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Mechanisms of martensitic phase transformations in body-centered cubic structural metals and alloys: Molecular dynamics simulations
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Acta Materialia 55 (2007) 6634–6641 www.elsevier.com/locate/actamat
Mechanisms of martensitic phase transformations in body-centered cubic structural metals and alloys: Molecular dynamics simulations Ya-Fang Guo b
a,*
, Yue-Sheng Wang a, Dong-Liang Zhao b, Wen-Ping Wu
a
a Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China Functional Materials Division, Central Iron and Steel Research Institute, Beijing 100081, China
Received 15 June 2007; received in revised form 30 July 2007; accepted 16 August 2007 Available online 4 October 2007
Abstract Two different mechanisms of the stress-induced martensitic phase transformation at the crack tip in body-centered cubic (bcc) structural metals and alloys have been studied by molecular dynamics simulations. For cracks with Æ1 0 0æ crack fronts, the bcc (B2) to facecentered cubic (fcc) (L10) phase transformation along the Bain stretch occurs. Whereas for cracks with Æ1 1 0æ crack fronts, either the bcc (B2) to fcc (L10) or the bcc (B2) to hexagonal close-packed (hcp) transformation is the candidate. We have found that the combination of local stress and crystal orientation plays an important role in the mechanism of the martensitic transformation. Thus a simple way to determine the mechanism of the martensitic transformation is developed. The complicated deformation behaviors at the crack tip in bcc iron and B2 NiAl are discussed in terms of this method. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Molecular dynamics simulations; Fracture; Martensitic transformation
1. Introduction Martensitic phase transformations are diffusionless lattice-distorting phase transformations of the crystal, in which atoms move cooperatively and often via a shear-like mechanism. In the past decades, martensitic transformations have been the subject of extensive experimental and theoretical investigations because they have an important effect on the physical and deformation properties of a material. Usually, martensitic transformations are temperatureinduced and may occur over a wide range of temperatures. However, stress-induced martensitic transformations are also widely observed and have a notable influence on the crack propagation properties and toughness of a material. For example, pioneering experiments [1] discovered that the ground state crystal structure of body-centered cubic *
Corresponding author. Fax: +86 10 51682094. E-mail address: [email protected] (Y.-F. Guo).
(bcc) iron undergoes a stress-induced martensitic phase transformation to a hexagonal close-packed (hcp) structure. For a certain range of pressure and temperature, bcc iron was also found to transform into a reversible face-centered cubic (fcc) structure, and other observations showed nanograin formation associated with stress-induced phase transformation in iron [2–4]. Moreover, martensites at crack tips in NiAl have been observed by transmission electron microscopy (TEM) [5] and atomic force microscopy [6]. The fracture toughness of NiAl was found to be influenced by the presence of thermally induced and stress-induced martensites [7]. Further TEM observations indicated that the martensite in NiAl alloys has a fine periodical microtwin structure with a typical stacking period of close-packed layers. The atomic configurations at macrotwin interfaces between microtwinned martensite plates were investigated in Refs. [8,9]. In addition to experimental observations, a detailed understanding of the martensitic transformation
1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.08.018
