Enabling microstructural changes of FCC/BCC alloys in 2D dislocation dynamics
EEnabling microstructural changes of FCC/BCC alloys in 2D dislocationdynamics
Ahmet Ilker Topuz
Materials innovation institute (M2i), 2600 GA Delft, The NetherlandsE-mail address: [email protected], [email protected]: +31 (0)15 278 2535Fax: +31 (0)15 278 2591
Abstract
Dimension reduction procedure is the recipe to represent defects in two dimensional dislo-cation dynamics according to the changes in the geometrical properties of the defects triggeredby different conditions such as radiation, high temperature, or pressure. In the present study,this procedure is extended to incorporate further features related to the presence of defects witha special focus on face-centered cubic/body-centered cubic alloys used for diverse engineeringpurposes. In order to reflect the microstructural state of the alloy on the computational cellof two dimensional dislocation dynamics, the distribution of the multi-type defects over sliplines is implemented by using corresponding strength and line spacing for each type of defect.Additionally, a simple recursive incremental relation is set to count the loop accumulation onthe precipitates. In the case of continuous resistance against the motion of edge dislocations onthe slip lines, an expression of friction is introduced to see its contribution on the yield strength.Each new property is applied independently on a different material by using experimental infor-mation about defect properties and grain sizes under the condition of plain strain deformation:both constant and dynamically increasing obstacle strength for precipitate coarsening in prime-aged and heat-treated copper-chromium-zirconium, internal friction in tantalum-2.5tungsten,and mixed hardening due to the presence of precipitates and prismatic loops in irradiated oxidedispersion strengthened EUROFER with 0.3% yttria.
Keywords:
Precipitates; Mixed hardening; Dynamic defect strength; Internal friction; Dimensionreduction; Dislocation dynamics
Combination of one or more metals or non-metals is an age-old technique to improve the mechanicalproperties such as yield strength. In addition to the alloying, the tensile response of the materialsis optimized by the use of supplementary processes like annealing or quenching which result in theevolution of the microstructure. In extreme environments, the initial state of the materials maynot be maintained due the operating conditions, and the performance of the materials may varyaccording to the perturbation in the intrinsic properties. At this point, the design of materials forfusion or generation IV reactors becomes an industrial necessity in order to deal with technologicalproblems.Experimental investigations on various nuclear alloys including but not limited to copper al-loys [1–4], austenitic steels [5–9], and ferritic/martensitic steels [10–13] have been widely performedin order to understand the relation between the evolution of defects and the constitutive behav-ior. As reported by these experiments, precipitates are commonly detected in FCC/BCC al-loys [1, 3, 4, 6, 8, 11, 12] at different irradiation levels [3, 8, 12] and periods of heat treatment [4],and significant changes due to these processes in the geometrical properties of precipitates result1 a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n n the variation of mechanical properties. There exists also certain types of defects which are in-duced by irradiation in specific crystal structures. While the formation of stacking-fault tetrahedra(SFT) occurs in some face-centered cubic (FCC) materials [2, 3], dislocation loops are the majorirradiation-induced defects in body-centered cubic (BCC) materials [12, 14]. At elevated tempera-tures (e.g. >
573 K [14] or >
623 K [6]), the population of voids is significant and contributes to thechange of the tensile properties.When dislocation-barrier interaction is considered, precipitates and related mechanisms in thealloys have been studied in 3D dislocation dynamics (DD) [15–18]. The themes of some exemplaryworks dedicated for the simulations of precipitates in 3D DD are precipitation-induced strengthen-ing in a Zr-1% Nb alloy proposing a mixture law [15], interactions between dislocations and Y O particles in PM2000 single crystals [16], combination of Orowan mechanism and forest hardeningin reactor pressure vessel (RPV) steels [17], and utilization of impenetrable facets for incoherentoxide particles together with shear-able facets for irradiation induced sessile dislocation loops inoxide dispersion strengthened (ODS) materials [18]. Although 3D DD delivers an illuminating de-scription about the motion of dislocations, it is considered as an expensive method of simulation interms of computation time. Moreover, current limitations such as simulation volumes far below thegrain size of many engineering materials imply that 3D DD may be desirable for the cases where3D setup is indispensable.In the present study, additional features based on dimension reduction procedure (DRP) [19]in 2D DD framework [20] are introduced in order to reflect the changes in the microstructure ofFCC/BCC alloys. In section 2, a brief description