, Saint Martin d’Hères, France
*To whom all correspondence should be addressed: Telephone: (+33) 1 69 08 22 70, fax: (+33) 1 69 08 71 67, email: [email protected]
Abstract Post-irradiation tensile straining is investigated by means of three-dimensional dislocation dynamics simulations adapted to body centred cubic Fe. Namely, 1 µm Fe grains are strained at various temperatures in the 100K-300K range, in absence and in presence of radiation-induced defect dispersions. The defect-induced hardening is consistent with the disperse barrier effect up to 5×1021 m-3 loops and is weakly dependent on the straining temperature. The dislocation-loops interaction rate augments with the accumulated plastic strain, loop density and strength; while it is mainly independent of the number of active slip systems and thermally activated screw dislocation mobility. An additional, radiation-induced hardening mechanism known as dislocation «decoration» is also implemented and tested for comparison. Those results show that the plastic flow localization transition depends on the total yield point rise rather than on the lone, dispersed loop density. The simulation results are then rationalized through an original micro-mechanical model relating the grain-scale stress-strain behaviour to dislocation sub-structure formation and spreading. That model combines strain dependent and strain independent hardening mechanisms, which both contribute to the associated stress-strain response and plastic flow spreading.
1. Introduction Ferritic alloys are widely used as structural nuclear materials, undergoing various types of dose-dependent evolutions including: embrittlement, initial yield point rise, ductility loss, swelling, ductile-to-brittle transition temperature (DBTT) shift, etc [1-3]. Besides, plastic flow is no longer uniform beyond a certain critical dose: it concentrates into scarce band-like regions and combines with a pronounced hardening-softening response (plastic instability). The above-described radiation-induced evolutions are usually ascribed to the formation of nano-sized defect cluster populations, mostly in the form of interstitial dislocation loops [4,5].
Dislocation-loop interaction has been investigated in details in the past few years, using molecular dynamics calculations. In the meantime, dislocation dynamics (DD) simulations have been developed and adapted to treat plastic straining in presence of loop (or loop-like) defect clusters. Dislocation-loop interactions are usually treated using pre-assumed loop absorption mechanisms, either based on MD simulation or experimental observations [6-9]. The transition from homogeneous to heterogeneous post-irradiation deformation mode has been revisited more recently, using DD simulations without predefined dislocation-loop interaction rules [10]. Further investigation effort is needed however, towards gaining a comprehensive description of the post-irradiation straining behaviour under various material conditions (cold-worked, solution-annealed, precipitation hardened, etc) and straining temperatures.
Our goal in this work is to describe plastic strain development in presence of radiation defect populations using three-dimensional DD simulations. Both thermally activated glide and cross-slip are accounted for, in the whole DBTT range of pure Fe grains (100K-300K). Special care is taken to develop and implement simplified dislocation-loop interaction rules
compatible with the grain-scale plasticity spreading. The paper outline is as follows. The simulation constitutive rules and setups are presented in details in Section 2. Plastic deformation of un-irradiated grains (section 3.1) is first compared with irradiated grains (section 3.2). The effect of various influential parameters on the quantitative dislocation-loop interaction rate is then discussed in section 3.3, for further validation. Conditions for plastic flow localisation are examined in section 3.4 and further analysed in section 4, where an original and comprehensive micro-model is presented.
2. DD simulations adapted to alpha-Fe grains: model description 2.1 Dislocation mobility rules and cross-slip algorithm The present 3D Dislocation Dynamics (DD) code «TRIDIS» treats the dislocation lines as a succession of discrete orthogonal edge and screw segments [11]. Twelve {110}<111> slip systems are implemented, in consistence with the particular temperature range and bcc-Fe grain size investigated herein (see experimental evidence in [12,13] and Section 2.2). The edge segment velocity vedge = τ*b/B, where τ* is the effective resolved shear stress and B the phonon drag coefficient given in Table 1 below.
[Insert Table 1 about here]
The mobility rules used in DD simulations of bcc crystal plasticity are based on nucleation and spreading of isolated double-kinks, along the whole screw line span [14-16]. This approach clearly breaks down beyond intermediate temperatures however (T ≥ 100K) and can lead to computational instabilities, i.e. strong dislocation velocity changes due to small stress increments. A possible improvement consists in introducing a correction term accounting for the mean free path of kink pairs, as proposed in reference [17]. The specific features implemented in our DD model are the following. (i) In calculating the screw
dislocation velocities, we consider the average distance X along a screw dislocation swept out by a kink before its annihilation at another kink or if not, its immobilisation at the end of the screw segment. This new rule naturally accounts for the possibility to nucleate simultaneously several double-kinks along a screw dislocation of finite length. (ii) The formation of critical kink pairs is stress-dependent. Screw dislocation velocity corresponding to rules (i) and (ii) is expressed by:
vscrew = hXJ
(1)
where h is the distance between two consecutive Peierls barriers, X the average annihilation distance for a kink pair through a L long screw dislocation and J is the double-kink nucleation rate per unit length of screw dislocation. Both X and J terms are calculated as explained in reference [18]:
J=
πτ∗ µBh
exp −
∆
(2)
where µ is the shear modulus, kB the Boltzmann’s constant, T the test temperature and ∆G the (stress-dependent) kink pair activation energy barrier [14]. The latter quantity is calculated using Kock’s expression [15]:
τ∗ τ
∆τ = ∆ 1 −
(3)
where ∆Η0 is the thermal activation barrier at 0°K, while the other parameters: τp (the Peierls stress) p and q, allow fitting the evolutions of Fe crystal yield stress, with temperature. The average annihilation distance is then:
X=
X∞ L
(4)
X∞ + L
where X∞ is the average distance swept by a kink pair before annihilation with another kink pair along an infinitely long screw dislocation:
X∞ = 2 J
½
(5)
where kink velocity vk = vedge (see above). The screw dislocation velocities calculated using expression-(1) and Table-1 are consistent with the measured values, across the whole DBTT range [19].
Cross-slip is treated according to a specific procedure where the glide plane of each screw segment is updated at each time step, depending on its physical selection probability. The cross-slip algorithm is described in details elsewhere (see expressions (8) and (9) from [20]). It must be pointed out that the proposed dislocation mobility and cross-slip rules are semiempirical and not adapted to predict single (atomic scale) kink nucleation or cross-slip events, for which minimum energy or maximum entropy criteria are, in principle, better suited. Rather, the adopted approach is dedicated to the meso-scale typical of DD modelling, both in terms of space (minimum glide distance is 10b) and time (typical DD time step is 10-10
seconds). These rules proved to be very robust and yield simulation results consistent with observations [21,22].
2.2
Simulation
volume
geometry,
boundary
conditions,
irradiation
defect
implementation The typical simulation volume geometry (cubical grain) and size (1 µm) are shown in figure 1a. The outer grain interfaces are taken as impenetrable obstacles to dislocation motion, as if for highly disordered grain boundaries. This approximation is a fairly accurate description of grain boundary effect at least up to εp = 2×10-3, the maximum plastic strain level achieved herein. It allows for a realistic of intra-granular stress field description and accompanying dislocation sub-structure spreading [21,22]. Information on plastic displacement position and amplitude is presented in the form of «plastic strain maps», calculated using a similar postprocessing method as presented in [23]. In practice, six different planar arrays (one per each grain surface) including 200×200 calculation points are inserted inside, parallel and close to the elastically rigid grain boundary surfaces. The mesh-to-boundary standoff distance corresponds to several times the discrete lattice spacing (plastic strain εp ≠ 0 everywhere inside the grain).
[Insert Figure 1 about here]
Defect clusters in bcc-Fe alloys mainly develop in the form of interstitial loops, during irradiation at 300-500°C [24]. Back to room temperature, these loops can be trapped by impurities and act as immobile obstacles, for the mobile dislocations. Loop interaction with screw dislocations leads to loop absorption in the form of helical jogs. Unlike in fcc metals, helical jogs in bcc-Fe are unstable and tend to collapse by emitting a new prismatic loop, after unpinning completion [4]. Edge-loop interaction is comparatively weaker compared to
screw-loop interaction and in general, loops are neither damaged nor displaced, after interaction completion [5]. The MD observed dislocation-loop reaction scheme is replicated in our DD simulations by inserting immobile obstacle dispersions, within the simulated cell (see figure 1b).
For simplicity, the obstacles are implemented in the form of planar, square-shaped surfaces (facets), oriented normal to the (001) direction. The facets have the same size (and resistance) as the radiation-induced loops and for this reason; they are called «facet-loops» throughout the present paper. Each time a possible dislocation/facet collision is detected, the total local stress acting on the considered dislocation segment is compared with the critical facet strength. If the effective shear stress acting on the dislocation segment exceeds the facet strength, the segment is allowed to cut through the obstacle. The adopted facet-loop strength is calibrated based on MD simulation results [4,5] and depends on the character of the incoming dislocation segment (see also Section 3.2.1, for calibration procedure details). Facet-loops thus act as soft obstacles: mobile dislocations can go through a given facet (or not), depending on whether a local stress criterion is satisfied.
The facet-loop implementation has been validated by comparison with high (space and time) resolution DD simulations using explicit prismatic loops, without pre-defined interaction rules [25]. The facet-loop method is much more efficient because: i-it does not account for all the details of MD calculations, ii-preserves the interaction characteristics affecting the upper scale response (loop-sized super-jogs), iii- allows using a much larger simulation time step (10-10 s instead of 10-12 s, in «high resolution» DD simulations).
2.3 Simulated cases
Simulations are performed at 300K and 100K, in uni-axial tension along (001) direction. The applied tensile loading yields the same resolved shear stress (RSS) factor (0.41) in both the primary (101)<-1-11> and cross-slip (1-10)[-1-11] systems. The RSS factor in the second cross-slip system (011)<-1-11> is equal to zero, so the stress acting thereof is entirely due the internal dislocation structure development. The applied stress magnitude is feedback controlled in order to keep the plastic deformation rate at the constant, pre-selected level: 103 s-1. Stress history, dislocation density evolution and slip activity were recorded for analysis. Unless otherwise specified, the simulated grains initially contain one L = 500 nm long initial dislocation source, in the primary slip system. The initial source position is exactly the same in all the cases, with a view to facilitate the analysis of the results.
Reference simulations with non-irradiated (loop-free) grains are performed first. Then, four different mono-modal loops densities are introduced and tested: 5×1020, 1×1021, 2×1021 and 5×1021 m-3 (see figure 2). The selected loop size (loop diameter = 20 nm) and densities are representative of moderate irradiation doses (< 1 dpa), carried out below 400°C [26]. In actual irradiated grains, defect clusters have different sizes and are distributed at random positions in the crystal. In the present simulations however, loop-facet clusters are regularly positioned in the form of a three-dimensional array. This simple arrangement avoids strain localisation in (random) lower particle density regions and hence, comparison between different simulated systems will only depend on implemented loop size, strength and density (see section 3.2). The intermediate loop densities (and dose) investigated here represent a compromise between the computational load and the need to investigate relatively high dose effect on plastic deformation.
