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Fracture resistance of textured polycrystalline Zr: A simulation study
Computational Materials Science 162 (2019) 304–313
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Fracture resistance of textured polycrystalline Zr: A simulation study a,⁎
b
G. Bertolino , M. Ruda , D. Farkas a b c
T
c
División Física de Metales, Centro Atómico Bariloche – CONICET – Instituto Balseiro, Bariloche, Argentina Depto. Fisicoquímica de Materiales, CAB-CNEA, Bariloche, Argentina Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Fracture Nano-crystalline Zr Molecular dynamics simulation
We report the results of large-scale molecular dynamics atomistic simulations of crack propagation in of αZirconium. The samples studied are polycrystalline columnar grains of 11–38 nm average diameter, and different textures. The focus is on deformation mechanisms in the crack tip region and the influence of texture, grain size and temperature on the fracture propagation. We found that the [1–100] texture is the most brittle with deformation at the crack tip occurring only through twinning with almost no dislocation activity. The more ductile textures are those where dislocation mechanisms are activated. For the basal orientation we observed < a > dislocations while < c + a > dislocations are dominant for the [11–20] texture. One important observation is that crack propagation can be hindered by the presence of the grain boundaries. As a result, samples with the smallest grain sizes are the more resistant to crack propagation. As expected, higher temperatures imply easier deformation, resulting in more ductile behavior for all orientations. At room temperature and higher, crack propagation occurs mostly intra granularly. However, at 77 K the [1–100] textured sample shows crack propagation through the nucleation of micro-voids ahead of the main crack.
1. Introduction Zirconium, an hexagonal close-packed (hcp) metal with excellent resistance to irradiation as well as good biocompatibility, has been extensively used in the nuclear industry [1]. Therefore, for many decades the mechanical behavior of Zr and its resistance to fracture has been the subject of intense research in order to evaluate endurance and safety of nuclear reactors. More recently, there is a strong interest in Zr and Zr alloys with a nanoscale microstructure that might offer improved mechanical properties. Atomistic simulation can provide an insight on the behavior of metallic materials with small grain sizes that cannot be easily accessed by experiment in order to guide the design of alloys with better mechanical performances. In particular, fracture and deformation mechanisms in nanocrystalline materials have been the subject of intensive research for the last two decades [2–9]. The evolution of a crack under increased stress intensity is fundamental for the understanding of fracture mechanisms and toughness in nanocrystalline materials. Molecular dynamics (MD) is a suitable tool for elucidating the atomistic mechanisms that control deformation and rupture of chemical bonds at nanoscale, and for relating this information to macroscopic failure phenomena (see, e.g., review articles and books [10–13] and references within) ⁎
For the specific case of Zr, there has been a significant effort that utilizes atomistic simulation for understanding deformation mechanisms [13–21]. These studies were based on empirical interatomic potentials and have included tensile deformation of samples with different textures and various grain sizes, as well as fracture propagation in single crystals with different textures [19–21]. The purpose of the present work is to continue our previous studies of the mechanical properties of α-Zirconium [19–21] by analyzing the mechanisms involved in crack propagation in polycrystalline α-Zirconium using molecular dynamics (MD). In recent work [21] we studied deformation mechanisms occurring in MD virtual tension/compression tests of textured polycrystalline Zr. In the present work, we use similar digital samples to address fracture resistance. We generated 50 grains self-similar columnar samples with [0 0 0 1], [11–20] and [1–100] textures and average grain sizes of 15, 27 and 38 nm. We performed the simulations at four different temperatures, 77 K, 150 K, 300 K and 650 K. Our focus is on understanding the effects of texture, grain size and temperature on crack propagation in these polycrystalline materials.
Corresponding author. E-mail address: [email protected] (G. Bertolino).
https://doi.org/10.1016/j.commatsci.2019.02.033 Received 29 November 2018; Received in revised form 4 February 2019; Accepted 23 February 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
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2. Methodology
and the grain shapes were also at random. To simulate columnar grains periodic boundary conditions were used in all directions, with the periodicity in the [0 0 0 1] direction for the basal and [11–20] and [1–100] directions for the prismatic textures being that of the crystalline lattice. All samples contained 50 grains. Samples with different grain sizes were generated as self-similar samples by changing the simulation domain dimensions while keeping the number and all geometrical parameters of the grains unchanged. The average grain sizes (ϕ) were 11, 27 and 38 nm calculated as in [21]. The samples contained up to 7 million atoms. All samples were relaxed with MD techniques and a Nose-Hoover barostat using the LAMMPS code [27]. The relaxation procedure achieved zero external pressure and consisted on first heating at high temperature (500 K or 700 K depending on the test) and then cooling to the temperature desired for each simulation (77 K, 150 K, 300 K and 650 K) until an equilibrated configuration is reached, typically 14 ps for the total procedure.
2.1. Interatomic potentials The description of the interactions between atoms is the basis for large-scale MD simulations. The issue of selecting an appropriate interatomic potential is challenging, particularly for the case of Zr. Because of their efficiency in the computational resources required we utilize standard Embedded Atom Method (EAM) potentials. In our previous work [21] we have compared the performance of two different EAM potentials at MD simulation of α-Zr mechanical properties: the Pasianot and Monti [22] (PM) modified for short interatomic distances and potential #3 of Mendelev-Ackland [23] (MA3). The deformation mechanisms predicted by both potentials are basically the same, but because the predicted ratio of the prismatic (γsp) and basal (γsb) stable fault energies (γsp/γsb) is larger for the PM than for the MA3 potential, the PM potential predicts softer behavior. Both potentials predict qualitatively similar trends in the analysis of deformation behavior for various textures. In the present work we utilize the MA3 potential to describe the interactions between atoms in our MD simulations. Because these are empirical potentials, the importance of our results is in understanding mechanisms and observing trends, rather than quantitative predictions. The EAM potential we utilize here is the most widely used in the literature for the simulation of α-Zr mechanical properties. This is because first-principles data on the interstitial and stacking-fault defect formation energies were included in the fitting procedure ensuring that the I1 prism plane fault has a lower energy than the I2 basal plane fault. This agrees with first-principles calculations and the experimentally determined dominant prismatic slip in Zr.