for the content of dimension reduction is given.Then, in section 3, representation of multi-type defects, internal friction, and dynamically increasingobstacle strength is shown. Section 4 describes BCC slip systems according to the suggestedconfiguration stated in another study [21] for plain strain deformation and explains the calculationof effective Burgers vector magnitude [22] in the corresponding slip systems. Finally, in section 5,each new property is applied separately on a different alloy by using experimental informationabout the geometrical properties of defects under the condition of plain strain deformation: bothconstant and dynamically increasing obstacle strength for precipitate coarsening in prime-agedand heat-treated FCC CuCrZr [4], internal friction in BCC Ta-2.5W [23], and mixed hardening inirradiated BCC ODS EUROFER with 0.3% yttria [24, 25]. DRP [19] of defect properties in 2D DD [20] is the transition of 3D geometrical information to thebasis of slip lines by using the principles of fractional conservation [26, 27] and barrier hardeningformulations such as Dispersed Barrier Hardening (DBH) [28] model or Bacon-Kocks-Scattergood(BKS) model [29]. This procedure is performed by the application of two consecutive slicingoperations on the objects of a certain number, size and geometry as described in Fig. 1. Firstslicing yields an average 2D profile in terms of density and size and it permits the determination ofdefect strength assuming that 2D average spacing is preserved inside the geometrical cell. At the endof the latter slicing, all 3D objects are reduced to the segments as a result of intersections between2D objects and slip lines. In this manner, line density which is defined as the number of segmentsper slip line is obtained to represent defects in 2D DD. For practical reasons, in lieu of segments,points are distributed over all slip lines. Moreover, prismatic dislocation loops are considered asrepresentative segments on the lateral cross section of the material, and a special treatment thatcounts the number of interactions between a mesh of both parallel and perpendicular lines andsegments of prismatic loops is applied [19]. 2 ” ” ¦ ¦ ¦ ¦ Figure 1: Transition of 3D geometrical properties of defects into 1D information for slip lineson which dislocations ( ⊥ ) are generated from Frank-Read (FR) sources ( (cid:13) ), can glide and bepinned/depinned at obstacles ( • ).For each type of defect, DRP yields an ordered pair of line density and strength varying with3D size and density at every level of any process. Density and spacing formulations delivered byDRP [19] are summarized in Table 1 with respect to their geometrical classification. While sphericaldefects may be referred to the precipitates or voids, SFT are apparently the examples of tetrahedralobjects. Table 1: Density and spacing formulations according to DRP [19] Defect type 2D density 2D spacing Line density Line spacingSpherical π ρ R * / (cid:113) π ρ R π ρ R /π ρ R Tetrahedral √ ρ a † / (cid:113) √ ρ a √ ρ a √ /ρ a Prismatic loops ρ d ‡ / (cid:113) ρ d T § T § (cid:80) (cid:80) intersections (cid:80) mesh length (cid:18) T T (cid:80) (cid:80) intersections (cid:80) mesh length (cid:19) − Average radius of spheres in 3D. † Average edge of regular tetrahedra in 3D. ‡ Average size of prismatic loops in 3D. § Number of realizations.
Strength of the obstacles which determines pinning/depinning of edge dislocations is computedvia DBH or BKS formulations depending on the nature of the corresponding defect. Substitutionof the 2D spacing terms stated in Table 1 into these formulations yields the expressions shown inTable 2. Table 2: Strength formulations according to DRP [19]
Spherical Tetrahedral Prismatic loopsStrength
Aµb (cid:104) ln (cid:16) Dr (cid:17) + B (cid:105) (cid:113) π ρ R . µb (cid:113) √ ρ a . µb (cid:113) ρ d If the parameters that constitute BKS formulation are examined, A is a coefficient dependingon the character of the dislocation, A = 1 / π (1 − ν ) for screw dislocation and A = 1 / π for edgedislocation, ν is Poisson’s ratio, µ is the shear modulus, b is the Burgers vector magnitude, B is3 coefficient which is equal to 0.7 for precipitates and 1.52 for voids, r is the line energy cut-offradius defining the elastic dislocation core size selected as b , and D is relative diameter defined as D = ( D − + L − ) − where L is the 2D spacing between the discs of spherical defects, and D is the disc size taken as 2 R = 4 R/π since R = 2 R/π on average [19]. In DBH model, thebarrier strength coefficient, α , which is scaled in the interval of [0.11, 1] [6] is select as 0.67 and0.33 for SFT and prismatic dislocation loops, respectively. In agreement with the experimental observations, the microstructure of alloys may not consist ofonly one type of defect. Depending on the nature of the process and the external load, differentkinds of defects such as precipitates and dislocation loops may be present at the same time in thecrystalline materials. Simultaneous existence of multi-type defects in the computational cell is theaccurate description to show the condition of the material. From the perspective of DRP [19], eachdefect is characterized