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mechanisms and the related phenomena of twinning and microstructure formation is necessary for the control, improvement and design of the materials properties. It is known that the martensitic transformations are purely displacing transformations. During the transformation, the atoms of a highly ordered parent crystal rearrange in a coordinated manner, leading to a new product crystalline phase. From a geometrical point of view, the martensitic transformation from cubic to compact structures is easily represented by the classical Bain relation between the bcc and fcc structures [10]. The different close-packed configurations are then discussed and interpreted as the result of suitable piecewise-linear deformations involving the Bain stretch variants [11–14]. In addition, a stress-induced martensitic phase transformation from bcc to hcp structures is clearly observed in iron. It is proposed that this martensitic transformation obeys certain crystallographic relations described by the Burgers scheme [15], which involves the opposite displacement of the adjacent {1 1 0} plane in the Æ1 1 0æ direction, composed of a stretching and a compression along the Æ1 1 0æ and Æ0 0 1æ directions, respectively. In our previous works, both twinning and phase transformation were observed at the crack tip in bcc iron by molecular dynamics simulations, and the mechanisms of phase transformation from bcc to hcp structures were carefully discussed [16,17]. More recently, we investigated the deformation behavior of the crack tip in B2 NiAl alloys, and a clear picture of the layered structures of martensites with a typical stacking period of close-packed layers was presented by atomistic simulations [18]. It is known that the local stress and composition conditions can play an important role in the final structures appearing in a material. Therefore, this work will compare the mechanisms of stress-induced martensitic transformations at the crack tip in bcc iron and B2 NiAl, with the aim of investigating the effects of local stress on the deformation behavior at the crack tip in bcc structural metals and alloys. 2. Conditions of computer simulations In the simulations, five types of mode I (opening mode) cracks both in bcc iron and in ordered B2 NiAl are studied, including types of {1 0 0}Æ1 0 0æ, {1 1 0}Æ1 0 0æ, {1 0 0}Æ1 1 0æ,
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{1 1 0}Æ1 1 0æ and {1 1 1}Æ1 1 0æ as shown in Fig. 1. Herein {1 0 0}, {1 1 0} and {1 1 1} indicate the directions of the crack planes; Æ1 0 0æ and Æ1 1 0æ indicate the directions of the crack fronts. The plane-strain problem is considered and the periodic boundary condition is applied in the direction parallel to the crack front. The outmost two (or four) layers of atoms in the model are identified to the boundary region (except for the crack faces) in atomistic simulations. The thickness of the boundary region is larger than the cutoff distance of the potential. For bcc iron, the Johnson pair potential [19] was used in the early atomistic simulations. Subsequently, many types of the embedded atom method (EAM) interatomic potentials [20–23] have been developed to describe the atomic interactions in bcc iron. In this work, the Finnis–Sinclair N-body potential [24] was employed, which has been widely used to study the deformation behavior, e.g. dislocation nucleation and emission, twinning, fracture and plasticity behavior, in bcc crystal [16,25,26]. It has been proved to be efficient for the calculation of deformation behavior in bcc crystals. For ordered B2 NiAl alloys, many-body interatomic potentials were used based on the EAM developed by Farkas et al. [27], which were built on the basis of the Ni and Al potentials from Voter and Chen [28]. In previous works, the potentials of Voter and Chen were successfully applied to simulations of the B2 to L10 martensitic transformation in NiAl [29–31]. However, these potential did not predict the B2 phase as the stable structure at the equi-atomic composition [32]. The potentials developed by Farkas et al. reproduce the properties of B2 NiAl and predict B2 as the stable phase for the equi-atomic composition. Before atomistic simulations, the pre-existing crack in the model is obtained according to continuum theory by shifting the atoms from their positions in a perfect crystal to positions specified by the anisotropic elastic continuum mechanics equations [33] for the desired stress intensity KI. After the initial configurations of the crack at different loads have been obtained, the atoms at the crack tip are fully relaxed for 4000 steps of magnitude 5 · 1015 s (or 2 · 1015 s for NiAl) with the fixed-displacement boundary condition, and the structure evolution of the crack tip at different loading levels is investigated. The temperature of
Fig. 1. Lattice structure of bcc crystal: (a) for {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks; (b) for {1 0 0}Æ1 1 0æ, {1 1 0}Æ1 1 0æ and {1 1 1}Æ1 1 0æ cracks.