2.3. Multi-scale simulation of fracture After relaxation, linear elastic fracture mechanics was utilized to introduce a semi-infinite sharp crack under a given Mode I stress intensity factor KI. The initial crack was an atomically sharp wedge with its tip located near the center of the simulation block as illustrated in Fig. 1(a). The simulation block was then divided into two different regions. The outer region, defined as the fixed region, contains atoms with positions determined by continuum theory. In the free region, defined at the center of the block, the atoms are free to move and rearrange following the laws governed by MD and the EAM potentials. MD simulations of the crack advance were performed with a version of the LAMMPS code [27] including a user-package SICRACK that we developed for fracture simulations. The SICRACK package follows linear elastic fracture mechanics equations, as described by Latapie and Farkas [4]. The fracture process in each sample is conducted by incrementally loading the semi-infinite Mode I crack by increasing the value of the applied stress intensity factor KI. Periodic boundary conditions are used in the direction parallel to the crack front, but in the other two orthogonal directions the boundary conditions are fixed. These boundary conditions bridge the length scales necessary to study the fracture process. On a typical simulation the loading was increased from KI = 0 to KI = 6 MPa m0.5 with ΔKI = 5.10−6 MPa m0.5. Under the fixed stress intensity factor at a certain temperature, the sample evolves constrained to the equations of motion and the temperature is kept constant by using a velocity scaling algorithm. As the simulation progresses and the stress intensity factor increases, the crack begins to advance. The resistance of the material to crack propagation can be
2.2. Sample generation For the present study of fracture behavior, we utilize similar samples as in our previous work [21] where we had already analyzed the deformation mechanisms in tension/compression virtual tests. To generate samples with the various desired microstructures of polycrystalline Zr we used a Voronoi construction technique, as described in our previous work [21]. These samples are subject to the known limitations of the Voronoi construction [24–26] but are adequate for the atomistic study of crack tip response and comparison with our previous deformation studies. Columnar grain samples were generated with three different textures: basal, with random misorientation angles around the [0 0 0 1] axis and two prismatic cases. In these cases the angles were orientated around the [11–20] and [1–100] axis (see Fig. 1). In all cases grain boundaries (GB) were of tilt character and randomly orientated
Fig. 1. (a) Relaxed self-similar columnar polycrystalline textured samples with initial crack tip. The boundary conditions at the colored region correspond to the continuum solution of Mode I crack loading. The enlargement zone shows the relative orientations of the grains: (b) basal, (c) prismatic around the [11–20] axis, (d) prismatic around the [1–100] axis. Self-similar samples were generated at 11, 27 and 38 nm average grain sizes. 305
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Video 1.
When the crack front is parallel to the (0 0 0 1) direction (basal texture, Fig. 3 (a)) < 11–20 > dislocations nucleate at the crack tip and travel towards the GB as can be observed from the middle column of Supplementary material Video 1. For the prismatic texture with the crack front in the (11–20) direction (Fig. 3(b)), < c + a > partial dislocations of 1/6[2–203] type are emitted from the crack tip and they travel towards the GB. After a few simulation steps {1–100} twins start to nucleate close to the crack tip and next to the partial dislocations and the crack advances with a blunted tip that helps to slow down the crack growth; however dislocations emissions seems to be the predominant deformation mechanism at this orientation. With regular hexagonal polycrystals and the COMB potentials [17–18] the same < c + a > partial dislocations formed in a tensile test at the same orientation. For the more brittle [1–100] prismatic texture samples twin formation at the crack tip is the plastic deformation mechanism. In Fig. 3(c) the insight shows that {11–21} twins develop at the crack tip and then grow until they interact with the GB. These twins make the crack tip sharper than the tip observed with the other textures, resulting in easier crack propagation. For the three textures, these deformation mechanisms at the crack tip agree well with our previous statics results in single crystal Zr [19]. Fig. 4 shows snapshots of the sample as the loading is increased. The snapshots show the local shear stress at KI = 0.2, 1.0 and 2.0 MPa m0.5 depicting the evolution of the crack in the same samples of Fig. 2 (27 nm grain size samples in the three textures at 300 K). At low loading levels the plastic deformation occurs at the zones where the continuum solution for the stress state at a crack tip predicts more deformation, regardless of the texture. At high loading levels significant plasticity events that occur at the crack tip strongly depend on the sample texture. Dislocations trajectories are seen as lines in the basal and in the [11–20] prismatic structures, twins are shown as wide shaded areas in the prismatic textures.
Fig. 2. Mode I stress intensity factor vs. crack tip advance at 300 K and 27 nm average grain size polycrystalline samples at different textures as specified on Fig. 1.
studied by plotting the applied tress intensity factor versus crack advance. We can also observe the mechanisms involved. Since the MD technique follows the actual forces on the atoms as they migrate, the fracture mechanisms can be determined by direct observation, without having any a priori assumptions. This required state of the art visualization, which was performed using the software package OVITO and the utilities within it [28].
3 Results and discussion 3.1. Deformation mechanisms at the crack tip Fig. 2 shows the Mode I stress intensity factor as a function of crack advance for the sample with 27 nm average grain size at 300 K and the three different textures studied. The basal and [11–20] configurations show a similar behavior; the [1–100] texture presents a more brittle response, with easier crack advance. This particular orientation was found to have the lowest fracture toughness in single crystal static simulations of fracture using the PM potential [19]. This most brittle [1–100] prismatic texture was found to be stronger than the basal texture in virtual tensile tests of polycrystalline samples performed with the MA3 potential [21]. In order to understand the deformation mechanisms at the crack tip that could explain the behavior of the textured samples in Fig. 2 we used OVITO [28] to visualize our results at the atomic level. Fig. 3 shows details of the deformation mechanisms at the crack tip. A dynamic version of these results is shown in the Supplementary material, Video 1. In the left column of that video we can observe the evolution of the different deformation mechanisms originated on the crack tip, by looking at a Common Neighbor Analysis (CNA) of the crystal structure. Displacement fields between 0 and 8 nm are depicted in the middle column allowing us to see the trajectory of the dislocations. In the right column of the video we can see the evolution of the shear strain for the three textures calculated from the atomic level strain tensors referred to the initial configuration of the sample. Fig. 3 shows snapshots of the sample at KI = 0.2, 1.0 and 2.0 MPa m0.5 depicting the evolution of shear strain in the same samples of Fig. 2.