by a value for strength and line density on the slip line, and different levels ofthe triggering mechanisms such as irradiation or heat treatment are translated to a pair of strengthand line density according to the changes in the 3D geometrical properties of the defects. Definingthat L line = 1 /ρ line where L line is the line spacing between obstacles and ρ line is the line density,the distribution of multi-type obstacles is shown in Fig. 2. ´ ´ ´ Τ , L Τ , L ´´ : Primary obstacle: Secondary obstacle ¦¦ ¦ ¦ Figure 2: Representation of multi-type defects according to their strength τ obs and line spacing L line on slip lines.Precipitates may be considered as primary defects in alloys since their potency to change theyield point is generally superior in comparison with the other defects. In accordance with the crystalstructure, SFT or dislocation loops may behave as secondary defects. It is worth to mention thatthis representation has no limitation for the number of defect type unless the total fraction isextremely high, and it is practical to enable the distribution of the multi-type defects as well asthe sub-types of any category on slip lines.The combined effect of multi-type defects has been investigated in different studies [15,17,30–32],and the suggested resulting strength due to i different defects is generally expressed by using the4ollowing expression: τ obsres = (cid:34)(cid:88) i (cid:104) τ obs i (cid:105) p (cid:35) /p (1)In spite of the existence of various values suggested for p in the interval of [1,2] (e.g. p =1 [30]), p = 2 has been theoretically approved [32], and it is experimentally observed [31]. It should benoted that line density, being an additional parameter in DRP for 2D DD, is as effective as strengthin the case of single type defects [19], but it is not present in the mixture hardening rule statedin Eq. 1. Another form of mixture law is the concentration-weighted superposition of two defectstrengths: τ obsres = (cid:104) ( C ) q ( τ obs1 ) p + ( C ) q ( τ obs2 ) p (cid:105) /p (2)where C and C are the concentration factors. ( p, q ) = (1 , .
5) [30] and ( p, q ) = (1 ,
1) [33] are someof the proposed values for the pair of p and q . A 3D DD study [15] defines concentration factors byusing the spacing between each type of obstacles, and suggests the following concentration basedmixture law: τ obsres = l τ obs1 + l τ obs2 (cid:112) l + l (3)where l and l are 2D spacing between obstacles of each type. According to DRP, line spacing L line is associated with the concentration of the obstacles since line density determines the populationof the obstacles on the slip lines. Therefore, one of the goals in the present study is to check if thebehavior of yield strength under the condition of consecutive alignment described in Fig. 2 showssimilarities with the mixture law in Eq. 3 when the concentration factors are defined in terms ofline spacing L line . Internal friction in 2D DD is the continuous resistance against the motion of edge dislocations onthe slip lines. Obstacles are discrete objects on which dislocations pin/depin depending on thecomparison between the strength of the obstacles and the absolute value of resolved shear stress τ of corresponding dislocations; however, internal friction in 2D DD is an implicit form of resistancethat either sets dislocations immobile or influences the velocity of the dislocation by comparing theinternal friction with | τ | . The velocity term of an edge dislocation with a resolved shear stress τ affected by the internal friction denoted as τ f gets the following form: v ( τ, τ f ) = (cid:40) ( τ − sgn( τ ) τ f ) bB if | τ | > τ f | τ | ≤ τ f (4)where B is the drag coefficient and b is the Burgers vector magnitude. Whilst the internal frictionmay depend on some variables in the form of τ f = τ f ( x, y, t, T ), which means it may be spatiallychanging ( x, y ) or specific to any slip system, temporal ( t ) or temperature-dependent ( T ), presentstudy assumes that τ f is constant in the computation cell during any simulation for internal friction.The reference value of internal friction is aimed to be the Peierls stress (PS) which is defined asthe minimal stress to move a dislocation at 0 K [34]. An FCC lattice exerts a very weak resistanceagainst the motion of dislocations: the PS lies within the order of 10 − µ or less [35], where µ isthe shear modulus; in contrast, PS is 900 MPa ( ≈ . × − µ ) [36] for screw dislocations in BCCIron. Not only the PS value for the edge dislocation in the simulation material, but also valuesaround this PS are investigated to depict the general behavior of yield strength under the effect ofinternal friction that slows down or immobilizes the edge dislocations.5 .3 Dynamically increasing defect strength When a dislocation bypasses a row of particles, a so-called Orowan loop is produced after therelease of this dislocation as illustrated in Fig. 3. Orowan loops on the particles act to increasethe required resolved shear stress of the next dislocation [17]; hence, loops together with particlesbehave as a combined barrier against the motion of the dislocations.Figure 3: Orowan loops on the precipitates.In order to incorporate the effects of