[Insert Figure 2 about here]
The simulation parameters (feedback loop control and time step) of the different runs are adjusted to obtain exactly the same strain rate conditions in each case, as explained in [20]. The resulting stress-strain evolutions then help in analysing the separate effect of loop dispersions, grain boundaries, dislocation-dislocation interactions or any combination of the above (see section 3). It must be pointed out; however, that the tensile response calculated here is not directly comparable to macroscopic stress-strain data on poly-crystalline specimens. In principle, such comparison would involve a whole set of different DD calculations, where the initial dislocation configurations, grain sizes, boundary conditions and applied stress tensor are systematically varied. This would need a separate investigation that is far beyond the scope of this paper.
3. Results 3.1 Tensile straining before irradiation: the loop-free grain case The familiar glide characteristics of bcc-Fe crystals are well captured by the dislocation mobility rules of Section-2.1, using Table-1 parameters. Firstly, the edge segments glide much faster than the screw segments: this is the well-known, thermally activated edge-screw mobility anisotropy. Secondly, pencil glide and wavy slip traces are observed here, instead of straight slip traces (see figure 3a and 3b). This results from screw dislocation cross-slip mechanism in bcc-Fe. Unlike in most fcc crystals, the dislocation core in bcc-Fe crystals is compact and for this reason, there is little energy barrier preventing dislocation hopping out of and back to its initial glide plane [20,27,28]. The dislocation densities increase linearly throughout the simulated time, reaching ~1013 m-2 at εp = 2×10-3. The same type of dislocation density evolution is observed in all our simulations, which directly relates to the adopted load control procedures (see section 2.3). A transient high stress regime is visible in figure 3c, where εp < 10-3. This initial behaviour is an artefact that can be totally suppressed by using a larger number of initial dislocation sources (> 10). It must be pointed out, however, that exactly the same steady state stress level is achieved regardless of the initial number of
dislocation sources. The adopted dislocation configuration is nevertheless kept as a single initial source, for simplicity and analysis purposes. It is worth noting that fast applied stress variations generally correspond to the spontaneous generation of new dislocation sources, though cross-kink wandering [27] or double cross-slip [29] mechanisms. Dislocation multiplication processes thus control the steady-state stress level achieved at the grainscale.
[Insert Figure 3 about here]
In the steady state regime (εp > 10-3), the population of mobile dislocations is much smaller than the population of immobile dislocations (estimated ratio 1:40), mostly stored in the form of tangles, near the grain boundaries. The stress fluctuations still present in the steady state regime (about 70 MPa) comes from the discrete character of plasticity in finite-sized, closed grains. Similar (though smaller) stress fluctuations are also observed in corresponding experiments, in polycrystalline low-alloy Fe [12]. The stress-strain curve presented in figure 3c exhibits a limited work hardening rate1 (≈ 20 GPa, for εp > 10-3), in agreement with available experimental evidence [12,30,31]. This behaviour is again ascribed to the abovedescribed cross-slip behaviour of bcc-Fe [32]. As a result, screw dislocations tend to follow the maximum resolved shear stress path through random cross-slip activation. This prevents the formation of pile-ups (as in fcc crystals) and gives rise to pencil glide and dislocation tangles.
In figure 4a, it is shown that the loaded slip systems 1 and 2 are simultaneously and almost equally active. Multiple (dual) slip conditions are compatible with homogenous plastic strain spreading, as confirmed by the plastic strain map of figure 3b, where no strain localisation is 1
The work hardening rate of 1 µm fcc grains is one order of magnitude larger: results to be published elsewhere.
present. The initial flow stress augmentation with decreasing temperature (see figure 4b: 100K) reflects the temperature dependent evolution of screw dislocation mobility, through quantity ∆G (see section 2.1). During straining at 100K, the dislocation population partition is practically the same as in figure 4a (not shown) and plastic strain also spreads homogeneously, across the whole grain.
[Insert Figure 4 about here]
3.2 Tensile straining after irradiation: the effect of facet-loop implementation 3.2.1 Micro-scale simulations: facet-loop strength calibration and reaction description Dislocation-loop interactions are emulated by implementing facet-loops of constant strength. The facet-loop strength is calibrated using the simple simulation setup as shown in figure 5, with a D = 20 nm loop diameter and 80 nm loop spacing and using an infinite facet strength. This tested configuration yields a critical Orowan shear stress of ∆τ = 95 MPa, in good agreement with the continuum theory [33]. MD calculations show that defect loop resistance to dislocation motion (the "loop strength") generally depends on the mobile dislocation character (see also section 2.2). For instance, the maximal loop strength is 2.26 times the Orowan stress for screw dislocations [4] and 0.7 times the Orowan stress for edge dislocations [5]. Accordingly, we set the loop-facet strength at ∆τ = 2.26×95 MPa×(L/D) = 860 MPa if the incoming dislocation segment is a screw. Finally, we set the facet strength to ∆τ = 0.70×95 MPa×(L/D) = 268 MPa, if the incoming dislocation segment is an edge.
The simple simulation setup adopted in this section also helps in analyzing the irradiated grain simulations, to be presented next. In the case of edge-loop interaction, dislocation glide is coplanar before and after the interaction completion. In the case of screw-loop interaction
however, facet-loops leads to the formation of non-coplanar super-jogs (see figure 5b, frames 3 and 4). The initial super-jogs size is comparable to the facet-loop diameter, before gliding along the screw line direction. In grain-scale simulations, the super-jogs migration can then carry on up to obstacles (grain boundaries, for example), where they accumulate and coarsen-up, by successive jog absorption [34,35].
[Insert Figure 5 about here]
3.2.2 Grain-scale simulation results: effect of facet-loop dispersion When the tensile loading is applied, the stress level keeps rising until the initial source is activated, by shear loop emission in the initial glide plane. Subsequent cross-slip activity spreads dislocations out, across the whole grain. Dislocation tangles then progressively develop, first from the grain boundaries and then, towards the grain interior (see figure 6a). Unlike in the loop-free grain case (see section 3.1), no initial transient regime is visible in figure 7a. This means that the generation of new dislocation source is facilitated by dislocation-loop interactions, throughout the simulated time. More specifically, loop-induced jogs form and coarsen-up until acting as new dislocation sources, as further discussed at the end of section 3.2.4. Local dislocation densities near the grain boundaries (within the tangles) are comparable to those observed in the loop-free grain case (compared figure 3a and 6a). This is consistent with the dose-independent work hardening behaviour as reported in [30,36].
[Insert Figure 6 about here]
The maximal loop-induced hardening obtained here is about ∆σ ≈ 150-200 MPa (see figure 7a for εp > 5×10-4). This level is consistent with the implemented loop density [30] and appears to be mostly independent of the straining temperature, in the 100K-300K range (compare figure 7a and 7b, for εp > 5×10-4). Interestingly, the implementation of up to 5×1021 loops/m3 has little effect on the meso-scale plastic deformation spreading across the grain. Namely, there is no evidence of loop-induced plastic strain localization (compare figures 6b and 3c). Correspondingly, the dislocation partition between the active slip systems is exactly the same as shown in figure 4 (dual slip) and temperature-independent as well.
[Insert Figure 7 about here]
3.2.3 Grain-scale
simulation results: factors influencing
the
dislocation-loop
interaction rate One of the long-term objectives of DD studies is to assist in the conception and validation of innovative continuum mechanics models [37,38]. Such models usually assume that irradiation-induced defects can be treated as local obstacles, namely by considering only their density and average strength, regardless of the defect type or the dislocation-loop reaction [39]. Actually, many interactions involve loop absorption by the mobile dislocation and for this reason; the residual loop population is expected to decrease with the accumulated number of dislocation-loop interactions [4,5,35]. Simulation inputs affecting the dislocation-loop interaction evolutions are examined in this section, using the procedure as explained in figure 8, with a view to validate the present DD simulation results.
[Insert Figure 8 about here]
The interaction number presented in figure 9a include edge-loop and screw-loop interactions altogether. It is found that the (maximal) number of interactions increases with the accumulated plastic strain, the dose (see figure 9a) and the loop strength (see figure 9b). The number of interactions depends very little on the straining temperature, the nature and number of active slip systems. The present counting results have been implemented in continuum-scale analytical models [40]. The corresponding stress-strain behaviour is consistent with the well-known experimental trends. The presented results and description remains valid as long as there is no strain localisation of the plastic flow, i.e. as long as the irradiation dose is low enough. Plastic flow localization conditions are examined in details, in the next section.
[Insert Figure 9 about here]
3.2.4 Grain-scale simulation results: plastic flow localisation transition The effect of loop density on post-irradiation deformation mechanism and stress-strain response are well-documented [41-44]. That experimentally-observed dependence has been recently analysed using 3D DD simulations [10], predicting a transition from homogeneous deformation (for loop densities up to 8.15×1020 m-3) to single-slip (for loop densities above 8.15×1021 m-3), due to 10 nm prismatic loop dispersions. Interestingly, they found that strain localisation is associated with plastic imbalance, where slip activity is suppressed in all but one slip system. The maximum (dispersed) loop density tested in the present paper is 5×1021 m-3, i.e. well below the loop density needed to initiate the strain localisation transition (8.15×1021 m-3). In other words, the absence of strain localisation visible in figure 6 seems consistent with [10] (see also Section 4, for detailed explanations). Classically, it is thought that the loop density affect the mechanical response of irradiated crystals through the disperse barrier hardening effect [45]. This model however neglects the fact that defect
atmospheres develop naturally around grown-in dislocations, as part of the irradiation process.
Dislocation decoration causes a yield stress increase that can be explained in terms dislocation unlocking from the decorating defects [46-50]. The decorating defect density and corresponding hardening effect naturally augments with the dose [49-51]. For simplicity, it can be assumed that heavily decorated lines are completely sessile except for short sections, where the decorating defect density is lower (or zero) [50]. The overall decoration effect is thus equivalent to a shortening of the initial dislocation sources. In an attempt to test this idea, new DD runs have been performed using an initial dislocation source L = 120 nm (instead of L = 500 nm), representing undecorated dislocation arms of a longer, decorated dislocation. The tested source length L is in agreement with TEM observations, for loop densities close to saturation conditions [51,52].
It is important to note that the decoration-assisted mechanical response shown in Figure-10 is associated with the fixed strain rate loading conditions as specified in Section-2.1: dεp/dt = ρmvmb = constant, where ρm and vm are the mobile dislocation density and average dislocation velocity, respectively. If ρm decreases due to dislocation decoration then, keeping the keeping the initial plastic strain rate level constant requires that vm (thus, the applied stress level: see expression (1)-(5)) increases in the same proportion. In other words, the initial yield point level relates not only to disperse barrier hardening effect but also to the ability to generate new mobile dislocations. The dislocation exhaustion effect observed here occurs because the line tension stress contribution (τLT ∝ L-1) strongly opposes the applied stress. It is important to note that this hardening contribution is obtained without any change on dispersed loop number density [50]. The other important result is a deformation mode transition towards the strain localization: glide in the cross-slip system is inhibited:
deformation involves mostly one active slip system indeed (compare figure 10a to figure 4a). The onset of plastic flow localisation is generally associated with a pronounced hardeningsoftening response [30], as shown in figure 10b.