3.2. Crack-GB interactions In all cases we observe that the plasticity mechanisms interact with GB affecting the crack propagation process. In a previous work [21] we performed an extensive analysis of the role of the GBs in the deformation mechanisms occurring in virtual tensile and compression tests with similar samples as in the present work but without the crack. We found that the GB́ s interact with defects either by absorbing or transmitting deformations to the next grain depending on the relative misorientation of neighboring grains. In the basal textured samples dislocations were absorbed by the GB and easily transmitted. In the [1–100] textured samples we found examples of twin absorbtion and transmission, as well as twin formation from the GB as a result of an absorbed dislocation. In order to understand the role of GB in crack advance, we now focus on analyzing the effects of grain size, and 306
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Fig. 3. Texture dependent main mechanisms for crack tip advance. (a) Basal [0 0 0 1] texture, < 11–20 > edge dislocations with Burgeŕs vector a. (b) Prismatic [11–20] texture, twins {1–100} and dislocations with Burgeŕs vector c + a. (c) Prismatic [1–100] texture, {11–21} twins. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: other.
Δa curves are smaller for the samples with larger grain sizes. As stated in the previous section, for samples with this orientation the main crack advance mechanisms are dissociated dislocations emitted from the crack tip (Fig. 3(a)). When they reach the GBs they are mainly absorbed and the crack tip remains blunted while the loading is increased. In the analysis we performed in [20] we found GB sliding, rotation and migration and shear band formation as the common deformation mechanisms operating in basal textured samples under tension. The
therefore the role of GB on the Mode I crack growth resistance curves for all textures. The results for the three textures considered at 300 K can be seen on Fig. 5. From these curves in all orientations we can observe that there is an influence of the grain size on the resistance to crack propagation. In all cases the smaller grain size sample show greater resistance to crack advance. That is, our results indicate that smaller grains yield more ductile behavior. For the basal texture (Fig. 5(a,d)) we see that the slopes of the KI vs.
Fig. 4. Atomic shear strain maps visualized for the three textures at KI = 0.2, 1.0 and 2.0 MPa m0.5 at 300 K and 27 nm grain size. 307
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Fig. 5. Grain size (ϕ) effect on Mode I stress intensity factor vs. crack tip advance curves at 300 K and three textures (a) [0 0 0 1], (b) [11–20], (c) [1–100], (d) slopes of the KI vs. Δa curves as a function of grain size.
mechanisms found in the present work are consistent with those findings and are also shown dynamically in Supplementary material Video 1 where the displacement fields of the basal textured samples with a crack propagating on Mode I are shown. Also on the shear strain snapshots of Video 1 shown on Fig. 4 we can see the regions corresponding to some GBs appear with higher values of shear strain, indicating GB sliding. These results also agree with recent [29] experimental results showing that, for basal textured α-Zr samples, the grains close to the crack rotate when the sample deforms by dislocation emission. The deformation mechanisms observed at the [11–20] textured samples interact in a different way with the GBs. Snapshots at different KI values of three samples with different grain sizes for this texture are shown in Fig. 6. Partial < c + a > dislocations and twins nucleate from the crack tip as mentioned before. As they reach the GBs new partial dislocations nucleate in the adjacent grains. This is in agreement with observations in previous studies of dislocation-grain boundary interactions without a crack [30–32]. Blunting of the crack tip occurs as a result of the interaction of the dislocations and twins with the GBs and additional loading is required to propagate the crack. For higher loading levels the crack starts to advance towards regions with higher concentration of defects, enhancing the propagation of the fracture.
This can be seen as a change of slope in the KI vs. Δa curves in Fig. 5(b). Fig. 5(d) shows the average slopes of the curves in Fig. 5(a–c) plotted as a function of grain size. The basal orientation, where plasticity occurs through dislocations, shows less variation with grain size. The prismatic [1–100] orientation, where plasticity is dominated by twining shows the greatest effects of grain size. The prismatic [11–20], where both plasticity mechanisms occur presents an intermediate grain size effect. We can conclude that the grain size effect on crack advance is more important when twin formation and propagation are the main deformation mechanism occurring at a crack tip. The crack tip configurations shown in Fig. 6 indicate more blunting for the smaller grain sizes. For the 11 nm sample the crack tip configuration is the most blunted, resulting in enhanced ductility. For larger grain sizes the crack tip configuration contains more sharp corners that act as regions of stress concentration. These sharp corners develop as a result of twining and are specifically located at the twin boundaries. The crack is seen to propagate along some of these boundaries. This suggests that deformation by twinning results in more brittle behavior than deformation by dislocation emission and glide. This is also seen in the snapshots of Fig. 7, taken for three samples with different grain sizes for the [1–100] texture. For that orientation, < 11–21 > twin formation is the main 308
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Fig. 6. Grain size effects on Mode I crack propagation mechanisms on [1–210] samples at 300 K, grain sizes = 11, 27 and 38 nm and KI = 0.1 MPa m0.5; 0.7 MPa m0.5; 1.4 MPa m0.5. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: other.
3.3. Effects of temperature
deformation mechanism observed. When these twins grow they reach and interact with GB. (See Fig. 7 for KI = 0.1 MPa m0.5). The crack tip develops a sharp triangular shape until the twins reach the GBs and interact with them. The twinned regions become wider and the crack still keeps a sharp crack tip (see Fig. 7 with KI = 0.7 MPa m0.5). All these changes in the propagation mechanisms are reflected by a change in the slope of the corresponding KI vs. Δa curve in Fig. 5(c). These interactions between twins and GBs are reproduced for all grain sizes, once the twin tip reaches the GB. Twin transmission to neighboring grains was also observed in the present simulations, as can be seen on Supplementary material Video 1 for the 27 nm sample at 300 K. This is consistent with the mechanism that was extensively analyzed on samples without cracks but subject to tensile charges in the literature [13–16] and also on our previous paper [17]. As the crack advanced, we also found some < p > partial dislocations with the Burgers vectors of the 1/3 < 1–100 > type as shown in Ref. [1]. These partial dislocations are not generated at the crack tip and because the separation between partials is of fcc nature they can be easily observed as a green embryo with the CNA analysis on the Supplementary material Video 1.