such a mechanism, a simple approach is applied using arecursive relation based on the number of dislocations released from precipitates. Specifying that n is the number of released dislocations (i.e. the number of accumulated Orowan loops on an obstacle)and ∆ τ def is the percentage increase of the defect strength due to each Orowan loop, the strengthof corresponding obstacle on which ( n + 1) th edge dislocation is about to pin is hypothetically: τ obs n +1 = τ obs n (1 + ∆ τ def ) n = 1 , , ... (5)Since these obstacles are precipitates, and BKS model [29] is used to compute the initial strengthof the precipitates, the relation between amplified strength due to the presence of n loops denotedas τ Inc n +1 against the ( n + 1) th interacting edge dislocation and initial strength τ BKS is expressed ina more generalized form: τ Inc n +1 = τ BKS (1 + ∆ τ def ) n n = 0 , , ... (6)After the parametrization of dynamically increasing defect strength in terms of the initial valueof the precipitate strength, the number of the released dislocations or accumulated loops, and avalue for percentage increase, it is necessary to define a level of defect strength to terminate thealgorithm. If there is such a value of saturation for strength, say τ Sat , there also exists a maximumnumber of loops N which determines the level of saturation according to Eq. (6): τ BKS (1 + ∆ τ def ) N = τ Sat (7)The progression of dynamically increasing precipitate strength is illustrated in Fig. 4. As it is hardto extract the value of the parameters in Eq. (7) from experiments, the sensitivity of the incrementvalue ∆ τ def is explored by solving Eq. (7) for N , assuming that the ratio τ Sat /τ BKS is known.6 ¦ ¦
I II N ¦¦ I. release ” Τ BKS (1 + DΤ def )II. release ” Τ BKS H + DΤ def L N. release ” Τ BKS H + DΤ def L N Saturation
Released dislocations Τ BKS ¦ ¦ : Precipitate: Frank-Read source Figure 4: Dynamically increasing defect strength.To summarize, the strength of point obstacle referred to a precipitate is increased after eachrelease of an edge dislocation from the corresponding barrier by imagining that an Orowan loopis formed, and this operation is terminated when the number of released dislocations reaches thesaturation level which is indicated by the maximum number N . The description of the plastic slip in a BCC crystal together with the suggested planes is originallytaken from another study [21] and it is given in Fig. 5. In this description, the following slip planesare included: (101), (121), and (¯12¯1) while the common slip direction in BCC is of (cid:104) (cid:105) type.7igure 5: Suggested [21] slip systems for BCC crystal configuration.The Burgers vector magnitude of any member from (cid:104) (cid:105) is: | b | = a √ a √
32 (8)where a is the size of the atomic unit cell. (cid:160) b (cid:164) (cid:160) b (cid:164)(cid:160) b (cid:164) (cid:160) b (cid:164) Figure 6: Slip directions together with their Burgers vector magnitude in 3D.One may determine the effective Burgers vector magnitude by projecting the corresponding8ector on the trace of effective slip plane [22]. In order to accomplish this operation, Fig. 6 isseparated into three planes. Considering that the edge of the cubic cell is a , (101) is a rectangularplane whose edges are a and √ a . a (cid:72) (cid:76) (cid:64) (cid:68)(cid:64) (cid:68) (cid:64) (cid:68)(cid:200) b (cid:200) (cid:200) b (cid:200) Figure 7: (101) plane with [¯1¯11] and [1¯1¯1] directions.The combination of equal slip on (101)[¯1¯11] and (101)[1¯1¯1] yields an effective slip on (101)[0¯10].The trace of the resulting slip direction is shown as the thick dashed line in Fig. 7. Since bothdirections provide equal slip, projection of one direction onto the trace of effective direction is enoughto calculate the effective Burgers vector magnitude. According to the geometry, cos([¯1¯11] ∠ [0¯10]) = √ /
3. Hence, effective Burgers vector magnitude on [0¯10] is | b |√ / | b | is the magnitude ofBurgers vector for [¯1¯11] and [1¯1¯1] directions. (cid:72) (cid:76) (cid:64) (cid:68)(cid:200) b (cid:200) (cid:72) (cid:76) (cid:64) (cid:68)(cid:200) b (cid:200) Figure 8: (121)[1¯11] and (¯12¯1)[¯1¯1¯1] slip systems.(121) and (¯12¯1) are the remaining isosceles triangular planes with two edges of a √ / √ a . Corresponding slip systems contribute individually to the global description.However, their contribution is similar since they are symmetrical around [101]. Fig. 8 shows theirgeometrical properties and Burgers vector direction. In both cases, Burgers vector direction in9D coincides with the trace of the effective slip plane shown as thick dashed lines; therefore, slipdirection and effective Burgers vector magnitude remain as identical.To conclude, [¯1¯11] and [1¯1¯1] on rectangular plane (101) result in an effective slip direction on(101)[0¯10] where the effective Burgers vector magnitude is | b |√ / | b | .These three slip systems in 2D are shown in Fig. 9 and enumerated asfollows:(i) is (101)[0¯10] with effective Burgers vector magnitude | b |√ / | b | ;(iii) is (¯12¯1)[¯1¯1¯1] with effective Burgers vector magnitude | b | . H ii L xy ¦ ¦ ¦ n n n ¦ ¦ ¦ H iii L H i L Figure 9: Three slip systems with their inclination angles.