[Insert Figure 10 about here]
The dislocations then spread across the simulated grain into a 150-200 nm thick band-like structure (see figure 10a, rightmost insert). Plastic strain spreading across the grain is therefore significantly less pronounced than in figure 6b, for the same plastic deformation level. Detailed examination of the dislocation structures reveals that channelling involves super-jog formation and subsequent coarsening (as shown in figure 5b and [34,53]). Additional dislocation sources are formed and spread the channel whenever the locally available amount of stress becomes comparable to the critical stress2 ∆τ ≈ (µb)/l. In practice, this happens whenever the super-jog size l becomes several times larger than the loop diameter D (l = 170 nm for ∆τ = 110 MPa), i.e. compatible with the observed channel thickness. This means the dislocation-loop interaction rules of sections 2.1 and 2.2 yield a fairly accurate description of grain-scale evolutions of the dislocation structures, regardless of the dislocation-loop reaction details and subsequent evolutions before l ∼ D.
4 Discussion: plastic strain spreading prediction based on DD simulation results The strain localisation transition is assisted by decoration-induced hardening and can take place for relatively small defect densities (see section 3.2.4). It then strongly depends on the stress level achieved at the onset of plastic flow, as pointed out in [54]. This situation will be
2
This is the critical source activation stress as per the line tension model approximation.
further explored hereafter, using a comprehensive micro-mechanical model linking the stress-strain response to the dislocation sub-structure evolutions.
Dislocation sub-structure development in bcc crystals Let first consider a single dislocation source in a finite-sized fcc grain. Whenever the applied load exceeds the line tension and lattice friction stresses, a shear loop forms and grows up to the grain boundaries. As the applied stress keeps rising, another loop is emitted and keeps expanding until piling-up against the first one, in the same glide plane. The stress-strain relation associated with this simple 2D mechanism can be described by a close-form analytical expression, where the applied shear stress increment ∆τapp is proportional to shear strain increment ∆γp:
∆τ =
!
µ
" !#ν
$ ∆γ
(6)
where µ is the shear modulus, ν the Poisson coefficient, S a dimensionless parameter characterizing the pile-up (or grain) geometry (cubic, circular, semi-infinite, etc). The bracketed term can be taken as the strain hardening modulus of a single 2D pile-up. A simple, yet more realistic strained grain configuration corresponds to the activation of a set of N parallel sources, separated by a constant spacing λ. At equilibrium, the i-th pile-up withstands: 1-the applied stress ∆τapp, 2-the line tension self-stress -∆τLT and 3-the mutual interactions due to the N-1 other pile-ups. The shear stress due to the j-th pile-up standing at a distance x from i-th pile-up is described by:
∆τ%→& = −∆τ '() −*+
|-| .
(7)
where k(z) is a positive function of coordinate z parallel to active slip plane, x is the distance normal to the initial slip plane and l the pile-up length characterizing the number of shear loops emitted by the i-th dislocation source [18]. A given set of N parallel equilibrated pile-ups can be described through a set of N balance equations, accounting the above 3 stress contributions. Solving-up this set of equations yields the N following ∆τi stress increments:
∆τ! = ∆τ/ =
!
!012#
∆τ5 = ∆τ/#! =
λ
34
!#12#
!012#
∆τ
λ
34
λ
34
(8a)
∆τ
(8b)
It is readily seen that outer pile-ups 1 and N (expression-8a) sustain more load (and strain) than the other inner pile-ups (expression-8b). Then, replacing expression-(6) into expression(8) makes it:
∆γ! = ∆γ/ =
"!#ν
∆γ5 = ∆γ/#! =
6
!012#
"!#ν 6
!
λ
34
∆τ
!#12# 3
(9a)
λ
!012#
4
λ
34
∆τ
The total, grain-scale shear strain ∆γ = ∑/ &8! ∆γ& and after rearranging, we finally get:
(9b)
∆τ =
!012# 34 6 9 : ∆γ " !#ν 50/#5!#12# λ !
λ
34
(10)
where k = k(z = l) = constant and Dg the grain size. If the pile-up separation is small then
λ→0 and we finally obtain the simpler expression:
! !
∆τ ≈ /
µ
" !#ν
$ ∆γ
(11)
The effective strain hardening modulus in expression-(11) thus depends on the number of active glide planes N involved in the deformation process and hence, on the degree of strain localization. In other words, a small N value corresponds to strong plastic flow localisation and conversely. Unlike in fcc materials however, work hardening in bcc Fe (and its alloys) is moderate and mostly independent of temperature [23] and dose [30]. This situation is associated with the specific cross-slip behaviour of bcc crystals (see section 3.1), preventing the formation of extended pile-up structures. Expression (10) and (11) predictions can be evaluated by combining with a complementary approach, to be described in the next subsection.
Stress-strain development in bcc crystals The strain hardening behaviour of bcc-Fe grains is analysed assuming the total dislocation density ρt is the sum of the mobile dislocations ρm plus the immobile or stored dislocations ρs
(i.e. ρt = ρm + ρs). The following balance equations are based on information coming from insitu examinations of bcc crystals:
<ρ= ⁄<> = @ρ= A − Bρ5= A 5 − Cρ= A
(12)
<ρD ⁄<> = Cρ= A
(13)
where V is the mobile dislocation velocity, M, A and E are constants characterizing dislocation multiplication, dislocation pair annihilation and mobile dislocation exhaustion, respectively [12]. The mobile dislocation density at saturation means < ρ= ⁄<> = 0 so from expression-(12), we get: (M – E) – AρmV = 0. Combining this result with Orowan’s expression yields:
Fε⁄FG H
= ρ= A =
I#J K
(14)
Combining expressions (12) and (13) (see [12]) with expression-(14) yields:
< ρG ⁄<> =
I#J K
@−B
I#J K
$=C
I#J K
=
JFε⁄FG H
(15)
The total dislocation population evolution is finally obtained by integrating expression-(15):
J
ρG = ρ + H ε
(16)
where ρ0 the initial dislocation density (in m-2). The corresponding stress increase is given by the Taylor expression, i.e. it is proportional to the square root of the total scalar dislocation density:
∆τ = αµMNρG = αµMOρ +
5J H
∆γ
(17)
where ∆τapp is the accumulated applied stress past the yield point, α is a dimensionless interaction strength coefficient and µ the shear modulus (in MPa). Quantity E is expressed in m-1 and can be interpreted as the reciprocal mean free path of mobile dislocations. Expression-(17) indicates two distinct and possibly independent contributions to the work hardening rate. The first contribution is associated with coefficient α and is independent of the plastic strain level. The second contribution is deformation-dependent and relates to parameter E, which directly fixes the amount of internal stress generated by the developing dislocation structures (in bcc Fe: tangles).
Stress-strain evolution in bcc crystals in relation with dislocation tangle development It is now possible to analyze the plastic strain localization conditions by combining expression-(17) with expression-(11), for simplicity:
αMOρ +
5J H
!
∆γ = /
!
"!#ν
$ ∆γ
(18)
Maximal flow localisation corresponds to N = 1: all the straining is accommodated by a single channel, in the whole 1 µm3 simulated grain (as described in section 3.2.4). Solving expression-(18) for α using S = 0.66 (cubic grains), ∆γp = 10-3, E = 6.7×106 m-1 and ρ0 = 5×1013 m-2 yields α ≈ 0.8. The selected E and ρ0 quantities are based on TEM observations performed before [12] and after irradiation3 [50]. A comparable α value can be obtained by combining expression-(17) with the grain-size dependent expression-(10), taking λ = Dg/N and P = O4RSMTγ5 (estimation based on [12]). This α value is relatively large however (see expression-(19) and related comment) and obviously includes the contributions of different hardening mechanisms, affecting the initial yield point level. Replacing α = 0.8 back into expression-(17) yields an initial critical hardening level ∆τ ≈ 110 MPa, causing strain localisation. This level is consistent with the simulation results presented in figure 7a, since ∆σ = ∆τ/Schmid-factor = 110 MPa/0.41 = 268 MPa. A critical loop density ncrit corresponding to ∆τcrit = 110 MPa can be evaluated as if the hardening is entirely due to the dispersed barrier effect. In that case:
∆τ = ∆τUVW = α.XXD µM√ZD
(19)
where n is the loop density, D the loop diameter and αloops the interaction strength coefficient, depending on dislocation-loop reaction type and the presence of segregating alloying elements around the loops (αloops is then comprised between 0.2 and 0.5). Taking αloops = 0.3 (a typical value for dislocation-dislocation interaction) and D = 20 nm yields a critical loop density ncrit = 1.9×1022 loops/m3. This number is larger yet, comparable to the critical loop 3
E thus, the dislocation tangle characteristics are assumed to be dose-independent. This particular point would need a separate investigation.
density reported in [10]. The difference can be explained, in part, by the use of closed grain boundary conditions, instead of periodic boundary conditions. Mobile dislocations accumulate at grain boundaries, which later contribute to generate extended structures (tangles). The latter (locally) strengthen the internal stress field and cross-slip activity, which spread dislocations throughout the whole of the grain. In these conditions, a larger number of stronger (bigger) loops is needed to achieve plastic flow localisation. More generally, the assumptions and simplifications adopted in our study (facet-loops instead of dislocation loops, only one facet-loop diameter, regular instead of random loop positions, isotropic linear elasticity, etc) may change the stress level at which plastic strain localisation occurs, but not the general tendency to localize above a critically high hardening level. Expression-(18) regarded as a preliminary micro-model, that inherently accounts for the initial material conditions, loop population and straining temperature, through a limited set of measurable quantities (α, ρ0 and E).
5. Summary Tensile straining of 1 µm Fe grains is simulated at 100K and 300K, using 3D dislocation dynamics calculations. Thermally activated screw dislocation motion and cross-slip rules were developed and implemented, yielding pencil (wavy) glide and dislocation storage in the form of non-planar 3D tangles, with accumulated plastic strain.
In absence of irradiation-induced defects, it is shown that: -
flow stress variations with temperature reflect the corresponding evolutions of quantity ∆G/kBT used to calculate the screw dislocation velocity,
-
dislocation population partition between the different slip systems is independent of the straining temperature.
Irradiation-induced loops (or defects) are introduced in the form of immobile internal obstacles called facet-loops, whose interaction strength depends on the mobile dislocation type (edge or screw). Dislocation-loop interactions result in the formation of super-jogs, which coarsen-up with accumulated plastic deformation.
Systematic comparisons between grains containing various loop densities were performed. The obtained loop-induced hardening level (∆τ ≈ +80 MPa) is consistent with the disperse barrier effect up to 5×1021 m-3 loops and weakly dependent on straining temperature (100K300K range). The number of loops interacting with mobile dislocations is monitored throughout the simulated time, for modelling analysis and validation purposes. The results show that: -
the number of loop interactions increase with the accumulated plastic strain, loop strength (at least up to 300 MPa loop strength) and loop density.