Generally, twinning is preferred to slip at low deformation temperatures; at higher temperatures, however, slip is favored [1]. Since twinning is the main deformation mechanism at [1–100] prismatic texture but not at the others textures, we decided to analyze the temperature effects on crack advance using this orientation. In Figs. 8 and 9 we present the results for the 27 nm sample with the [1–100] texture at different temperatures. As expected, deformation is easier and all our samples are more ductile at higher temperatures and more brittle at lower temperatures. The deformation mechanisms involved in the crack advance do not change significantly from room temperature to 650 K. These mechanisms are mostly intragranular crack propagation, with some crack arrest at the GB. Blunting increases with temperature. The mechanisms change at 77 K, with crack propagation occurring through the nucleation of new intergranular microcracks in the vicinity of the main crack tip. Note that no changes in the slope of the resistance curves are seen at 300 K and 650 K where the crack advances linearly with the applied stress intensity factor KI. However the 150 K and the 77 K curves show a series of plateaus indicating fast crack growth followed by steep regions where KI needs to increase substantially in order for the crack to advance. The first plateau of both curves corresponds to the generation and 309
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Fig. 7. Crack tip advance at prismatic [1–100] textured at 11 nm, 27 nm and 38 nm samples and K = 0.1 MPa m0.5; 0.7 MPa m0.5; 1.4 MPa m0.5. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: other.
displacement of that sample are depicted simultaneously to allow a better comprehension of the crack propagation mechanism. In the lower right hand side part of the Supplementary material Video 2 we can see again the twin transmission between neighboring grains. The out-of-plane displacements video shows that all the twins at this texture are of the shuffling type as we had stated before [21].
growth of {11–21} twins from the crack tip as discussed previously. At 150 K the plateau ends when the crack tip reaches the GB and new twins develop in the neighboring grain. At the 77 K curve the plateau also ends when a twin reaches the GB, as indicated in Fig. 8, insert A, but the crack blunts due to the twin-GB interaction (Fig. 8, insert B) and the stress concentration promotes the formation of an intergranular micro-void (Fig. 8, inserts C and D). After the void is formed the crack advances rapidly changing its direction towards the location of the void (Fig. 8, plateau, inserts D-E). When the crack reaches the void its tip is reoriented towards a new void that was formed between other grains (upper right side of insert F, Fig. 8). This can be corroborated by looking at the upper part of Fig. 9 where we can see the localization of the shear strain corresponding to KI values of 0.2 MPa m0.5 (Fig. 8 insert A), 1.0 MPa m0.5 (Fig. 8 insert D) and 2.5 MPa m0.5 (Fig. 8 insert F). Because twins are the main deformation mechanism for this texture, shear strain is localized not only at the crack tip but also at zones where twins generate and grow, including regions ahead of the crack tip. Voids are formed at triple junctions and at the crossing of shear bands where the strain concentration is highest, as was observed experimentally in Zr alloys [33–34]. In the Supplementary material Video 2 we can see the crack propagation at 77 K for the 27 nm sample with the [1–100] texture. The Common Neighbor Analysis, atomic shear strain and out of plane
Video 2.
4. Summary and conclusions We have performed MD atomistic simulations of crack propagation in of α-Zirconium polycrystalline textured columnar samples and analyzed the details of the deformation mechanisms at the crack tip. We found that these are influenced by texture, grain size and temperature. 310
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Fig. 8. Temperature effects on Mode I crack advance on [1–100] texture 27 nm samples, MA3 potentials. Inserts correspond to the CNA analysis of the crack tip region at different stress intensity factor levels of the 77 K curve as indicated in the plot. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: unspecified.
As expected, higher temperatures imply easier plastic deformation that results in a more ductile behavior for all orientations. Crack propagation occurs mostly intra granularly, except at the lowest temperature tested of 77 K. At this low temperature the [1–100] textured sample shows crack propagation through the nucleation of intergranular micro-voids in regions of stress concentration ahead of the main crack. Higher temperatures induce increased atomic mobility, and thereby atom reorganization, which allows twin formation or dislocation formation, releasing the stress at the crack tip. At the lowest temperature tested this is completely inhibited and there is nucleation of nano-void ahead of the main crack tip, similar to what is observed in fcc nanocrystalline materials [36]. Fracture toughness of an irradiated Zr-2.5Nb alloy pressure tube has been evaluated experimentally [38]. These studies found that fracture toughness increased with increasing in temperature up to 523 K. Experiments reported by Daunys et al. [36] also showed that higher temperatures lead to increased toughness.
We found the more brittle orientation to be the [1–100] and this is related to the fact that in this orientation deformation at the crack tip occurs only through twinning with almost no dislocation activity. The other two orientations are more ductile and deformation proceeds through different dislocation mechanisms. For the basal orientation these are < a > dislocations while < c + a > dislocations are dominant for the [11–20] texture. Some partial < p > dislocations with the Burgers vectors of the 1/3 < 1–100 > type appear at the GB́ s during the propagation of the crack at the [1–100] texture. These results suggest that deformation through twinning results in a more brittle behavior than deformation through dislocation emission and glide. In our simulations this seems to be related to the fact that deformation by twinning results in crack tip blunting that still has sharp corners, favoring stress concentration. The deformation mechanisms observed at the crack tip for the various orientations tested agree well with the deformation mechanisms that were found in our previous work under tensile testing [21]. In a recent simulation study, Singh et al. [35] also found that deformation in single crystal Zr was governed either by twinning or the emission of dislocations from the crack tip. In agreement with our current work their results suggest that deformation via twinning leads to easier crack propagation, whereas dislocation mechanisms are more effective in improving toughness. We found that the samples with the smallest grain sizes are the more resistant to crack propagation. This is particularly evident in the prismatic orientations, and is due to the grain boundaries constituting barriers to crack propagation. These results are consistent with previous studies of fracture response in nanocrystalline materials in fcc [36] as well as bcc [35,37,38] materials. These effects of grain size are more significant for the cases where deformation occurs through twinning than in the orientations where deformation occurs through dislocation emission and glide.
CRediT authorship contribution statement G. Bertolino: Investigation, Writing - review & editing. M. Ruda: Investigation, Writing - review & editing. D. Farkas: Investigation, Writing - review & editing.
Acknowledgement The authors thank Dr. Enzo Dari for computational resources and help at CAB-CNEA.
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Fig. 9. Atomic shear strain maps as the crack advances on [1–100] textured 27 nm samples at temperatures 77 K, 150 K, 300 K and 650 K and three values of KI: 0.2 MPa m0.5, 1.0 MPa m0.5 and 2.5 MPa m0.5.
Data availability
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The raw data required to reproduce these findings are available upon request by email to the corresponding author: [email protected]. gov.ar. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.commatsci.2019.02.033. References [1] E. Tenckhoff, Deformation Mechanisms, Texture and Anisotropy in Zirconium and Zircaloy, American Society for Testing and Materials, Philadelphia, 1988. [2] D. Farkas, Atomistic simulations of metallic microstructures, Curr. Opin. Sol. Stat. Mat. Sci. 17 (2013) 284–297, https://doi.org/10.1016/j.cossms.2013.11.002.