The first material under investigation is CuCrZr [4, 37] which is considered for the utilization inthe first wall and divertor components of ITER [4]. This material is subjected to certain processessuch as prime aging (PA) and heat treatment (HT) for different periods of time in order to realizethe modification in the precipitate properties. The reason of this operation is mentioned [4] as theinability for the inhibition of dislocation motion during plastic deformation due to small size ofprecipitates [4], and annealing is applied on PA CuCrZr in order to coarsen the precipitates.Simulations are performed in a single grain of CuCrZr of size 30 µ m [37] by using three cases,which are PA without HT, PA and HT for one hour (PA+873K/1h), and PA and HT for four hours10PA+873K/4h) at the temperature of 873 K. Common input parameters such as elastic modulus E and inter-planar distance s [19], which are used in all simulations of CuCrZr are given in Table 3.Table 3: Common input in all CuCrZr simulations E (GPa) ν b (nm) Strain rate (s − ) s Grain size ( µ m) FR density ( µ m − )123 0.34 0.25 2500 200 b
30 1Initially, DRP is applied to the precipitates of CuCrZr by using the size (i.e. diameter) anddensity data delivered from an experimental study [4]. Table 4 presents the experimental datatogether with the line density and strength values obtained via dimension reduction.Table 4: Precipitate properties in CuCrZrProcess Size (nm) 3D density ( µ m − ) Line density ( µ m − ) Strength (MPa)PA 2.2 2 . × . × . × PAPA + (cid:144) + (cid:144) ¶ Σ @ M P a D Figure 10: Precipitate coarsening in single grain CuCrZr using constant defect strength.11t is seen that the yield strength of the material is led by properties of the precipitates andit is lowered by the coarsening due to HT. According to Fig. 10, PA+873K/4h shows a moresignificant yield drop in comparison with PA+873K/1h. Whereas the value of the line densityin PA+873K/1h indicates an inconsequential change with respect to PA, PA+873K/1h has a re-markably lower number of defect points than both PA and PA+873K/1h cases. The strengthof precipitates is reduced in PA+873K/1h as well as PA+873K/4h, but the amount of decline ishigher in PA+873K/4h. The stress-strain curves shown in the experimental study [4] differs fromFig. 10 in the position of PA+873K/1h. According to these experimental tensile results [4], thegap between the yield strength of PA and PA+873K/1h is large; on the other hand, the differ-ence between PA+873K/4h and PA+873K/1h is much smaller than this gap. There may existseveral causes for this dissimilarity, nevertheless different perspectives may interpret this differencein a variety of ways. For instance, it is prevalent to associate the defect strength with the yieldstrength of the material by means of a linear relation between the change in yield strength anddefect strength. Even if the line density, which is a fundamental parameter in the DRP is ignored,the defect strengths shown in Table 4 do not support the experimental proximity of PA+873K/1hto PA+873K/4h. On the contrary, defect strength values computed via BKS model imply thatthe gap between PA+873/1h and PA+873K/4h may be approximately 1.6 times greater than thedifference between PA and PA+873/1h. Over and above, formulations for the estimations of defectstrength are not unique [15, 17, 28, 29], and contradicting as well as supporting formulations maybe found for this specific case, but they may reverse their roles in another case.Dynamically increasing defect strength is also applied on CuCrZr by using the same precipi-tate properties mentioned in Table 4. The parameters of the first trial are inspired from anotherstudy [17] dedicated to the Orowan mechanism. The ratio between the saturation level and theinitial strength value τ Sat /τ BKS is assumed to be 2, and ∆ τ def is set to be 10%. When these inputsare substituted into Eq. (7), the maximum number of loops N after which the process of increaseis terminated is found to be around 7. The result of this simulation is given in Fig. 11.12 A, DΤ def = % PA + (cid:144) DΤ def = % PAPA + (cid:144) DΤ def = % PA + (cid:144) + (cid:144) ¶ Σ @ M P a D Figure 11: Dynamically increasing precipitate strength in single grain CuCrZr with the use of defectstrength increment 10% and saturation ratio 2.The dynamic increase of precipitate strength causes a significant rise in the yield strength, andthe contribution of this feature proportionally increases with respect