-
the number of interacting loops is independent of the number of active slip systems and thermally activated screw dislocation mobility (100K-300K range).
The above conclusions hold true despite the simplified dislocation-loop interaction treatment adopted herein, neglecting the loop-induced internal stress field and the complex, MDobserved reaction mechanisms. Modelling the grain-scale response thus critically depends on the ability to capture the appropriate initial yield point level and the loop-sized jog evolutions (by multiple recombinations).
No plastic flow spreading transition (channelling) is observed up to 5×1021 loops.m-3, if radiation-induced hardening is entirely due to the dispersed barrier effect. Flow localisation takes place using shorter initial sources (and 5×1021 loop/m3), i.e. in presence of a significant decoration hardening contribution. Overall, a «loop-density versus irradiation-hardening» (n-
H) correlation can be drawn if the decoration effect is weak and fails; if the decoration effect is strong. Experimentally, the n-H correlation indeed pertains to doses below the onset saturation [41-44] and fails beyond saturation [45], i.e. where decoration effect builds up. In other words, the deformation mode transition possibly correlates with total hardening level (including both DBH and decoration contributions), rather than with the sole, dispersed loop density number n.
The simulation results are then analysed using an original micro-mechanical model, relating the stress-strain behaviour to dislocation arrangement formation and multiplication. That model accounts for deformation-dependent and deformation-independent parameters (measurable quantities: α, ρ0 and E) and apply to various initial materials conditions (grain size, prior deformation), damage densities and types (defect dispersion and decoration), straining temperatures, etc...
6. Acknowledgements This work is supported by an Eurofusion Grant, from the Radiation Effects Modelling and Experimental Validation (IREMEV) Work-Package; and the Materials Research Program RMATE of the Nuclear Energy Division of the French Atomic Energy Commission (CEA/DEN).
7. References
[1] M. Matijasevic, E. Lucon, A. Almazouzi, J. Nucl. Mater. 377 (2008) 101-108. [2] A. Kohyama, A. Hishinuma, D.S. Gelles, R.L. Klueh, W. Dietz, K. Ehrlich, J. Nucl. Mater. 233-237 (1996) 138-147. [3] N. Baluc, D.S. Gelles, S. Jitsukawa, A. Kimura, R.H. Klueh, G.R. Odette, B. van der Schaaf, Y. Jinnan, J. Nucl. Mater. 367-370 (2007) 33-41. [4] D. Terentyev, D.J. Bacon, Y.N. Osetsky, Phil. Mag. iFirst (2009) 1-15. [5] D. Terentyev, P. Grammatikopoulos, D.J. Bacon, Y.N. Osetsky, Acta Mater. 56 (2008) 5034-5046. [6] T. Diaz de la Rubia, H.M. Zbib, T. Khraishi, B.D. Wirth, M. Victoria, M.J. Caturla, Nature 406 (2000) 871-874. [7] N.M. Ghoniem, S.H. Tong, Phil. Mag. A 81 (2001) 2743-2764. [8] J. Huang, N.M. Ghoniem, Comp Mater. Sci. 23 (2002) 225-234. [9] T.A. Khraishi, H.M. Zbib, T. Diaz de la Rubia, M. Victoria, Metall Mater Trans B 33B (2002) 285. [10] A. Arsenlis, M. Rhee, G. Hommes, R. Cook, J. Marian, Acta Mater. 60 (2012) 37483757. [11] M. Verdier, M. Fivel, I. Groma, Mod. Sim. Mat. Sc. Eng. 6 (1998) 755-770. [12] C. Robertson, K. Obrtlik, B. Marini, J. Nucl. Mater. 366 (2007) 58-69. [13] K. Obrtlik, C.F. Robertson, B. Marini, J. Nucl. Mater. 342 (2005) 35-41. [14] D. Caillard, J-L Martin, (2003) Thermally Activated Mechanisms in Crystal Plasticity (Pergamon Press) Oxford.
[15] U.F. Kocks, A.S. Argon, M.F. Ashby, Prog Mater. Sci. 19 (1975) 1-288. [16] M. Libert, Doctoral (2007) Doctoral Thesis Manuscript Ecole Centrale de Paris SRMA/CEA/Saclay. [17] J.P. Hirth, J. Lothe, (1992) Theory of Dislocations (Krieger Malabar). [18] C. Robertson, K. Gururaj, J. Nucl. Mater. 415 (2011) 167-178. [19] N. Urabe, J. Weertman, Mat. Sci. Eng. 18 (1975) 41-49. [20] J. Chaussidon, C. Robertson, D. Rodney, M. Fivel, Acta Mater. 56 (2008) 5466-5476. [21] J. Chaussidon, C. Robertson, M. Fivel, B. Marini, Model. Simul. Mater. Sci. Eng. 18 (2010) 025003. [22] D. Garcia-Rodriguez (2011) Doctoral Thesis Manuscript, INP Grenoble. [23] C. Déprés, M. Fivel, Philosophical Magazine, Phil. Mag. A 94 No. 28 (2014) 3206-3214. [24] M.L. Jenkins, Z. Yao, M. Hernandez-Mayoral, M.A. Kirk, J. Nucl. Mater. 389 (2009) 197202. [25] K. Gururaj, C. Robertson, (2010) October 11-13 Communication to 2nd Asian Nuclear Prospect Conference Chennai India. [26] M. Hernandez-Mayoral, D. Gomew-Briceno, J. Nucl. Mater. 399 (2010) 146-153. [27] F. Louchet, H. Saka, Mater. Sci. and Eng. A352 (2003) 71-75. [28] B.W. Lagow, I.M. Robertson, M. Jouiad, D.H. Lassila, T.C. Lee, H.K. Birnbaum, Mater. Sci. Eng. A309-310 (2001) 445-450. [29] C. Déprés, C.F. Robertson, M.C. Fivel, Phil. Mag. A 84 (2004) 2257-2275. [30] K. Farrell, T. S. Byun, and N. Hashimoto, Mapping flow localization process in deformation of irradiated reactor structural alloys-final report, Nuclear Energy Research initiative Program No. MSF99-0072, ORNL/TM-2003/63.
[31] S.A. Maloy, M.R. James, G. Willcutt, W.F. Sommer, M. Sokolov, L.L. Snead, et al., J. Nucl. Mater. 296 (2001) 119-128. [32] J. Chaussidon, M. Fivel, D. Rodney, Acta Mater. 54 (2006) 3407-3416. [33] D.J. Bacon, U.F. Kocks, R.O. Scattergood, Phil. Mag. 28 (1973) 1241-1263. [34] T. Nogaret, D. Rodney, M. Fivel, C. Robertson, Problems in Atomic Science and Technology, 4 (2009) 97-108. [35] T. Nogaret, C. Robertson, D. Rodney, Phil. Mag. A 87 No. 6 (2007) 945-966. [36] T. S. Byun, N. Hashimoto, J. Nucl. Mater. 354 (2006) 123-130. [37] G. Monnet, B. Devincre, L. Vincent, (2011) Crystal plasticity constitutive law for irradiated RPV steel Deliverable D1-2.9 Euratom Project PERFORM60. [38] G. Monnet, L Vincent, B. Devincre, Acta Mater. 61 No. 16 (2013) 6178-6190. [39] G. Monnet, L. Vincent, B. Devincre, Crystal plasticity constitutive law for irradiated RPV steel, Deliverable D1-2.9, Euratom 7th PCRD, Project PERFORM60, November 2011. [40] C. Robertson, M. Fivel, Dislocation-defect interaction in alpha-Fe grain under postirradiation plastic straining: a 3D dislocation dynamics investigation. Deliverable D1.2-4-1, Euratom 7th PCRD, Project PERFORM60, December 2011. [41] M. Victoria, N. Baluc, C. Bailat, Y. Dai, M.I. Luppo, R. Schaeublin, J. Nucl. Mater. 276 (2000) 114-122. [42] SJ. Zinkle, BN. Singh, J. Nucl. Mater. 351 (2006) 269-284. [43] Y. Chen, P. Spaetig, M. Victoria, J. Nucl. Mater. 271-272 (1999) 128-132. [44] T. Byun, N. Hashimoto, Nucl. Eng. Technol. 38 (2006) 619-638. [45] C Pokor, Y Bréchet, P Dubuisson, J.-P Massoud, X Averty, J. Nucl. Mater. 326 No. 1 (2004) 30-37. [46] B.N. Singh, A.J.E. Foreman, H. Trinkaus, J. Nucl. Mater. 249 (1997) 103-115.
[47] H. Trinkaus, B.N. Singh, A.J.E. Foreman, J. Nucl. Mater. 249 (1997) 91-102. [48] H. Trinkaus, B.N. Singh, A.J.E. Foreman, J. Nucl. Mater. 251 (1997) 172-187. [49] C. Robertson (1998) Doctoral Thesis Manuscript, Université de Paris 6. [50] J.S. Robach, I.M. Robertson, B.D.Wirth, A. Arsenlis, Phil. Mag. 83 No. 8 (2003) 955-967. [51] C. Robertson, E. Meslin, Experimental analysis of the plastic behaviour of ion-irradiated bainitic RPV steel, Deliverable D1-2.2, Euratom 7th PCRD, Project PERFORM60, December 2010. [52] E. Meslin, D. Brimbal, Communication to Euratom 7th PCRD, Project Longlife, WP3, Saclay, France, 26-28 February 2013. [53] J. Sharp, Phil. Mag. 16 (1967) 77-96. [54] T. Nogaret, D. Rodney, M. Fivel, C. Robertson, J. Nucl. Mater. 380 (2008) 22-29.
Table and figure captions
Table 1. Simulation parameters describing bcc-Fe single crystal plasticity. Screw dislocation mobility depends on the activation energy ∆G, calculated using the ∆H0,τp, p and q values provided herewith. ∆H0 and τp are the thermal activation barrier and friction stress at 0°K whereas p and q are fitting parameters, following the yield stress evolution with temperature. The phonon drag coefficient B is used to calculate the edge dislocation mobility; the Burgers vector b is used to calculate the crystal lattice parameter; E and ν are the elastic constants; ∆t the typical discrete time step.
Figure 1. DD simulation setup and radiation-induced defect implementation. (a) Simulated space representing a 1 µm Fe-grain, including one initial dislocation source. b) Mobile dislocations interact with dispersed, planar obstacles called facet-loops.
Figure 2. DD simulation setup: the different alpha-Fe grain cases. a) Unirradiated loop-free grain case, (b) Irradiated alpha-Fe grain case containing 5×1020 loops/m3, c) 1021 loops/m3, d) 2×1021 loops/m3, e) 5×1021 loops/m3.
Figure 3. The loop-free grain evolutions under tensile loading conditions at 300K. a) The 3D dislocation structure at εp = 2×10-3 and b) corresponding 3D plastic strain mapping. c) Stressstrain response.