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Fracture resistance of textured polycrystalline Zr: A simulation study a,⁎
b
G. Bertolino , M. Ruda , D. Farkas a b c
T
c
División Física de Metales, Centro Atómico Bariloche – CONICET – Instituto Balseiro, Bariloche, Argentina Depto. Fisicoquímica de Materiales, CAB-CNEA, Bariloche, Argentina Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Fracture Nano-crystalline Zr Molecular dynamics simulation
We report the results of large-scale molecular dynamics atomistic simulations of crack propagation in of αZirconium. The samples studied are polycrystalline columnar grains of 11–38 nm average diameter, and different textures. The focus is on deformation mechanisms in the crack tip region and the influence of texture, grain size and temperature on the fracture propagation. We found that the [1–100] texture is the most brittle with deformation at the crack tip occurring only through twinning with almost no dislocation activity. The more ductile textures are those where dislocation mechanisms are activated. For the basal orientation we observed < a > dislocations while < c + a > dislocations are dominant for the [11–20] texture. One important observation is that crack propagation can be hindered by the presence of the grain boundaries. As a result, samples with the smallest grain sizes are the more resistant to crack propagation. As expected, higher temperatures imply easier deformation, resulting in more ductile behavior for all orientations. At room temperature and higher, crack propagation occurs mostly intra granularly. However, at 77 K the [1–100] textured sample shows crack propagation through the nucleation of micro-voids ahead of the main crack.
1. Introduction Zirconium, an hexagonal close-packed (hcp) metal with excellent resistance to irradiation as well as good biocompatibility, has been extensively used in the nuclear industry [1]. Therefore, for many decades the mechanical behavior of Zr and its resistance to fracture has been the subject of intense research in order to evaluate endurance and safety of nuclear reactors. More recently, there is a strong interest in Zr and Zr alloys with a nanoscale microstructure that might offer improved mechanical properties. Atomistic simulation can provide an insight on the behavior of metallic materials with small grain sizes that cannot be easily accessed by experiment in order to guide the design of alloys with better mechanical performances. In particular, fracture and deformation mechanisms in nanocrystalline materials have been the subject of intensive research for the last two decades [2–9]. The evolution of a crack under increased stress intensity is fundamental for the understanding of fracture mechanisms and toughness in nanocrystalline materials. Molecular dynamics (MD) is a suitable tool for elucidating the atomistic mechanisms that control deformation and rupture of chemical bonds at nanoscale, and for relating this information to macroscopic failure phenomena (see, e.g., review articles and books [10–13] and references within) ⁎
For the specific case of Zr, there has been a significant effort that utilizes atomistic simulation for understanding deformation mechanisms [13–21]. These studies were based on empirical interatomic potentials and have included tensile deformation of samples with different textures and various grain sizes, as well as fracture propagation in single crystals with different textures [19–21]. The purpose of the present work is to continue our previous studies of the mechanical properties of α-Zirconium [19–21] by analyzing the mechanisms involved in crack propagation in polycrystalline α-Zirconium using molecular dynamics (MD). In recent work [21] we studied deformation mechanisms occurring in MD virtual tension/compression tests of textured polycrystalline Zr. In the present work, we use similar digital samples to address fracture resistance. We generated 50 grains self-similar columnar samples with [0 0 0 1], [11–20] and [1–100] textures and average grain sizes of 15, 27 and 38 nm. We performed the simulations at four different temperatures, 77 K, 150 K, 300 K and 650 K. Our focus is on understanding the effects of texture, grain size and temperature on crack propagation in these polycrystalline materials.
Corresponding author. E-mail address: [email protected] (G. Bertolino).
https://doi.org/10.1016/j.commatsci.2019.02.033 Received 29 November 2018; Received in revised form 4 February 2019; Accepted 23 February 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
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2. Methodology
and the grain shapes were also at random. To simulate columnar grains periodic boundary conditions were used in all directions, with the periodicity in the [0 0 0 1] direction for the basal and [11–20] and [1–100] directions for the prismatic textures being that of the crystalline lattice. All samples contained 50 grains. Samples with different grain sizes were generated as self-similar samples by changing the simulation domain dimensions while keeping the number and all geometrical parameters of the grains unchanged. The average grain sizes (ϕ) were 11, 27 and 38 nm calculated as in [21]. The samples contained up to 7 million atoms. All samples were relaxed with MD techniques and a Nose-Hoover barostat using the LAMMPS code [27]. The relaxation procedure achieved zero external pressure and consisted on first heating at high temperature (500 K or 700 K depending on the test) and then cooling to the temperature desired for each simulation (77 K, 150 K, 300 K and 650 K) until an equilibrated configuration is reached, typically 14 ps for the total procedure.
2.1. Interatomic potentials The description of the interactions between atoms is the basis for large-scale MD simulations. The issue of selecting an appropriate interatomic potential is challenging, particularly for the case of Zr. Because of their efficiency in the computational resources required we utilize standard Embedded Atom Method (EAM) potentials. In our previous work [21] we have compared the performance of two different EAM potentials at MD simulation of α-Zr mechanical properties: the Pasianot and Monti [22] (PM) modified for short interatomic distances and potential #3 of Mendelev-Ackland [23] (MA3). The deformation mechanisms predicted by both potentials are basically the same, but because the predicted ratio of the prismatic (γsp) and basal (γsb) stable fault energies (γsp/γsb) is larger for the PM than for the MA3 potential, the PM potential predicts softer behavior. Both potentials predict qualitatively similar trends in the analysis of deformation behavior for various textures. In the present work we utilize the MA3 potential to describe the interactions between atoms in our MD simulations. Because these are empirical potentials, the importance of our results is in understanding mechanisms and observing trends, rather than quantitative predictions. The EAM potential we utilize here is the most widely used in the literature for the simulation of α-Zr mechanical properties. This is because first-principles data on the interstitial and stacking-fault defect formation energies were included in the fitting procedure ensuring that the I1 prism plane fault has a lower energy than the I2 basal plane fault. This agrees with first-principles calculations and the experimentally determined dominant prismatic slip in Zr.