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the system is kept invariant at 5 K throughout the simulation, which is achieved by scaling the instantaneous velocities of all atoms with the appropriate Maxwell–Boltzmann distribution at a specified temperature. The XMD program is used for the atomistic simulations [34]. According to anisotropic linear elastic continuum theory, the displacement and stress fields of a straight crack for the plane-strain condition [33] are given by fug ¼ K I ðrÞ
1=2
ff ðhÞg;
ð1=2Þ
frg ¼ K I ðrÞ
ð1Þ
fgðhÞg;
ð2Þ
where fug ¼ fux ; uy gT ;
ff g ¼ ffx ; fy gT ;
frg ¼ frx ; ry ; rxy gT ;
fgg ¼ fgx ; gy ; gxy gT ;
ð3Þ
and K I ¼ rðapÞ
1=2
:
ð4Þ
In Eqs. (1)–(4), KI is the stress intensity factor; r and h are the polar coordinates of the atom for which the displacement is calculated; r is the distance from the atom to the crack tip; g is a function of the angle h between r and the crack plane; r is the applied load; and a is the length of the crack. In the present study, the stress intensity factor KI is used to express the corresponding external load r, which can be defined as nKIC, where KIC is the critical stress intensity factor. Several systems of different sizes were tested beforehand, and it was verified that the basic results of the structure evolution at the crack tip in the simulation are neither affected by the presence of the fixed boundary nor insensitive to the change in the system size if the simulated atomistic region is sufficiently large. For bcc iron, we select systems with dimensions of 35 · 35 · 0.8 (or 1.1) nm (for the Æ0 0 1æ or Æ1 1 0æ crack front, respectively), which contain 97,200–138,528 atoms in total. The initial crack length in ˚ . For B2 NiAl, the choour calculation model is about 37 A sen systems had dimensions of 33 · 33 · 1.4 (or 1.6) nm, which contain about 128,000–143,360 atoms for different ˚. crack systems. The initial crack length is about 33 A 3. Stress-induced martensitic phase transformation in bcc iron
Fig. 2 shows the atomistic configuration of the crack tip at 5 K under a load of 1.9KIC for the {1 0 0}Æ1 0 0æ, {1 1 0}Æ1 0 0æ and {1 0 0}Æ1 1 0æ cracks; two long twin strips form at the stressed crack tip for these three types of cracks. In Fig. 2a and b for the {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks, we find that the cubic structure unit transforms to a tetragonal unit at the twinned region, and a bcc-to-fcc phase transformation occurs. A sharp habit plane is observed between the parent and transformed structures, and the twinned martensite strips at the crack tip grow on the habit planes of ð1 1 0Þ and (1 1 0) along the [1 1 0] and ½1 1 0 directions separately. The resulting twin boundary is a prior {1 1 0}bcc plane. The (0 1 1)fcc plane of the twinned martensite unit originates from the former (0 0 1)bcc plane of the bcc crystal. We observe that this martensitic transformation occurs along the Bain path. As illustrated in Fig. 3, by varying the ratio c/a, the bcc unit is continuously transformed into an fcc unit along the Bain path. When a bcc lattice unit undergoes a stretching along the Æ0 1 0æ direction, the bcc-to-fcc phase transformation p ffiffiffi occurs, and the perfect ratio c/a of the Bain stretch is 2. However, owing to the volume compatibility conditions between the parent and product phases, we notice that in our present work, the atomic distance in the [0 1 0] direction ˚ to approximately 3.60 A ˚ after the increases from 2.8665 A phase transformation, while the atomic distance in the ˚ to approximately [1 0 0] direction decreases from 2.8665 A ˚ . Thus the ratio c/a of the formed fcc unit is about 2.69 A 1.38, pffiffiffi a little smaller than the perfect Bain stretch value of 2. In Fig. 2c for the {1 0 0}Æ1 1 0æ crack, although the representative structure unit in the {1 1 0} plane is different from that in the {1 0 0} plane, the same bcc-to-fcc transformation occurring at the twinned martensite strips at the crack tip is observed. However, the twin boundary is a prior {1 2 1}bcc plane, instead of the {1 1 0}bcc plane as shown in Fig. 2a and b. The twinned martensite strips at the crack tip grow on the habit planes of f1 2 1g and f1 2 1g along the ½1 1 1 and ½1 1 1 directions, respectively. The (0 0 1)fcc plane of the fcc unit originates from the former (0 1 1)bcc plane, thus the (0 1 1)bcc plane of the parent bcc crystal is identified as the (0 01·)fcc plane of the formed fcc lattice. The ratio c/a of the formed fcc unit is about 1.33, a little smaller than the perfect Bain stretch value. Moreover, crack blunting is also observed due to new grain nucleation at the crack tip in Fig. 2c.