Figure 4. Dislocation density evolutions in a loop-free Fe grain. a) Symmetrical dual slip is clearly evidenced: dislocation density evolutions are nearly identical in slip system 1 and slip
system 2. Slip system 3 is mostly inactive. The slip system definitions are provided in the main text. b) Effect of temperature on the stress-train response. Curve : un-irradiated or loop-free Fe grain strained at 300K. Curve : un-irradiated or loop-free Fe grain strained at 100K.
Figure 5. Test DD simulations: interaction between one dislocation and 2 facet-loops. The calculations are performed under controlled plastic strain rate loading conditions. The loop diameter is D =20 nm and centre-to centre loop inter-spacing is 80 nm. The initial L = 150 nm screw segment is pinned at its extremities and belongs to the (101)[-1-11] slip system. a) Edge-loop interaction case. Frame 1: initial configuration. Frame 2: pinning at loops. Frame 3: edge arm breaking free by loop shearing, leaving 2 screw segments behind. b) Screw-loop interaction case. Frame 1: initial configuration. Frames 2 and 3: pinning at loops 1 and 2. Frame 4: break through loop 2 and super-jog formation at loop 1, due to local cross-slip activation. This jog size is comparable to the loop diameter. Frame 5: unpinning from loop 1, super-jog gliding and coarsening, during subsequent jog absorption. Frame 6: the initial screw arm is gliding free (out of sight), leaving 2 edge arms behind. Post-interaction debris are visible, near loop 1.
Figure 6. Irradiated grain evolutions under tensile loading conditions at 300K. (a) 3D dislocation and loop structures and (b) corresponding 3D plastic strain map.
Figure 7. Comparative stress-strain responses of unirradiated and irradiated grains under tensile loading conditions. Stress-strain response: a) at 300K, : loop-free grain, : grain including 5×1021 loops/m3, b) at 100K, : loop-free grain, : grain including 5×1021 loops/m3. The loop-induced hardening level (∼100 MPa) is mostly independent of the straining temperature.
Figure 8. Dislocation-loop interaction evolutions with plastic deformation. The facet-loops pictured here are those which have interacted at least one time with the mobile dislocations. In this example, the interaction number evolution associated with frames 1 through 6 is: 2, 3, 5, 14, 31, 40. The initial loop density is 5×1021 loops/m3. The grain boundaries are not displayed for clarity.
Figure 9. Factors affecting the dislocation-loop interaction rate during straining at 300K. a) Effect of various loop densities , , on the dislocation-loop interaction rate. b) Effect of variable loop strength , , , on the dislocation-loop interaction rate, for a fixed loop density of 5×1021 loops/m3.
Figure 10. Effect of initial dislocation source shortening on the stress-strain response. Loops density = 5×1021 loops/m3. a) Dislocation density evolutions in the different slip systems. The signature of strain localization is clearly evidenced: single slip in slip system 1 (compare with figure 4a). The corresponding dislocation structure (a single dislocation channel) is inserted for illustration. The facet-loops are not shown for clarity. b) Effect of the initial source length on the stress-strain response. Note the strong hardening-softening effect obtained in case : L = 120 nm.
Figure_1
Figure_2
Figure_3
Figure_4
Figure_5
Figure_6
Figure_7
Figure_8
Figure_9
Figure_10
Table 1
H0
p
(eV)
(MPa)
p
q
B Phonon drag coefficient
Burgers vector magnitude -10
-5
(10 Pa s)
0.76
360
0.593
1.223
10.5
b (10
2.5
m)
Young modulus
Poisson ratio
Time step
E
t
(GPa)
210
(s)
0.3
510-10
S0022-3115(15)00035-5 http://dx.doi.org/10.1016/j.jnucmat.2015.01.031 NUMA 48889
To appear in:
Journal of Nuclear Materials
Received Date: Accepted Date:
20 May 2014 13 January 2015
Please cite this article as: K. Gururaj, C. Robertson, M. Fivel, Post-irradiation plastic deformation in bcc Fe grains investigated by means of 3D dislocation dynamics simulations, Journal of Nuclear Materials (2015), doi: http:// dx.doi.org/10.1016/j.jnucmat.2015.01.031
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Post-irradiation plastic deformation in bcc Fe grains investigated by means of 3D dislocation dynamics simulations
K. Gururaj 1, C. Robertson2*, M. Fivel3
1
Materials Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, 603 102 Tamil Nadu, India 2
CEA, DEN, SRMA, Building-455, F91191 Gif-sur-Yvette, France
3
SIMaP-GPM2, Grenoble INP, CNRS/UJF, 101 Rue de la Physique, BP 46, 38402, Saint Martin d’Hères, France
*To whom all correspondence should be addressed: Telephone: (+33) 1 69 08 22 70, fax: (+33) 1 69 08 71 67, email: [email protected]
Abstract Post-irradiation tensile straining is investigated by means of three-dimensional dislocation dynamics simulations adapted to body centred cubic Fe. Namely, 1 µm Fe grains are strained at various temperatures in the 100K-300K range, in absence and in presence of radiation-induced defect dispersions. The defect-induced hardening is consistent with the disperse barrier effect up to 5×1021 m-3 loops and is weakly dependent on the straining temperature. The dislocation-loops interaction rate augments with the accumulated plastic strain, loop density and strength; while it is mainly independent of the number of active slip systems and thermally activated screw dislocation mobility. An additional, radiation-induced hardening mechanism known as dislocation «decoration» is also implemented and tested for comparison. Those results show that the plastic flow localization transition depends on the total yield point rise rather than on the lone, dispersed loop density. The simulation results are then rationalized through an original micro-mechanical model relating the grain-scale stress-strain behaviour to dislocation sub-structure formation and spreading. That model combines strain dependent and strain independent hardening mechanisms, which both contribute to the associated stress-strain response and plastic flow spreading.
1. Introduction Ferritic alloys are widely used as structural nuclear materials, undergoing various types of dose-dependent evolutions including: embrittlement, initial yield point rise, ductility loss, swelling, ductile-to-brittle transition temperature (DBTT) shift, etc [1-3]. Besides, plastic flow is no longer uniform beyond a certain critical dose: it concentrates into scarce band-like regions and combines with a pronounced hardening-softening response (plastic instability). The above-described radiation-induced evolutions are usually ascribed to the formation of nano-sized defect cluster populations, mostly in the form of interstitial dislocation loops [4,5].
Dislocation-loop interaction has been investigated in details in the past few years, using molecular dynamics calculations. In the meantime, dislocation dynamics (DD) simulations have been developed and adapted to treat plastic straining in presence of loop (or loop-like) defect clusters. Dislocation-loop interactions are usually treated using pre-assumed loop absorption mechanisms, either based on MD simulation or experimental observations [6-9]. The transition from homogeneous to heterogeneous post-irradiation deformation mode has been revisited more recently, using DD simulations without predefined dislocation-loop interaction rules [10]. Further investigation effort is needed however, towards gaining a comprehensive description of the post-irradiation straining behaviour under various material conditions (cold-worked, solution-annealed, precipitation hardened, etc) and straining temperatures.
Our goal in this work is to describe plastic strain development in presence of radiation defect populations using three-dimensional DD simulations. Both thermally activated glide and cross-slip are accounted for, in the whole DBTT range of pure Fe grains (100K-300K). Special care is taken to develop and implement simplified dislocation-loop interaction rules
compatible with the grain-scale plasticity spreading. The paper outline is as follows. The simulation constitutive rules and setups are presented in details in Section 2. Plastic deformation of un-irradiated grains (section 3.1) is first compared with irradiated grains (section 3.2). The effect of various influential parameters on the quantitative dislocation-loop interaction rate is then discussed in section 3.3, for further validation. Conditions for plastic flow localisation are examined in section 3.4 and further analysed in section 4, where an original and comprehensive micro-model is presented.
2. DD simulations adapted to alpha-Fe grains: model description 2.1 Dislocation mobility rules and cross-slip algorithm The present 3D Dislocation Dynamics (DD) code «TRIDIS» treats the dislocation lines as a succession of discrete orthogonal edge and screw segments [11]. Twelve {110}<111> slip systems are implemented, in consistence with the particular temperature range and bcc-Fe grain size investigated herein (see experimental evidence in [12,13] and Section 2.2). The edge segment velocity vedge = τ*b/B, where τ* is the effective resolved shear stress and B the phonon drag coefficient given in Table 1 below.
[Insert Table 1 about here]
The mobility rules used in DD simulations of bcc crystal plasticity are based on nucleation and spreading of isolated double-kinks, along the whole screw line span [14-16]. This approach clearly breaks down beyond intermediate temperatures however (T ≥ 100K) and can lead to computational instabilities, i.e. strong dislocation velocity changes due to small stress increments. A possible improvement consists in introducing a correction term accounting for the mean free path of kink pairs, as proposed in reference [17]. The specific features implemented in our DD model are the following. (i) In calculating the screw
dislocation velocities, we consider the average distance X along a screw dislocation swept out by a kink before its annihilation at another kink or if not, its immobilisation at the end of the screw segment. This new rule naturally accounts for the possibility to nucleate simultaneously several double-kinks along a screw dislocation of finite length. (ii) The formation of critical kink pairs is stress-dependent. Screw dislocation velocity corresponding to rules (i) and (ii) is expressed by:
vscrew = hXJ
(1)
where h is the distance between two consecutive Peierls barriers, X the average annihilation distance for a kink pair through a L long screw dislocation and J is the double-kink nucleation rate per unit length of screw dislocation. Both X and J terms are calculated as explained in reference [18]:
J=
πτ∗ µBh
exp −
∆
(2)
where µ is the shear modulus, kB the Boltzmann’s constant, T the test temperature and ∆G the (stress-dependent) kink pair activation energy barrier [14]. The latter quantity is calculated using Kock’s expression [15]:
τ∗ τ
∆τ = ∆ 1 −
(3)
where ∆Η0 is the thermal activation barrier at 0°K, while the other parameters: τp (the Peierls stress) p and q, allow fitting the evolutions of Fe crystal yield stress, with temperature. The average annihilation distance is then:
X=
X∞ L
(4)
X∞ + L
where X∞ is the average distance swept by a kink pair before annihilation with another kink pair along an infinitely long screw dislocation:
X∞ = 2 J
½
(5)
where kink velocity vk = vedge (see above). The screw dislocation velocities calculated using expression-(1) and Table-1 are consistent with the measured values, across the whole DBTT range [19].
Cross-slip is treated according to a specific procedure where the glide plane of each screw segment is updated at each time step, depending on its physical selection probability. The cross-slip algorithm is described in details elsewhere (see expressions (8) and (9) from [20]). It must be pointed out that the proposed dislocation mobility and cross-slip rules are semiempirical and not adapted to predict single (atomic scale) kink nucleation or cross-slip events, for which minimum energy or maximum entropy criteria are, in principle, better suited. Rather, the adopted approach is dedicated to the meso-scale typical of DD modelling, both in terms of space (minimum glide distance is 10b) and time (typical DD time step is 10-10
seconds). These rules proved to be very robust and yield simulation results consistent with observations [21,22].