2.3. Multi-scale simulation of fracture After relaxation, linear elastic fracture mechanics was utilized to introduce a semi-infinite sharp crack under a given Mode I stress intensity factor KI. The initial crack was an atomically sharp wedge with its tip located near the center of the simulation block as illustrated in Fig. 1(a). The simulation block was then divided into two different regions. The outer region, defined as the fixed region, contains atoms with positions determined by continuum theory. In the free region, defined at the center of the block, the atoms are free to move and rearrange following the laws governed by MD and the EAM potentials. MD simulations of the crack advance were performed with a version of the LAMMPS code [27] including a user-package SICRACK that we developed for fracture simulations. The SICRACK package follows linear elastic fracture mechanics equations, as described by Latapie and Farkas [4]. The fracture process in each sample is conducted by incrementally loading the semi-infinite Mode I crack by increasing the value of the applied stress intensity factor KI. Periodic boundary conditions are used in the direction parallel to the crack front, but in the other two orthogonal directions the boundary conditions are fixed. These boundary conditions bridge the length scales necessary to study the fracture process. On a typical simulation the loading was increased from KI = 0 to KI = 6 MPa m0.5 with ΔKI = 5.10−6 MPa m0.5. Under the fixed stress intensity factor at a certain temperature, the sample evolves constrained to the equations of motion and the temperature is kept constant by using a velocity scaling algorithm. As the simulation progresses and the stress intensity factor increases, the crack begins to advance. The resistance of the material to crack propagation can be
2.2. Sample generation For the present study of fracture behavior, we utilize similar samples as in our previous work [21] where we had already analyzed the deformation mechanisms in tension/compression virtual tests. To generate samples with the various desired microstructures of polycrystalline Zr we used a Voronoi construction technique, as described in our previous work [21]. These samples are subject to the known limitations of the Voronoi construction [24–26] but are adequate for the atomistic study of crack tip response and comparison with our previous deformation studies. Columnar grain samples were generated with three different textures: basal, with random misorientation angles around the [0 0 0 1] axis and two prismatic cases. In these cases the angles were orientated around the [11–20] and [1–100] axis (see Fig. 1). In all cases grain boundaries (GB) were of tilt character and randomly orientated
Fig. 1. (a) Relaxed self-similar columnar polycrystalline textured samples with initial crack tip. The boundary conditions at the colored region correspond to the continuum solution of Mode I crack loading. The enlargement zone shows the relative orientations of the grains: (b) basal, (c) prismatic around the [11–20] axis, (d) prismatic around the [1–100] axis. Self-similar samples were generated at 11, 27 and 38 nm average grain sizes. 305
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Video 1.
When the crack front is parallel to the (0 0 0 1) direction (basal texture, Fig. 3 (a)) < 11–20 > dislocations nucleate at the crack tip and travel towards the GB as can be observed from the middle column of Supplementary material Video 1. For the prismatic texture with the crack front in the (11–20) direction (Fig. 3(b)), < c + a > partial dislocations of 1/6[2–203] type are emitted from the crack tip and they travel towards the GB. After a few simulation steps {1–100} twins start to nucleate close to the crack tip and next to the partial dislocations and the crack advances with a blunted tip that helps to slow down the crack growth; however dislocations emissions seems to be the predominant deformation mechanism at this orientation. With regular hexagonal polycrystals and the COMB potentials [17–18] the same < c + a > partial dislocations formed in a tensile test at the same orientation. For the more brittle [1–100] prismatic texture samples twin formation at the crack tip is the plastic deformation mechanism. In Fig. 3(c) the insight shows that {11–21} twins develop at the crack tip and then grow until they interact with the GB. These twins make the crack tip sharper than the tip observed with the other textures, resulting in easier crack propagation. For the three textures, these deformation mechanisms at the crack tip agree well with our previous statics results in single crystal Zr [19]. Fig. 4 shows snapshots of the sample as the loading is increased. The snapshots show the local shear stress at KI = 0.2, 1.0 and 2.0 MPa m0.5 depicting the evolution of the crack in the same samples of Fig. 2 (27 nm grain size samples in the three textures at 300 K). At low loading levels the plastic deformation occurs at the zones where the continuum solution for the stress state at a crack tip predicts more deformation, regardless of the texture. At high loading levels significant plasticity events that occur at the crack tip strongly depend on the sample texture. Dislocations trajectories are seen as lines in the basal and in the [11–20] prismatic structures, twins are shown as wide shaded areas in the prismatic textures.
Fig. 2. Mode I stress intensity factor vs. crack tip advance at 300 K and 27 nm average grain size polycrystalline samples at different textures as specified on Fig. 1.
studied by plotting the applied tress intensity factor versus crack advance. We can also observe the mechanisms involved. Since the MD technique follows the actual forces on the atoms as they migrate, the fracture mechanisms can be determined by direct observation, without having any a priori assumptions. This required state of the art visualization, which was performed using the software package OVITO and the utilities within it [28].
3 Results and discussion 3.1. Deformation mechanisms at the crack tip Fig. 2 shows the Mode I stress intensity factor as a function of crack advance for the sample with 27 nm average grain size at 300 K and the three different textures studied. The basal and [11–20] configurations show a similar behavior; the [1–100] texture presents a more brittle response, with easier crack advance. This particular orientation was found to have the lowest fracture toughness in single crystal static simulations of fracture using the PM potential [19]. This most brittle [1–100] prismatic texture was found to be stronger than the basal texture in virtual tensile tests of polycrystalline samples performed with the MA3 potential [21]. In order to understand the deformation mechanisms at the crack tip that could explain the behavior of the textured samples in Fig. 2 we used OVITO [28] to visualize our results at the atomic level. Fig. 3 shows details of the deformation mechanisms at the crack tip. A dynamic version of these results is shown in the Supplementary material, Video 1. In the left column of that video we can observe the evolution of the different deformation mechanisms originated on the crack tip, by looking at a Common Neighbor Analysis (CNA) of the crystal structure. Displacement fields between 0 and 8 nm are depicted in the middle column allowing us to see the trajectory of the dislocations. In the right column of the video we can see the evolution of the shear strain for the three textures calculated from the atomic level strain tensors referred to the initial configuration of the sample. Fig. 3 shows snapshots of the sample at KI = 0.2, 1.0 and 2.0 MPa m0.5 depicting the evolution of shear strain in the same samples of Fig. 2.