3.1. Phase transformation from bcc to fcc structures 3.2. Mechanisms of martensitic transformation in bcc iron In our previous work [17], phase transformation from bcc to hcp structures has been observed at the crack tip, and the mechanism of the martensitic transformation has been discussed. Moreover, previous works [16] have shown that the twinning and the cleavage fracture at low temperatures or at high strain rates are cooperating processes in bcc iron. However, the twinning at the crack tip is found to be accompanied by a bcc-to-fcc phase transformation in the present work.
We have found two different mechanisms of the stressinduced martensitic phase transformation at the crack tip in bcc iron for different crack orientations: (i) a bcc-to-fcc transformation caused by a stretch in the [0 1 0] direction along the classical Bain path; (ii) a bcc-to-hcp transformation through a shear in the {1 1 0} plane and relative displacement of atoms in different atomic layers [17], as previously proposed by Mao et al. in 1967 [35]. For cracks
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Fig. 2. Atomistic configuration of crack at 5 K under a load of 1.9 KIC for: (a) {1 0 0}Æ1 0 0æ crack; (b) {1 1 0}Æ1 0 0æ crack; and (c) {1 0 0}Æ1 1 0æ crack. Solid and open circles indicate atoms in A and B layers; small squares indicate the representative structure units.
Fig. 3. Bain relation between bcc and fcc structures.
with Æ1 0 0æ crack fronts, such as the {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks, only the bcc-to-fcc transformation along the Bain path occurs. For cracks with Æ1 1 0æ crack fronts, two types of martensitic transformation can occur. For example, both the bcc-to-fcc and bcc-to-hcp transformations occur at the crack tip for the {1 1 1}Æ1 1 0æ crack. As shown in Fig. 4, a new grain nucleates above the crack plane, accompanying a bcc-to-hcp phase transformation. The (0 1 1)bcc plane of the parent bcc crystal is identified to the (0 0 0 1)hcp plane of the formed hcp crystal. Below the crack plane, a twinned martensite strip is observed lying along the ½1 1 1 direction with a ð1 2 1Þ twin plane, and a bcc-to-fcc transformation occurs. The (0 1 1)bcc plane of the parent bcc crystal is identified as the (0 0 1)fcc plane, this is exactly consistent with the results in Fig. 2c for the {1 0 0}Æ1 1 0æ crack. Experimental results have revealed that for a certain combination of pressure and temperature, both the bccto-fcc and bcc-to-hcp phase transformations can be
Fig. 4. Details of the crack tip region at 5 K under a load of 2.0 KIC: (a) {1 1 0}Æ1 1 0æ crack after 1000 steps of relaxation; (b) {1 1 1}Æ1 1 0æ crack after 400 steps of relaxation. Solid and open circles indicate atoms in A and B layers.