2.2
Simulation
volume
geometry,
boundary
conditions,
irradiation
defect
implementation The typical simulation volume geometry (cubical grain) and size (1 µm) are shown in figure 1a. The outer grain interfaces are taken as impenetrable obstacles to dislocation motion, as if for highly disordered grain boundaries. This approximation is a fairly accurate description of grain boundary effect at least up to εp = 2×10-3, the maximum plastic strain level achieved herein. It allows for a realistic of intra-granular stress field description and accompanying dislocation sub-structure spreading [21,22]. Information on plastic displacement position and amplitude is presented in the form of «plastic strain maps», calculated using a similar postprocessing method as presented in [23]. In practice, six different planar arrays (one per each grain surface) including 200×200 calculation points are inserted inside, parallel and close to the elastically rigid grain boundary surfaces. The mesh-to-boundary standoff distance corresponds to several times the discrete lattice spacing (plastic strain εp ≠ 0 everywhere inside the grain).
[Insert Figure 1 about here]
Defect clusters in bcc-Fe alloys mainly develop in the form of interstitial loops, during irradiation at 300-500°C [24]. Back to room temperature, these loops can be trapped by impurities and act as immobile obstacles, for the mobile dislocations. Loop interaction with screw dislocations leads to loop absorption in the form of helical jogs. Unlike in fcc metals, helical jogs in bcc-Fe are unstable and tend to collapse by emitting a new prismatic loop, after unpinning completion [4]. Edge-loop interaction is comparatively weaker compared to
screw-loop interaction and in general, loops are neither damaged nor displaced, after interaction completion [5]. The MD observed dislocation-loop reaction scheme is replicated in our DD simulations by inserting immobile obstacle dispersions, within the simulated cell (see figure 1b).
For simplicity, the obstacles are implemented in the form of planar, square-shaped surfaces (facets), oriented normal to the (001) direction. The facets have the same size (and resistance) as the radiation-induced loops and for this reason; they are called «facet-loops» throughout the present paper. Each time a possible dislocation/facet collision is detected, the total local stress acting on the considered dislocation segment is compared with the critical facet strength. If the effective shear stress acting on the dislocation segment exceeds the facet strength, the segment is allowed to cut through the obstacle. The adopted facet-loop strength is calibrated based on MD simulation results [4,5] and depends on the character of the incoming dislocation segment (see also Section 3.2.1, for calibration procedure details). Facet-loops thus act as soft obstacles: mobile dislocations can go through a given facet (or not), depending on whether a local stress criterion is satisfied.
The facet-loop implementation has been validated by comparison with high (space and time) resolution DD simulations using explicit prismatic loops, without pre-defined interaction rules [25]. The facet-loop method is much more efficient because: i-it does not account for all the details of MD calculations, ii-preserves the interaction characteristics affecting the upper scale response (loop-sized super-jogs), iii- allows using a much larger simulation time step (10-10 s instead of 10-12 s, in «high resolution» DD simulations).
2.3 Simulated cases
Simulations are performed at 300K and 100K, in uni-axial tension along (001) direction. The applied tensile loading yields the same resolved shear stress (RSS) factor (0.41) in both the primary (101)<-1-11> and cross-slip (1-10)[-1-11] systems. The RSS factor in the second cross-slip system (011)<-1-11> is equal to zero, so the stress acting thereof is entirely due the internal dislocation structure development. The applied stress magnitude is feedback controlled in order to keep the plastic deformation rate at the constant, pre-selected level: 103 s-1. Stress history, dislocation density evolution and slip activity were recorded for analysis. Unless otherwise specified, the simulated grains initially contain one L = 500 nm long initial dislocation source, in the primary slip system. The initial source position is exactly the same in all the cases, with a view to facilitate the analysis of the results.
Reference simulations with non-irradiated (loop-free) grains are performed first. Then, four different mono-modal loops densities are introduced and tested: 5×1020, 1×1021, 2×1021 and 5×1021 m-3 (see figure 2). The selected loop size (loop diameter = 20 nm) and densities are representative of moderate irradiation doses (< 1 dpa), carried out below 400°C [26]. In actual irradiated grains, defect clusters have different sizes and are distributed at random positions in the crystal. In the present simulations however, loop-facet clusters are regularly positioned in the form of a three-dimensional array. This simple arrangement avoids strain localisation in (random) lower particle density regions and hence, comparison between different simulated systems will only depend on implemented loop size, strength and density (see section 3.2). The intermediate loop densities (and dose) investigated here represent a compromise between the computational load and the need to investigate relatively high dose effect on plastic deformation.
[Insert Figure 2 about here]
The simulation parameters (feedback loop control and time step) of the different runs are adjusted to obtain exactly the same strain rate conditions in each case, as explained in [20]. The resulting stress-strain evolutions then help in analysing the separate effect of loop dispersions, grain boundaries, dislocation-dislocation interactions or any combination of the above (see section 3). It must be pointed out; however, that the tensile response calculated here is not directly comparable to macroscopic stress-strain data on poly-crystalline specimens. In principle, such comparison would involve a whole set of different DD calculations, where the initial dislocation configurations, grain sizes, boundary conditions and applied stress tensor are systematically varied. This would need a separate investigation that is far beyond the scope of this paper.
3. Results 3.1 Tensile straining before irradiation: the loop-free grain case The familiar glide characteristics of bcc-Fe crystals are well captured by the dislocation mobility rules of Section-2.1, using Table-1 parameters. Firstly, the edge segments glide much faster than the screw segments: this is the well-known, thermally activated edge-screw mobility anisotropy. Secondly, pencil glide and wavy slip traces are observed here, instead of straight slip traces (see figure 3a and 3b). This results from screw dislocation cross-slip mechanism in bcc-Fe. Unlike in most fcc crystals, the dislocation core in bcc-Fe crystals is compact and for this reason, there is little energy barrier preventing dislocation hopping out of and back to its initial glide plane [20,27,28]. The dislocation densities increase linearly throughout the simulated time, reaching ~1013 m-2 at εp = 2×10-3. The same type of dislocation density evolution is observed in all our simulations, which directly relates to the adopted load control procedures (see section 2.3). A transient high stress regime is visible in figure 3c, where εp < 10-3. This initial behaviour is an artefact that can be totally suppressed by using a larger number of initial dislocation sources (> 10). It must be pointed out, however, that exactly the same steady state stress level is achieved regardless of the initial number of
dislocation sources. The adopted dislocation configuration is nevertheless kept as a single initial source, for simplicity and analysis purposes. It is worth noting that fast applied stress variations generally correspond to the spontaneous generation of new dislocation sources, though cross-kink wandering [27] or double cross-slip [29] mechanisms. Dislocation multiplication processes thus control the steady-state stress level achieved at the grainscale.
[Insert Figure 3 about here]
In the steady state regime (εp > 10-3), the population of mobile dislocations is much smaller than the population of immobile dislocations (estimated ratio 1:40), mostly stored in the form of tangles, near the grain boundaries. The stress fluctuations still present in the steady state regime (about 70 MPa) comes from the discrete character of plasticity in finite-sized, closed grains. Similar (though smaller) stress fluctuations are also observed in corresponding experiments, in polycrystalline low-alloy Fe [12]. The stress-strain curve presented in figure 3c exhibits a limited work hardening rate1 (≈ 20 GPa, for εp > 10-3), in agreement with available experimental evidence [12,30,31]. This behaviour is again ascribed to the abovedescribed cross-slip behaviour of bcc-Fe [32]. As a result, screw dislocations tend to follow the maximum resolved shear stress path through random cross-slip activation. This prevents the formation of pile-ups (as in fcc crystals) and gives rise to pencil glide and dislocation tangles.
In figure 4a, it is shown that the loaded slip systems 1 and 2 are simultaneously and almost equally active. Multiple (dual) slip conditions are compatible with homogenous plastic strain spreading, as confirmed by the plastic strain map of figure 3b, where no strain localisation is 1
The work hardening rate of 1 µm fcc grains is one order of magnitude larger: results to be published elsewhere.
present. The initial flow stress augmentation with decreasing temperature (see figure 4b: 100K) reflects the temperature dependent evolution of screw dislocation mobility, through quantity ∆G (see section 2.1). During straining at 100K, the dislocation population partition is practically the same as in figure 4a (not shown) and plastic strain also spreads homogeneously, across the whole grain.
[Insert Figure 4 about here]
3.2 Tensile straining after irradiation: the effect of facet-loop implementation 3.2.1 Micro-scale simulations: facet-loop strength calibration and reaction description Dislocation-loop interactions are emulated by implementing facet-loops of constant strength. The facet-loop strength is calibrated using the simple simulation setup as shown in figure 5, with a D = 20 nm loop diameter and 80 nm loop spacing and using an infinite facet strength. This tested configuration yields a critical Orowan shear stress of ∆τ = 95 MPa, in good agreement with the continuum theory [33]. MD calculations show that defect loop resistance to dislocation motion (the "loop strength") generally depends on the mobile dislocation character (see also section 2.2). For instance, the maximal loop strength is 2.26 times the Orowan stress for screw dislocations [4] and 0.7 times the Orowan stress for edge dislocations [5]. Accordingly, we set the loop-facet strength at ∆τ = 2.26×95 MPa×(L/D) = 860 MPa if the incoming dislocation segment is a screw. Finally, we set the facet strength to ∆τ = 0.70×95 MPa×(L/D) = 268 MPa, if the incoming dislocation segment is an edge.
The simple simulation setup adopted in this section also helps in analyzing the irradiated grain simulations, to be presented next. In the case of edge-loop interaction, dislocation glide is coplanar before and after the interaction completion. In the case of screw-loop interaction
however, facet-loops leads to the formation of non-coplanar super-jogs (see figure 5b, frames 3 and 4). The initial super-jogs size is comparable to the facet-loop diameter, before gliding along the screw line direction. In grain-scale simulations, the super-jogs migration can then carry on up to obstacles (grain boundaries, for example), where they accumulate and coarsen-up, by successive jog absorption [34,35].
[Insert Figure 5 about here]
3.2.2 Grain-scale simulation results: effect of facet-loop dispersion When the tensile loading is applied, the stress level keeps rising until the initial source is activated, by shear loop emission in the initial glide plane. Subsequent cross-slip activity spreads dislocations out, across the whole grain. Dislocation tangles then progressively develop, first from the grain boundaries and then, towards the grain interior (see figure 6a). Unlike in the loop-free grain case (see section 3.1), no initial transient regime is visible in figure 7a. This means that the generation of new dislocation source is facilitated by dislocation-loop interactions, throughout the simulated time. More specifically, loop-induced jogs form and coarsen-up until acting as new dislocation sources, as further discussed at the end of section 3.2.4. Local dislocation densities near the grain boundaries (within the tangles) are comparable to those observed in the loop-free grain case (compared figure 3a and 6a). This is consistent with the dose-independent work hardening behaviour as reported in [30,36].