to precipitate properties asobserved in Fig. 11. A remarkable property of this case is that the use of ∆ τ def = 10% togetherwith τ Sat /τ BKS = 2 does not change the behavior of stress-strain curves sharply except a shift inthe yield strength. In order to test the effect of increment value, ∆ τ def = 3% is employed with thesame ratio, for which the number of dislocation releases, N , needed to terminate the strengtheningprocess is calculated as 23. The simulation output is illustrated in Fig. 12. It is observed thatwhen ∆ τ def = 3% is applied instead of 10%, the hardening characteristic is distinct, and the 0.1%offset yield point gets lower values when compared with the previous case. In short, according toboth Fig. 11 and Fig. 12, augmentation of precipitate strength based on dislocation release showsnotable effect on the variation of yield strength.13 A, DΤ def = % PA + (cid:144) DΤ def = % PAPA + (cid:144) + (cid:144) DΤ def = % PA + (cid:144) ¶ Σ @ M P a D Figure 12: Dynamically increasing precipitate strength in single grain CuCrZr with the use of defectstrength increment 3% and saturation ratio 2.
Tantalum and its alloys are defined as refractory metals thanks to their excellent resistance to cor-rosion and heat. These beneficial properties make them preferable also in nuclear applications [38].Therefore, single grain Ta-2.5W alloy of grain size 40 µ m (taken instead of 45 µ m [23]) is selected forthe simulations of internal friction which are performed by the implementation of friction-dependentmobility law stated in Eq. (4). The reference point is decided to be 24.8 MPa which is the PS valueused in a 3D DD study [39] for edge dislocations in Ta. Not only this PS value, but also 0 . E (GPa) ν b (nm) Strain rate (s − ) s Grain size ( µ m) FR density ( µ m − )186 0.34 0.25 2500 400 b
40 1The results obtained for three non-zero friction values including the PS value are sketchedin Fig. 13, and it is demonstrated that all τ f values result in parallel shifts in plastic response.Furthermore, when the yield points are determined by plotting the offset at 0.2% plastic strain, σ . = { . , . , . , . } is acquired for τ f = { , . , . , . } . Defining that σ is the σ . of frictionless case, the fraction ( σ . − σ ) /τ f yields 1.35, 1.72, and 1.79 for τ f (cid:54) = 0. Thus, thecontribution of internal friction is not entirely linear. However, a coarse approximation such that14 . ≈ σ + βτ f , where β is around 1.75 for an FR density of 1 µ m − in Ta-2.5W, may be obtainedfor this specific case. The PS value causes a very considerable rise in the yield strength by anamount of 93%. The simulations done for internal friction explicitly indicate that friction is aneffective term which shifts the yield strength of the material, hence the presence/absence of frictionin the mobility laws makes a significant difference. However, it is not always straightforward topredict the values of friction, and it may require a multi-scale approach. Τ f = Τ f = Τ f = Τ f = ¶ Σ @ M P a D Figure 13: Internal friction τ f in single grain Ta-2.5W. % yttria Ferritic/martensitic alloys, especially ODS steels, have been taken into account as candidate ma-terials [25] for the components of nuclear systems such as generation-IV as well as fusion reactorsthanks to their resistance occasioned by the existence of highly dispersed oxide particles to swellingand low damage accumulation [40]. For this reason, polycrystalline ODS EUROFER of grain size5 µ m [41, 42] with 0.3% yttria in which constituents are 8.9 wt% Cr, 1.1 wt% W, 0.47 wt% Mn,0.2 wt% V, 0.14 wt% Ta, 0.11 wt% C and Fe for the balance is selected for the simulations ofmixed hardening. ODS steel samples are irradiated with 590 MeV protons to 0.3, 1, and 2 dpa,and the properties of the irradiation-induced defects are chosen by using the observations of trans-mission electron microscopy (TEM) in an experimental study [24]. The dispersion properties inunirradiated material are included from another experimental study [25] where small angle neutronscattering (SANS) is the technique to investigate the particle properties. In both experiments, sam-ples are provided from Plansee [24, 25]. The reason of this combination is the possible variation ofdispersion properties with the initiation of irradiation, since another experimental method reportsconcretely different results, and there exists a remarkable rise in the ultimate tensile strength ofthe material [40]. 