observed in bcc iron [1–4]. Moreover, multimillion-atom molecular dynamics simulations [36] were used to investigate the shock-induced phase transformation of solid iron, and both the bcc-to-fcc and bcc-to-hcp phase transformations were observed under shock loading. The results indi-
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cated that the dynamics and orientation of the developing close-packed grains depend on the shock strength and the crystallographic shock direction. More recently, some experimental works have focused on the role of shear stresses in the critical pressure for phase transformation [37,38]. The results indicated that the combination of the high strain-rate sensitivity and dominant shear loading conditions seem to trigger the phase transition [39]. Atomistic calculations have also shown the significant contribution of shear strain to the bcc-to-hcp transition [40]. In our results, we have found that both the twinning and phase transformation are formed by the emission of 16 h1 1 1i partial dislocations along the {1 1 2} slip plane, which is actually caused by the shear stress at the crack tip region [17]. Thus we surmise that the combination of the local shear stress and the crystal orientation plays an important role on the mechanism of the martensitic transformation. Next we will develop a simple way to determine the deformation mechanism at the crack tip based on the analyses of the crystal structure and the local stress. 3.3. Method to determine the mechanisms of martensitic transformation First we have to select the representative structure units in the {1 0 0} plane and {1 1 0} plane, respectively. As shown in Fig. 2a and b, a cubic unit in the {1 0 0}bcc plane is selected for cracks with Æ1 0 0æ crack fronts. For cracks with Æ1 1 0æ crack fronts, a tetragonal unit in the {1 1 0}bcc plane is selected as shown in Figs. 2c and 4. Secondarily, the local stress applied on the selected unit which causes the structural transformation is simplified to a simple two-dimensional shear stress. As shown in Fig. 5, for cracks with Æ1 0 0æ crack fronts, a (1 0 0) lattice unit deforms under a simple two-dimensional shear of ½1 1 0ð 1 1 0Þ. The direction of the horizontal axis is parallel to the shear direction of [1 1 0], and the vertical axis is perpendicular to the ð 1 1 0Þ habit plane. Fig. 6a shows that for cracks with Æ1 1 0æ crack fronts, a (1 1 0) tetragonal lattice unit deforms under a simple two-dimensional shear of ½1 1 1ð 1 2 1Þ. In Fig. 6b, an opposite shear of ½ 1 1 1ð 1 2 1Þ is applied. The direction of the horizontal axis is parallel to the shear direction of ½1 1 1 or ½ 1 1 1, respectively. The vertical axis is perpendicular to the ð 1 2 1Þ habit plane.
Fig. 6. Lattice deformation of a (1 0 1) unit under simple two-dimensional shears of (a) ½1 1 1ð1 2 1Þ; and (b) ½1 1 1ð1 2 1Þ. Dot lines and solid lines indicates representative units before and after deformation, respectively.
We define the shorter axis of the tetragonal unit as the c axis in Fig. 6, the angle between the c axis and the shear direction is a. We find in Fig. 6a that, when the angle a is smaller than 90, shear deformation occurs easily, and the tetragonal unit transforms to a square unit along the Bain stretch. However, if the angle a is larger than 90, shear deformation becomes very difficult due to the higher deformation resistance, thus the Bain stretch is not the preferred path. Therefore, for cracks with Æ1 1 0æ crack fronts, two types of martensitic transformation mechanisms are optional. One is the bcc-to-fcc transformation along the Bain path as we have mentioned in Section 3.1; the other is the bcc-to-hcp transformation. For the {1 0 0}Æ1 1 0æ crack in Fig. 2c, the angle a is smaller than 90, and a bcc-to-fcc transformation occurs along the Bain path. Whereas for the {1 1 0}Æ1 1 0æ crack, the angle a is larger than 90, thus the bcc-to-hcp transformation is preferred. For the {1 1 1}Æ1 1 0æ crack in Fig. 4, both types of martensitic transformation occur. Above the crack plane, the angle a is larger than 90, and a new grain nucleates accompanying the bcc-to-hcp transformation; below the crack plane, the angle a is smaller than 90, and a twinned martensite strip forms along the Bain path. In Fig. 5 for cracks with Æ1 0 0æ crack fronts, a bcc-to-fcc transformation can occur along the Bain path either under the shear of [1 1 0] or the opposite ½1 1 0. Thus in Fig. 2a and b for the {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks, a bccto-fcc transformation occurs along the Bain path accompanying long, twinned martensite strips formed at the crack tip. From the above analyses, we conclude that this method can be used to determine the mechanisms of the martensitic transformation in bcc iron. In the following section, we will apply this method to analyze martensitic phase transformations in ordered B2 NiAl alloys. 4. Discussion on the deformation behavior in B2 NiAl 4.1. For cracks with Æ1 0 0æ crack fronts
Fig. 5. Lattice deformation of a (0 0 1) unit under a simple twodimensional shear of ½1 1 0ð1 1 0Þ. Dot lines and solid lines indicate representative units before and after deformation, respectively.