[Insert Figure 6 about here]
The maximal loop-induced hardening obtained here is about ∆σ ≈ 150-200 MPa (see figure 7a for εp > 5×10-4). This level is consistent with the implemented loop density [30] and appears to be mostly independent of the straining temperature, in the 100K-300K range (compare figure 7a and 7b, for εp > 5×10-4). Interestingly, the implementation of up to 5×1021 loops/m3 has little effect on the meso-scale plastic deformation spreading across the grain. Namely, there is no evidence of loop-induced plastic strain localization (compare figures 6b and 3c). Correspondingly, the dislocation partition between the active slip systems is exactly the same as shown in figure 4 (dual slip) and temperature-independent as well.
[Insert Figure 7 about here]
3.2.3 Grain-scale
simulation results: factors influencing
the
dislocation-loop
interaction rate One of the long-term objectives of DD studies is to assist in the conception and validation of innovative continuum mechanics models [37,38]. Such models usually assume that irradiation-induced defects can be treated as local obstacles, namely by considering only their density and average strength, regardless of the defect type or the dislocation-loop reaction [39]. Actually, many interactions involve loop absorption by the mobile dislocation and for this reason; the residual loop population is expected to decrease with the accumulated number of dislocation-loop interactions [4,5,35]. Simulation inputs affecting the dislocation-loop interaction evolutions are examined in this section, using the procedure as explained in figure 8, with a view to validate the present DD simulation results.
[Insert Figure 8 about here]
The interaction number presented in figure 9a include edge-loop and screw-loop interactions altogether. It is found that the (maximal) number of interactions increases with the accumulated plastic strain, the dose (see figure 9a) and the loop strength (see figure 9b). The number of interactions depends very little on the straining temperature, the nature and number of active slip systems. The present counting results have been implemented in continuum-scale analytical models [40]. The corresponding stress-strain behaviour is consistent with the well-known experimental trends. The presented results and description remains valid as long as there is no strain localisation of the plastic flow, i.e. as long as the irradiation dose is low enough. Plastic flow localization conditions are examined in details, in the next section.
[Insert Figure 9 about here]
3.2.4 Grain-scale simulation results: plastic flow localisation transition The effect of loop density on post-irradiation deformation mechanism and stress-strain response are well-documented [41-44]. That experimentally-observed dependence has been recently analysed using 3D DD simulations [10], predicting a transition from homogeneous deformation (for loop densities up to 8.15×1020 m-3) to single-slip (for loop densities above 8.15×1021 m-3), due to 10 nm prismatic loop dispersions. Interestingly, they found that strain localisation is associated with plastic imbalance, where slip activity is suppressed in all but one slip system. The maximum (dispersed) loop density tested in the present paper is 5×1021 m-3, i.e. well below the loop density needed to initiate the strain localisation transition (8.15×1021 m-3). In other words, the absence of strain localisation visible in figure 6 seems consistent with [10] (see also Section 4, for detailed explanations). Classically, it is thought that the loop density affect the mechanical response of irradiated crystals through the disperse barrier hardening effect [45]. This model however neglects the fact that defect
atmospheres develop naturally around grown-in dislocations, as part of the irradiation process.
Dislocation decoration causes a yield stress increase that can be explained in terms dislocation unlocking from the decorating defects [46-50]. The decorating defect density and corresponding hardening effect naturally augments with the dose [49-51]. For simplicity, it can be assumed that heavily decorated lines are completely sessile except for short sections, where the decorating defect density is lower (or zero) [50]. The overall decoration effect is thus equivalent to a shortening of the initial dislocation sources. In an attempt to test this idea, new DD runs have been performed using an initial dislocation source L = 120 nm (instead of L = 500 nm), representing undecorated dislocation arms of a longer, decorated dislocation. The tested source length L is in agreement with TEM observations, for loop densities close to saturation conditions [51,52].
It is important to note that the decoration-assisted mechanical response shown in Figure-10 is associated with the fixed strain rate loading conditions as specified in Section-2.1: dεp/dt = ρmvmb = constant, where ρm and vm are the mobile dislocation density and average dislocation velocity, respectively. If ρm decreases due to dislocation decoration then, keeping the keeping the initial plastic strain rate level constant requires that vm (thus, the applied stress level: see expression (1)-(5)) increases in the same proportion. In other words, the initial yield point level relates not only to disperse barrier hardening effect but also to the ability to generate new mobile dislocations. The dislocation exhaustion effect observed here occurs because the line tension stress contribution (τLT ∝ L-1) strongly opposes the applied stress. It is important to note that this hardening contribution is obtained without any change on dispersed loop number density [50]. The other important result is a deformation mode transition towards the strain localization: glide in the cross-slip system is inhibited:
deformation involves mostly one active slip system indeed (compare figure 10a to figure 4a). The onset of plastic flow localisation is generally associated with a pronounced hardeningsoftening response [30], as shown in figure 10b.
[Insert Figure 10 about here]
The dislocations then spread across the simulated grain into a 150-200 nm thick band-like structure (see figure 10a, rightmost insert). Plastic strain spreading across the grain is therefore significantly less pronounced than in figure 6b, for the same plastic deformation level. Detailed examination of the dislocation structures reveals that channelling involves super-jog formation and subsequent coarsening (as shown in figure 5b and [34,53]). Additional dislocation sources are formed and spread the channel whenever the locally available amount of stress becomes comparable to the critical stress2 ∆τ ≈ (µb)/l. In practice, this happens whenever the super-jog size l becomes several times larger than the loop diameter D (l = 170 nm for ∆τ = 110 MPa), i.e. compatible with the observed channel thickness. This means the dislocation-loop interaction rules of sections 2.1 and 2.2 yield a fairly accurate description of grain-scale evolutions of the dislocation structures, regardless of the dislocation-loop reaction details and subsequent evolutions before l ∼ D.
4 Discussion: plastic strain spreading prediction based on DD simulation results The strain localisation transition is assisted by decoration-induced hardening and can take place for relatively small defect densities (see section 3.2.4). It then strongly depends on the stress level achieved at the onset of plastic flow, as pointed out in [54]. This situation will be
2
This is the critical source activation stress as per the line tension model approximation.
further explored hereafter, using a comprehensive micro-mechanical model linking the stress-strain response to the dislocation sub-structure evolutions.
Dislocation sub-structure development in bcc crystals Let first consider a single dislocation source in a finite-sized fcc grain. Whenever the applied load exceeds the line tension and lattice friction stresses, a shear loop forms and grows up to the grain boundaries. As the applied stress keeps rising, another loop is emitted and keeps expanding until piling-up against the first one, in the same glide plane. The stress-strain relation associated with this simple 2D mechanism can be described by a close-form analytical expression, where the applied shear stress increment ∆τapp is proportional to shear strain increment ∆γp:
∆τ =
!
µ
" !#ν
$ ∆γ
(6)
where µ is the shear modulus, ν the Poisson coefficient, S a dimensionless parameter characterizing the pile-up (or grain) geometry (cubic, circular, semi-infinite, etc). The bracketed term can be taken as the strain hardening modulus of a single 2D pile-up. A simple, yet more realistic strained grain configuration corresponds to the activation of a set of N parallel sources, separated by a constant spacing λ. At equilibrium, the i-th pile-up withstands: 1-the applied stress ∆τapp, 2-the line tension self-stress -∆τLT and 3-the mutual interactions due to the N-1 other pile-ups. The shear stress due to the j-th pile-up standing at a distance x from i-th pile-up is described by:
∆τ%→& = −∆τ '() −*+
|-| .
(7)
where k(z) is a positive function of coordinate z parallel to active slip plane, x is the distance normal to the initial slip plane and l the pile-up length characterizing the number of shear loops emitted by the i-th dislocation source [18]. A given set of N parallel equilibrated pile-ups can be described through a set of N balance equations, accounting the above 3 stress contributions. Solving-up this set of equations yields the N following ∆τi stress increments:
∆τ! = ∆τ/ =
!
!012#
∆τ5 = ∆τ/#! =
λ
34
!#12#
!012#
∆τ
λ
34
λ
34
(8a)
∆τ
(8b)
It is readily seen that outer pile-ups 1 and N (expression-8a) sustain more load (and strain) than the other inner pile-ups (expression-8b). Then, replacing expression-(6) into expression(8) makes it:
∆γ! = ∆γ/ =
"!#ν
∆γ5 = ∆γ/#! =
6
!012#
"!#ν 6
!
λ
34
∆τ
!#12# 3
(9a)
λ
!012#
4
λ
34
∆τ
The total, grain-scale shear strain ∆γ = ∑/ &8! ∆γ& and after rearranging, we finally get:
(9b)
∆τ =
!012# 34 6 9 : ∆γ " !#ν 50/#5!#12# λ !
λ
34
(10)
where k = k(z = l) = constant and Dg the grain size. If the pile-up separation is small then
λ→0 and we finally obtain the simpler expression:
! !
∆τ ≈ /
µ
" !#ν
$ ∆γ
(11)
The effective strain hardening modulus in expression-(11) thus depends on the number of active glide planes N involved in the deformation process and hence, on the degree of strain localization. In other words, a small N value corresponds to strong plastic flow localisation and conversely. Unlike in fcc materials however, work hardening in bcc Fe (and its alloys) is moderate and mostly independent of temperature [23] and dose [30]. This situation is associated with the specific cross-slip behaviour of bcc crystals (see section 3.1), preventing the formation of extended pile-up structures. Expression (10) and (11) predictions can be evaluated by combining with a complementary approach, to be described in the next subsection.
Stress-strain development in bcc crystals The strain hardening behaviour of bcc-Fe grains is analysed assuming the total dislocation density ρt is the sum of the mobile dislocations ρm plus the immobile or stored dislocations ρs
(i.e. ρt = ρm + ρs). The following balance equations are based on information coming from insitu examinations of bcc crystals:
<ρ= ⁄<> = @ρ= A − Bρ5= A 5 − Cρ= A
(12)
<ρD ⁄<> = Cρ= A
(13)
where V is the mobile dislocation velocity, M, A and E are constants characterizing dislocation multiplication, dislocation pair annihilation and mobile dislocation exhaustion, respectively [12]. The mobile dislocation density at saturation means < ρ= ⁄<> = 0 so from expression-(12), we get: (M – E) – AρmV = 0. Combining this result with Orowan’s expression yields:
Fε⁄FG H
= ρ= A =
I#J K
(14)
Combining expressions (12) and (13) (see [12]) with expression-(14) yields:
< ρG ⁄<> =
I#J K
@−B
I#J K
$=C
I#J K
=
JFε⁄FG H
(15)
The total dislocation population evolution is finally obtained by integrating expression-(15):
J
ρG = ρ + H ε
(16)
where ρ0 the initial dislocation density (in m-2). The corresponding stress increase is given by the Taylor expression, i.e. it is proportional to the square root of the total scalar dislocation density:
∆τ = αµMNρG = αµMOρ +
5J H
∆γ
(17)
where ∆τapp is the accumulated applied stress past the yield point, α is a dimensionless interaction strength coefficient and µ the shear modulus (in MPa). Quantity E is expressed in m-1 and can be interpreted as the reciprocal mean free path of mobile dislocations. Expression-(17) indicates two distinct and possibly independent contributions to the work hardening rate. The first contribution is associated with coefficient α and is independent of the plastic strain level. The second contribution is deformation-dependent and relates to parameter E, which directly fixes the amount of internal stress generated by the developing dislocation structures (in bcc Fe: tangles).