3.2. Crack-GB interactions In all cases we observe that the plasticity mechanisms interact with GB affecting the crack propagation process. In a previous work [21] we performed an extensive analysis of the role of the GBs in the deformation mechanisms occurring in virtual tensile and compression tests with similar samples as in the present work but without the crack. We found that the GB́ s interact with defects either by absorbing or transmitting deformations to the next grain depending on the relative misorientation of neighboring grains. In the basal textured samples dislocations were absorbed by the GB and easily transmitted. In the [1–100] textured samples we found examples of twin absorbtion and transmission, as well as twin formation from the GB as a result of an absorbed dislocation. In order to understand the role of GB in crack advance, we now focus on analyzing the effects of grain size, and 306
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Fig. 3. Texture dependent main mechanisms for crack tip advance. (a) Basal [0 0 0 1] texture, < 11–20 > edge dislocations with Burgeŕs vector a. (b) Prismatic [11–20] texture, twins {1–100} and dislocations with Burgeŕs vector c + a. (c) Prismatic [1–100] texture, {11–21} twins. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: other.
Δa curves are smaller for the samples with larger grain sizes. As stated in the previous section, for samples with this orientation the main crack advance mechanisms are dissociated dislocations emitted from the crack tip (Fig. 3(a)). When they reach the GBs they are mainly absorbed and the crack tip remains blunted while the loading is increased. In the analysis we performed in [20] we found GB sliding, rotation and migration and shear band formation as the common deformation mechanisms operating in basal textured samples under tension. The
therefore the role of GB on the Mode I crack growth resistance curves for all textures. The results for the three textures considered at 300 K can be seen on Fig. 5. From these curves in all orientations we can observe that there is an influence of the grain size on the resistance to crack propagation. In all cases the smaller grain size sample show greater resistance to crack advance. That is, our results indicate that smaller grains yield more ductile behavior. For the basal texture (Fig. 5(a,d)) we see that the slopes of the KI vs.
Fig. 4. Atomic shear strain maps visualized for the three textures at KI = 0.2, 1.0 and 2.0 MPa m0.5 at 300 K and 27 nm grain size. 307
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Fig. 5. Grain size (ϕ) effect on Mode I stress intensity factor vs. crack tip advance curves at 300 K and three textures (a) [0 0 0 1], (b) [11–20], (c) [1–100], (d) slopes of the KI vs. Δa curves as a function of grain size.
mechanisms found in the present work are consistent with those findings and are also shown dynamically in Supplementary material Video 1 where the displacement fields of the basal textured samples with a crack propagating on Mode I are shown. Also on the shear strain snapshots of Video 1 shown on Fig. 4 we can see the regions corresponding to some GBs appear with higher values of shear strain, indicating GB sliding. These results also agree with recent [29] experimental results showing that, for basal textured α-Zr samples, the grains close to the crack rotate when the sample deforms by dislocation emission. The deformation mechanisms observed at the [11–20] textured samples interact in a different way with the GBs. Snapshots at different KI values of three samples with different grain sizes for this texture are shown in Fig. 6. Partial < c + a > dislocations and twins nucleate from the crack tip as mentioned before. As they reach the GBs new partial dislocations nucleate in the adjacent grains. This is in agreement with observations in previous studies of dislocation-grain boundary interactions without a crack [30–32]. Blunting of the crack tip occurs as a result of the interaction of the dislocations and twins with the GBs and additional loading is required to propagate the crack. For higher loading levels the crack starts to advance towards regions with higher concentration of defects, enhancing the propagation of the fracture.
This can be seen as a change of slope in the KI vs. Δa curves in Fig. 5(b). Fig. 5(d) shows the average slopes of the curves in Fig. 5(a–c) plotted as a function of grain size. The basal orientation, where plasticity occurs through dislocations, shows less variation with grain size. The prismatic [1–100] orientation, where plasticity is dominated by twining shows the greatest effects of grain size. The prismatic [11–20], where both plasticity mechanisms occur presents an intermediate grain size effect. We can conclude that the grain size effect on crack advance is more important when twin formation and propagation are the main deformation mechanism occurring at a crack tip. The crack tip configurations shown in Fig. 6 indicate more blunting for the smaller grain sizes. For the 11 nm sample the crack tip configuration is the most blunted, resulting in enhanced ductility. For larger grain sizes the crack tip configuration contains more sharp corners that act as regions of stress concentration. These sharp corners develop as a result of twining and are specifically located at the twin boundaries. The crack is seen to propagate along some of these boundaries. This suggests that deformation by twinning results in more brittle behavior than deformation by dislocation emission and glide. This is also seen in the snapshots of Fig. 7, taken for three samples with different grain sizes for the [1–100] texture. For that orientation, < 11–21 > twin formation is the main 308
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Fig. 6. Grain size effects on Mode I crack propagation mechanisms on [1–210] samples at 300 K, grain sizes = 11, 27 and 38 nm and KI = 0.1 MPa m0.5; 0.7 MPa m0.5; 1.4 MPa m0.5. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: other.
3.3. Effects of temperature
deformation mechanism observed. When these twins grow they reach and interact with GB. (See Fig. 7 for KI = 0.1 MPa m0.5). The crack tip develops a sharp triangular shape until the twins reach the GBs and interact with them. The twinned regions become wider and the crack still keeps a sharp crack tip (see Fig. 7 with KI = 0.7 MPa m0.5). All these changes in the propagation mechanisms are reflected by a change in the slope of the corresponding KI vs. Δa curve in Fig. 5(c). These interactions between twins and GBs are reproduced for all grain sizes, once the twin tip reaches the GB. Twin transmission to neighboring grains was also observed in the present simulations, as can be seen on Supplementary material Video 1 for the 27 nm sample at 300 K. This is consistent with the mechanism that was extensively analyzed on samples without cracks but subject to tensile charges in the literature [13–16] and also on our previous paper [17]. As the crack advanced, we also found some < p > partial dislocations with the Burgers vectors of the 1/3 < 1–100 > type as shown in Ref. [1]. These partial dislocations are not generated at the crack tip and because the separation between partials is of fcc nature they can be easily observed as a green embryo with the CNA analysis on the Supplementary material Video 1.