The deformation behavior at the crack tip in B2 NiAl has been investigated previously [18]. In this section, based on the analysis of the martensitic transformation mechanisms in bcc iron, the deformation mechanisms at the crack
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tip in B2 NiAl will be discussed. As shown in Fig. 7 for cracks with Æ1 0 0æ crack fronts in B2 NiAl, we have observed a similar bcc (B2)-to-fcc (L10) structure transformation at the crack tip. After a stretching along the Æ1 0 0æ direction, the transformation of the B2 phase into the L10 martensite phase occurs along the Bain path. The habit plane is also the {1 1 0} plane. The ratio c/a of the transformed L10 martensite phase is about 1.25, showing pffiffiaffi clear deviation from the perfect Bain stretch value of 2. It is also a little smaller than the ratio c/a of the formed fcc unit in iron. Moreover, there is a great difference between bcc iron and B2 NiAl in terms of the bcc (B2)-to-fcc (L10) structure transformations. The transformed martensites in B2 NiAl are found with layered structures. The faulted lines lying on the {1 0 0} plane can be seen clearly in Fig. 7. These internally faulted martensites formed at the crack tip in NiAl alloys are different from the twinned martensite strips in pure metallic iron. It is known that the primary slip plane in NiAl is the {1 0 0} plane, which is consistent with the faulted martensites plates in Fig. 7. Thus we assume that the formation of this internally faulted structure is closely related with the atoms bonding in different atoms layers, as well as the alloy composition. The relatively close-packed {1 1 0} plane is the primary cleavage plane in NiAl alloys, which is different from the {1 0 0} cleavage plane in bcc iron. Thus we observe the dislocation emission perpendicular to the crack surface in Fig. 7a, which causes the crack blunting. Moreover, due to the dislocation emission along the ½0 1 0 direction, we find that the shear direction in the left side of the dislocations is different to that in the right side. Thus in the left region of Fig. 7a, just below the crack plane, we can observe that the Bain stretch occurs along the ½ 1 0 0 direction. A faulted line appears on the (1 0 0) plane, which is perpendicular to those (0 1 0) faulted planes. A crossing type of martensites units is observed just below the crack tip region.
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4.2. For cracks with Æ1 1 0æ crack fronts As discussed in Section 3, two types of martensitic transformation mechanisms are possible for cracks with Æ1 1 0æ crack fronts in bcc iron. In Fig. 8 for the Æ1 1 0æ cracks in ordered B2 NiAl alloys, we have only observed a bcc (B2)-to-hcp transformation at the crack tip. In the top left of Fig. 8b for the {1 1 0}Æ1 1 0æ crack, a representative unit with the ½1 1 1 shear direction is marked out. The angle a is larger than 90, and thus the phase transformation from B2 to hcp occurs. This martensitic transformation is caused by atomic relative displacement in different atomic layers and a shear in the {1 1 0} plane, which is consistent with the mechanism in bcc iron. Moreover, two long martensite strips appear at the crack tip accompanying crack cleavage along the ½1 0 1 plane, which is completely different from the result in bcc iron in which a new martensite grain nucleates at the crack tip accompanying crack branching. In Fig. 8a for the {1 0 0}Æ1 1 0æ crack, crack blunting occurs due to the dislocations emission. At the blunted crack tip region, some changes that are not distinguishable are observed. The stress concentration at the crack tip is relaxed by dislocation emission and crack blunting, thus no martensite strip exists at the crack tip. In the left and right sides of the dislocation, we present two representative units, respectively. The shear directions for these two representative units are opposite due to the dislocation emission. For the left side unit, the angle a is larger than 90. Thus just below the crack surface, the martensitic phase transformation from the B2 structure to a hcp structure occurs, which is different from the result of the {1 0 0}Æ1 1 0æ crack in bcc iron that a bcc-to-fcc transition occurs along the Bain path. It can be concluded from this result that the deformation mechanisms at the crack tip depend strongly on the condition of the local stress. It should be mentioned that the cases of phase transformations below the crack surface, separately shown in Figs. 7a and 8a, may not agree well with the experimental results
Fig. 7. Details of the crack tip region at 5 K under a load of (a) 2.0 KIC for a {1 0 0}Æ1 0 0æ crack and 3.0 KIC a {1 1 0}Æ1 0 0æ crack. Open circles represent Ni atoms, while solid circles represent Al atoms.