15 olycrystalline Σ , ¶ Σ , ¶ Figure 14: Polycrystalline ODS EUROFER with 0.3% yttria for plain strain deformation.The polycrystalline computational cell of ODS EUROFER with 0.3% yttria for plain straindeformation is shown in Fig. 14. The polycrystalline material is composed of eight grains, andthe size of each grain is 5 µ m. Slip systems are randomly generated by conserving the anglesbetween slip lines, and effective Burgers vector magnitudes are used. The input parameters of thissimulation are given in Table 6.Table 6: Basic input for ODS EUROFER with 0.3% yttria simulations E (GPa) ν b (nm) Strain rate (s − ) s Grain size ( µ m) FR density ( µ m − )211.7 0.34 0.286 2500 200 b µ m − ) Loop size (nm) Loop density ( µ m − )0 3.8 11 . × - -0.3 6 −
10 4 . × − . × −
10 4 . × − . × −
10 4 . × −
10 5 . × DRP is applied on the properties of defects mentioned in Table 7 in accordance with theformulations stated in Table 1. BKS model is used to calculate the strength of precipitates whereasthe strength of prismatic dislocation loops is determined by DBH model together with α = 0 .
33 [43].16able 8: Line density and defect strength according to the data in Table 7Dispersion Prismatic loopIrra. dose (dpa) Line density ( µ m − ) Strength (MPa) Line density ( µ m − ) Strength (MPa)0 2.14 268 - -0.3 3.72 303 0.03 441 3.72 303 0.17 782 3.72 303 1.74 146A comparison of line density and strength values in Table 8 aids to predict the yield behaviorof the present material. Starting with the unirradiated case, the yield strength of four cases arelabeled as σ , σ . , σ , and σ respectively. When the dispersion data of unirradiated material iscompared with the case of 0.3 dpa, it is seen that the line density changes significantly, whereasthe strength of the dispersions varies slightly. This change guarantees hardening; thus, σ . > σ .The contribution of dislocation loops at this level is inactive due to very low line density andweak strength. For the next irradiation level, 1 dpa, there is no change in the dispersion data.A remarkable difference in the properties of irradiation induced defects is observed; however, linedensity is still very low. Therefore, one may expect that σ ≈ σ . . For the final irradiation level,besides the same dispersion properties, it is expected that irradiation induced defects result in aweak but visible hardening since almost every slip line has one or more interactions with theseloops and the strength of the defects is far from the strength of FR sources. To conclude, yieldstrength of these four cases can be sorted as follows: σ < σ . ≈ σ < σ . x x x x x x x x x x x x x x x x x x x x x x x x x x x x æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æá á á á á á á á á á á á á á á á á á á á á á á á á á á áó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ¶ Σ @ M P a D irradiationdose = = = óáæ x Figure 15: Plain strain response of polycrystalline ODS EUROFER with 0.3% yttria using Tables 6and 8. 17he simulation results reported in Fig. 15 confirm the initial expectations stated above, anddislocation loops make a weak contribution to hardening. The outcome of these simulations is alsoqualitatively consistent with the concentration based mixture law stated in Eq.3 if the concentrationfactors are expressed in terms of line spacing L line . When the secondary defect is much weaker andless populated than the primary defect, the resulting defect strength is not far from that of theprimary defect. Consequently, the yield strength is principally governed by the strength of theprimary defect.In order to see the effect of stronger secondary defects, an artificial case may be imagined byconsidering that barrier strength coefficient α in the strength formulation of prismatic loops statedin Table 2 may get greater values than 0.33. Therefore, the final simulation is dedicated for theinvestigation of α = { . , , . , . } by the use of geometrical properties stated for irradiationdose of 2 dpa in Table 7. x x x x x x x x x x x x x x x x x x x x x x x x x x x x æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æá á á á á á á á á á á á á á á á á á á á á á á á á á á áó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó óø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ¶ Σ @ M P a D øóáæ x Τ Disper =
303 MPa, Τ Loop =
300 MPa Τ Disper =
303 MPa, Τ Loop =
250 MPa Τ Disper =
303 MPa, Τ Loop =
200 MPa Τ Disper =
303 MPa, Τ Loop =
146 MPa Τ Disper =
303 MPa, no loop
Figure 16: Mixed hardening in the presence of dispersions with ρ lineDisper = 3 .