Stress-strain evolution in bcc crystals in relation with dislocation tangle development It is now possible to analyze the plastic strain localization conditions by combining expression-(17) with expression-(11), for simplicity:
αMOρ +
5J H
!
∆γ = /
!
"!#ν
$ ∆γ
(18)
Maximal flow localisation corresponds to N = 1: all the straining is accommodated by a single channel, in the whole 1 µm3 simulated grain (as described in section 3.2.4). Solving expression-(18) for α using S = 0.66 (cubic grains), ∆γp = 10-3, E = 6.7×106 m-1 and ρ0 = 5×1013 m-2 yields α ≈ 0.8. The selected E and ρ0 quantities are based on TEM observations performed before [12] and after irradiation3 [50]. A comparable α value can be obtained by combining expression-(17) with the grain-size dependent expression-(10), taking λ = Dg/N and P = O4RSMTγ5 (estimation based on [12]). This α value is relatively large however (see expression-(19) and related comment) and obviously includes the contributions of different hardening mechanisms, affecting the initial yield point level. Replacing α = 0.8 back into expression-(17) yields an initial critical hardening level ∆τ ≈ 110 MPa, causing strain localisation. This level is consistent with the simulation results presented in figure 7a, since ∆σ = ∆τ/Schmid-factor = 110 MPa/0.41 = 268 MPa. A critical loop density ncrit corresponding to ∆τcrit = 110 MPa can be evaluated as if the hardening is entirely due to the dispersed barrier effect. In that case:
∆τ = ∆τUVW = α.XXD µM√ZD
(19)
where n is the loop density, D the loop diameter and αloops the interaction strength coefficient, depending on dislocation-loop reaction type and the presence of segregating alloying elements around the loops (αloops is then comprised between 0.2 and 0.5). Taking αloops = 0.3 (a typical value for dislocation-dislocation interaction) and D = 20 nm yields a critical loop density ncrit = 1.9×1022 loops/m3. This number is larger yet, comparable to the critical loop 3
E thus, the dislocation tangle characteristics are assumed to be dose-independent. This particular point would need a separate investigation.
density reported in [10]. The difference can be explained, in part, by the use of closed grain boundary conditions, instead of periodic boundary conditions. Mobile dislocations accumulate at grain boundaries, which later contribute to generate extended structures (tangles). The latter (locally) strengthen the internal stress field and cross-slip activity, which spread dislocations throughout the whole of the grain. In these conditions, a larger number of stronger (bigger) loops is needed to achieve plastic flow localisation. More generally, the assumptions and simplifications adopted in our study (facet-loops instead of dislocation loops, only one facet-loop diameter, regular instead of random loop positions, isotropic linear elasticity, etc) may change the stress level at which plastic strain localisation occurs, but not the general tendency to localize above a critically high hardening level. Expression-(18) regarded as a preliminary micro-model, that inherently accounts for the initial material conditions, loop population and straining temperature, through a limited set of measurable quantities (α, ρ0 and E).
5. Summary Tensile straining of 1 µm Fe grains is simulated at 100K and 300K, using 3D dislocation dynamics calculations. Thermally activated screw dislocation motion and cross-slip rules were developed and implemented, yielding pencil (wavy) glide and dislocation storage in the form of non-planar 3D tangles, with accumulated plastic strain.
In absence of irradiation-induced defects, it is shown that: -
flow stress variations with temperature reflect the corresponding evolutions of quantity ∆G/kBT used to calculate the screw dislocation velocity,
-
dislocation population partition between the different slip systems is independent of the straining temperature.
Irradiation-induced loops (or defects) are introduced in the form of immobile internal obstacles called facet-loops, whose interaction strength depends on the mobile dislocation type (edge or screw). Dislocation-loop interactions result in the formation of super-jogs, which coarsen-up with accumulated plastic deformation.
Systematic comparisons between grains containing various loop densities were performed. The obtained loop-induced hardening level (∆τ ≈ +80 MPa) is consistent with the disperse barrier effect up to 5×1021 m-3 loops and weakly dependent on straining temperature (100K300K range). The number of loops interacting with mobile dislocations is monitored throughout the simulated time, for modelling analysis and validation purposes. The results show that: -
the number of loop interactions increase with the accumulated plastic strain, loop strength (at least up to 300 MPa loop strength) and loop density.
-
the number of interacting loops is independent of the number of active slip systems and thermally activated screw dislocation mobility (100K-300K range).
The above conclusions hold true despite the simplified dislocation-loop interaction treatment adopted herein, neglecting the loop-induced internal stress field and the complex, MDobserved reaction mechanisms. Modelling the grain-scale response thus critically depends on the ability to capture the appropriate initial yield point level and the loop-sized jog evolutions (by multiple recombinations).
No plastic flow spreading transition (channelling) is observed up to 5×1021 loops.m-3, if radiation-induced hardening is entirely due to the dispersed barrier effect. Flow localisation takes place using shorter initial sources (and 5×1021 loop/m3), i.e. in presence of a significant decoration hardening contribution. Overall, a «loop-density versus irradiation-hardening» (n-
H) correlation can be drawn if the decoration effect is weak and fails; if the decoration effect is strong. Experimentally, the n-H correlation indeed pertains to doses below the onset saturation [41-44] and fails beyond saturation [45], i.e. where decoration effect builds up. In other words, the deformation mode transition possibly correlates with total hardening level (including both DBH and decoration contributions), rather than with the sole, dispersed loop density number n.
The simulation results are then analysed using an original micro-mechanical model, relating the stress-strain behaviour to dislocation arrangement formation and multiplication. That model accounts for deformation-dependent and deformation-independent parameters (measurable quantities: α, ρ0 and E) and apply to various initial materials conditions (grain size, prior deformation), damage densities and types (defect dispersion and decoration), straining temperatures, etc...
6. Acknowledgements This work is supported by an Eurofusion Grant, from the Radiation Effects Modelling and Experimental Validation (IREMEV) Work-Package; and the Materials Research Program RMATE of the Nuclear Energy Division of the French Atomic Energy Commission (CEA/DEN).
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Table and figure captions
Table 1. Simulation parameters describing bcc-Fe single crystal plasticity. Screw dislocation mobility depends on the activation energy ∆G, calculated using the ∆H0,τp, p and q values provided herewith. ∆H0 and τp are the thermal activation barrier and friction stress at 0°K whereas p and q are fitting parameters, following the yield stress evolution with temperature. The phonon drag coefficient B is used to calculate the edge dislocation mobility; the Burgers vector b is used to calculate the crystal lattice parameter; E and ν are the elastic constants; ∆t the typical discrete time step.
Figure 1. DD simulation setup and radiation-induced defect implementation. (a) Simulated space representing a 1 µm Fe-grain, including one initial dislocation source. b) Mobile dislocations interact with dispersed, planar obstacles called facet-loops.
Figure 2. DD simulation setup: the different alpha-Fe grain cases. a) Unirradiated loop-free grain case, (b) Irradiated alpha-Fe grain case containing 5×1020 loops/m3, c) 1021 loops/m3, d) 2×1021 loops/m3, e) 5×1021 loops/m3.
Figure 3. The loop-free grain evolutions under tensile loading conditions at 300K. a) The 3D dislocation structure at εp = 2×10-3 and b) corresponding 3D plastic strain mapping. c) Stressstrain response.
Figure 4. Dislocation density evolutions in a loop-free Fe grain. a) Symmetrical dual slip is clearly evidenced: dislocation density evolutions are nearly identical in slip system 1 and slip
system 2. Slip system 3 is mostly inactive. The slip system definitions are provided in the main text. b) Effect of temperature on the stress-train response. Curve : un-irradiated or loop-free Fe grain strained at 300K. Curve : un-irradiated or loop-free Fe grain strained at 100K.
Figure 5. Test DD simulations: interaction between one dislocation and 2 facet-loops. The calculations are performed under controlled plastic strain rate loading conditions. The loop diameter is D =20 nm and centre-to centre loop inter-spacing is 80 nm. The initial L = 150 nm screw segment is pinned at its extremities and belongs to the (101)[-1-11] slip system. a) Edge-loop interaction case. Frame 1: initial configuration. Frame 2: pinning at loops. Frame 3: edge arm breaking free by loop shearing, leaving 2 screw segments behind. b) Screw-loop interaction case. Frame 1: initial configuration. Frames 2 and 3: pinning at loops 1 and 2. Frame 4: break through loop 2 and super-jog formation at loop 1, due to local cross-slip activation. This jog size is comparable to the loop diameter. Frame 5: unpinning from loop 1, super-jog gliding and coarsening, during subsequent jog absorption. Frame 6: the initial screw arm is gliding free (out of sight), leaving 2 edge arms behind. Post-interaction debris are visible, near loop 1.
Figure 6. Irradiated grain evolutions under tensile loading conditions at 300K. (a) 3D dislocation and loop structures and (b) corresponding 3D plastic strain map.
Figure 7. Comparative stress-strain responses of unirradiated and irradiated grains under tensile loading conditions. Stress-strain response: a) at 300K, : loop-free grain, : grain including 5×1021 loops/m3, b) at 100K, : loop-free grain, : grain including 5×1021 loops/m3. The loop-induced hardening level (∼100 MPa) is mostly independent of the straining temperature.
Figure 8. Dislocation-loop interaction evolutions with plastic deformation. The facet-loops pictured here are those which have interacted at least one time with the mobile dislocations. In this example, the interaction number evolution associated with frames 1 through 6 is: 2, 3, 5, 14, 31, 40. The initial loop density is 5×1021 loops/m3. The grain boundaries are not displayed for clarity.
Figure 9. Factors affecting the dislocation-loop interaction rate during straining at 300K. a) Effect of various loop densities , , on the dislocation-loop interaction rate. b) Effect of variable loop strength , , , on the dislocation-loop interaction rate, for a fixed loop density of 5×1021 loops/m3.
Figure 10. Effect of initial dislocation source shortening on the stress-strain response. Loops density = 5×1021 loops/m3. a) Dislocation density evolutions in the different slip systems. The signature of strain localization is clearly evidenced: single slip in slip system 1 (compare with figure 4a). The corresponding dislocation structure (a single dislocation channel) is inserted for illustration. The facet-loops are not shown for clarity. b) Effect of the initial source length on the stress-strain response. Note the strong hardening-softening effect obtained in case : L = 120 nm.
Figure_1
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Table 1
H0
p
(eV)
(MPa)
p
q
B Phonon drag coefficient
Burgers vector magnitude -10
-5
(10 Pa s)
0.76
360
0.593
1.223
10.5
b (10
2.5
m)
Young modulus
Poisson ratio
Time step
E
t
(GPa)
210
(s)
0.3
510-10