Generally, twinning is preferred to slip at low deformation temperatures; at higher temperatures, however, slip is favored [1]. Since twinning is the main deformation mechanism at [1–100] prismatic texture but not at the others textures, we decided to analyze the temperature effects on crack advance using this orientation. In Figs. 8 and 9 we present the results for the 27 nm sample with the [1–100] texture at different temperatures. As expected, deformation is easier and all our samples are more ductile at higher temperatures and more brittle at lower temperatures. The deformation mechanisms involved in the crack advance do not change significantly from room temperature to 650 K. These mechanisms are mostly intragranular crack propagation, with some crack arrest at the GB. Blunting increases with temperature. The mechanisms change at 77 K, with crack propagation occurring through the nucleation of new intergranular microcracks in the vicinity of the main crack tip. Note that no changes in the slope of the resistance curves are seen at 300 K and 650 K where the crack advances linearly with the applied stress intensity factor KI. However the 150 K and the 77 K curves show a series of plateaus indicating fast crack growth followed by steep regions where KI needs to increase substantially in order for the crack to advance. The first plateau of both curves corresponds to the generation and 309
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Fig. 7. Crack tip advance at prismatic [1–100] textured at 11 nm, 27 nm and 38 nm samples and K = 0.1 MPa m0.5; 0.7 MPa m0.5; 1.4 MPa m0.5. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: other.
displacement of that sample are depicted simultaneously to allow a better comprehension of the crack propagation mechanism. In the lower right hand side part of the Supplementary material Video 2 we can see again the twin transmission between neighboring grains. The out-of-plane displacements video shows that all the twins at this texture are of the shuffling type as we had stated before [21].
growth of {11–21} twins from the crack tip as discussed previously. At 150 K the plateau ends when the crack tip reaches the GB and new twins develop in the neighboring grain. At the 77 K curve the plateau also ends when a twin reaches the GB, as indicated in Fig. 8, insert A, but the crack blunts due to the twin-GB interaction (Fig. 8, insert B) and the stress concentration promotes the formation of an intergranular micro-void (Fig. 8, inserts C and D). After the void is formed the crack advances rapidly changing its direction towards the location of the void (Fig. 8, plateau, inserts D-E). When the crack reaches the void its tip is reoriented towards a new void that was formed between other grains (upper right side of insert F, Fig. 8). This can be corroborated by looking at the upper part of Fig. 9 where we can see the localization of the shear strain corresponding to KI values of 0.2 MPa m0.5 (Fig. 8 insert A), 1.0 MPa m0.5 (Fig. 8 insert D) and 2.5 MPa m0.5 (Fig. 8 insert F). Because twins are the main deformation mechanism for this texture, shear strain is localized not only at the crack tip but also at zones where twins generate and grow, including regions ahead of the crack tip. Voids are formed at triple junctions and at the crossing of shear bands where the strain concentration is highest, as was observed experimentally in Zr alloys [33–34]. In the Supplementary material Video 2 we can see the crack propagation at 77 K for the 27 nm sample with the [1–100] texture. The Common Neighbor Analysis, atomic shear strain and out of plane
Video 2.
4. Summary and conclusions We have performed MD atomistic simulations of crack propagation in of α-Zirconium polycrystalline textured columnar samples and analyzed the details of the deformation mechanisms at the crack tip. We found that these are influenced by texture, grain size and temperature. 310
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Fig. 8. Temperature effects on Mode I crack advance on [1–100] texture 27 nm samples, MA3 potentials. Inserts correspond to the CNA analysis of the crack tip region at different stress intensity factor levels of the 77 K curve as indicated in the plot. Atoms colored on line according to crystal structure from Common Neighbor Analysis (CNA): grey: hcp, green: fcc, blue: unspecified.
As expected, higher temperatures imply easier plastic deformation that results in a more ductile behavior for all orientations. Crack propagation occurs mostly intra granularly, except at the lowest temperature tested of 77 K. At this low temperature the [1–100] textured sample shows crack propagation through the nucleation of intergranular micro-voids in regions of stress concentration ahead of the main crack. Higher temperatures induce increased atomic mobility, and thereby atom reorganization, which allows twin formation or dislocation formation, releasing the stress at the crack tip. At the lowest temperature tested this is completely inhibited and there is nucleation of nano-void ahead of the main crack tip, similar to what is observed in fcc nanocrystalline materials [36]. Fracture toughness of an irradiated Zr-2.5Nb alloy pressure tube has been evaluated experimentally [38]. These studies found that fracture toughness increased with increasing in temperature up to 523 K. Experiments reported by Daunys et al. [36] also showed that higher temperatures lead to increased toughness.
We found the more brittle orientation to be the [1–100] and this is related to the fact that in this orientation deformation at the crack tip occurs only through twinning with almost no dislocation activity. The other two orientations are more ductile and deformation proceeds through different dislocation mechanisms. For the basal orientation these are < a > dislocations while < c + a > dislocations are dominant for the [11–20] texture. Some partial < p > dislocations with the Burgers vectors of the 1/3 < 1–100 > type appear at the GB́ s during the propagation of the crack at the [1–100] texture. These results suggest that deformation through twinning results in a more brittle behavior than deformation through dislocation emission and glide. In our simulations this seems to be related to the fact that deformation by twinning results in crack tip blunting that still has sharp corners, favoring stress concentration. The deformation mechanisms observed at the crack tip for the various orientations tested agree well with the deformation mechanisms that were found in our previous work under tensile testing [21]. In a recent simulation study, Singh et al. [35] also found that deformation in single crystal Zr was governed either by twinning or the emission of dislocations from the crack tip. In agreement with our current work their results suggest that deformation via twinning leads to easier crack propagation, whereas dislocation mechanisms are more effective in improving toughness. We found that the samples with the smallest grain sizes are the more resistant to crack propagation. This is particularly evident in the prismatic orientations, and is due to the grain boundaries constituting barriers to crack propagation. These results are consistent with previous studies of fracture response in nanocrystalline materials in fcc [36] as well as bcc [35,37,38] materials. These effects of grain size are more significant for the cases where deformation occurs through twinning than in the orientations where deformation occurs through dislocation emission and glide.
CRediT authorship contribution statement G. Bertolino: Investigation, Writing - review & editing. M. Ruda: Investigation, Writing - review & editing. D. Farkas: Investigation, Writing - review & editing.
Acknowledgement The authors thank Dr. Enzo Dari for computational resources and help at CAB-CNEA.
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Fig. 9. Atomic shear strain maps as the crack advances on [1–100] textured 27 nm samples at temperatures 77 K, 150 K, 300 K and 650 K and three values of KI: 0.2 MPa m0.5, 1.0 MPa m0.5 and 2.5 MPa m0.5.
Data availability
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