72 and prismatic loopswith ρ lineLoop = 1 .
74 when α = { . , , . , . } .In Fig. 16, it is shown that secondary defects may contribute effectively when their strength isnot far from the strength of primary defects. The present study focuses on the extension of DRP which is based on the representation of defectsin 2D DD by using fractional conservation laws and barrier hardening models. Existing 3D defectsproperties are translated to 1D basis with the aid of equivalence of fractions, and the resistance ofeach defect type is determined by using BKS or DBH model according to the observations basedon the outcomes of external computational studies. However, any other suggested or modifiedstrength expression may be preferred since these calculations are independent of 2D DD simulationframework. This setup is built on practical algorithms that provide realistic computational domains,18ontrol/track ability for its elements, and consequently opportunity to combine many propertiesby merely using points of interaction and analytical estimations.Within the content of this study, it is seen that the geometrical properties of defects are ex-tremely crucial in the governance of tensile characteristics since not only distribution of theseobjects, but also corresponding features depend on the size and density of defects. However, thisinformation is supposed to be provided from either experimental or external computational studies,and DRP is liable to the existence and accuracy of defect properties. The present model has alsolimitations due to the nature of 2D DD. Screw dislocations have not been mentioned frequentlybecause the dynamics of screw dislocations cannot be represented in 2D DD framework. Absenceof mechanisms specific to screw dislocations such as kink-pair migration or cross-slip may causeinconsistency for BCC alloys for temperatures between 0-300 K since the plasticity of BCC mate-rials is considered as governed by mainly screw dislocations. However, under the condition of plainstrain deformation and in higher temperature intervals where screw dislocations are not mastering,barrier-dislocation interaction and related features may be suitable for simulations with the presentframework.Although the recipe of defect representation and principal features related to defects have beenexhibited, enabling microstructural changes in 2D DD is started, but it is not over. Temperaturedependence of the mobility laws is supposed to be implemented in either Arrhenius or non-Arrheniusform to efficiently test the performance of the materials although it requires a multi-scale approachfor each material. There exists also substantial defect-concerned traits that can be easily done:grain boundary penetration by distributing finite strength obstacles on the interfaces of the grains,defect annihilation by dynamically setting the strength of corresponding defects to zero, or defectmultiplication by initially distributing zero-strength defect points that will gain an effective strengthin the further time steps of the simulation.
The present study reveals out the following properties on the basis of the additional features ofDRP for 2D DD: • Representation of multi-type defects by using consecutively distributed points of interac-tion on slip lines shows consistency with the concentration-balanced superposition. Pre-determined barrier strength for each type of defect provides exceptional opportunity to speedup the simulations and to create flexibility for the usage of different barrier hardening formu-lations. • Dynamically increasing defect strength ends up with a significant rise in the yield strength.Pre-defined variation of barrier strength broadens the practicality of dimension reduction, andit leads to a strategy to deal with similar mechanisms such as defect annihilation. However,the parameters of these mechanisms may not be trivial and may require a careful treatment. • Static internal friction is an effective term which results in parallel shifts in the plastic responseof the material. An enhanced function for friction may also be used to control the velocityof edge dislocations when necessary, but the origin of friction should be identified with lowerscale methods or experimental data. • A unique asset of DRP is the capability to perform large grain deformation. When thematerials at the service of industry are taken into account, DRP for 2D DD is able to cope19ith realistic grain sizes as well as major heterogeneities in these grains. Therefore, DRPadvances the feasibility of dislocation dynamics for a wide range of operative materials. • The main disadvantages of present framework are the restriction to the plain strain deforma-tion and the absence of screw dislocations. Plain strain deformation is the principal conditionunder which 2D DD is employed; however, deformation types which load screw dislocationsare not suitable with 2D DD. Additionally, in the cases where yield strength strongly de-pends on the dominant mechanisms and properties related to screw dislocations, the outcomeof DRP for 2D DD may be unsatisfying. • Modeling of other mechanical phenomena like Mode I crack propagation or 2D nanoindenta-tion is also suitable for the application of DRP.
Acknowledgments
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