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Fig. 8. Details of the crack tip region at 5 K under a load of (a) 1.8 KIC for a {1 0 0}Æ1 1 0æ crack and (b) 2.0 KIC for a {1 1 0}Æ1 1 0æ crack. Big circles represent Ni atoms, while small circles represent Al atoms. Solid and open circles indicate atoms in A and B layers, respectively.
in an actual material. These phase transformations are caused by the combined effects of dislocation emission and the restriction of the fixed boundary due to the simulation condition. However, we can conclude from above simulation results that the deformation mechanisms at the local positions are finally determined by the local stress and composition condition. In an actual material under the influences of the external load and temperature, the local stress and composition condition are very complicated. Therefore, very complicated deformation behaviors have been observed in the experimental results. For example, the high-resolution TEM observations have revealed a crossing-type macrotwin with different orientations of microtwin invariants in NiAl martensite [8]. Although our simulated results, such as the formation the crossing type of martensite units as shown in Fig. 7a, are too simple compared with this experimental phenomenon, it provides a useful comparison with the experimental results. 5. Conclusions We have investigated the low-temperature deformation at the crack tip in bcc iron by molecular dynamics simulations. Two different mechanisms of martensitic phase transformations for different types of cracks are discussed in particular. One is the bcc-to-fcc transformation caused by a stretch in the Æ0 1 0æ direction along the Bain path. The other is the bcc-to-hcp transformation through a shear in the {1 1 0} plane and atomic relative displacement in different atomic layers. The choice of the mechanisms of the martensitic transformations depends on the local stress at the crack tip and the crystal orientation. A simple way to determine the deformation behavior at the crack tip is presented. We apply this method to analyze the martensitic phase transformation in the ordered B2 NiAl alloys. The complicated deformation behavior at the crack tip is discussed.
Moreover, based on the studies of the deformation behavior at the crack tip in bcc structural metals and alloys, we find that the crack propagation and the fracture toughness of a material are closely related to the structural evolution at the crack tip region. Crack cleavage usually accompanies with the formation of twinned or martensite strips at the crack tip, while crack blunting occurs due to the dislocation emission or a new grain nucleation at the crack tip. Acknowledgements The research was supported by Chinese Nature Science Foundation (Grant No. 10672016) and NJTU Science Foundation (No. 2005SM0035). References [1] Bancroft D, Peterson E, Minshall S. J Appl Phys 1956;27:291. [2] Fujiwara H, Inomoto H, Sanada R, Ameyama K. Scripta Mater 2001;44:2039. [3] Zhang J, Guyot F. Phys Chem Minerals 1999;26:419. [4] Ivanisenko Yu, MacLaren I, Sauvage X, Valiev RZ, Fecht HJ. Solid State Phen 2006;114:133. [5] Schryvers D, Tanner L. Micro Microsc Acta 1989;20:153. [6] Hangen UD, Sauthoff G. Micro Mater. In: Proceedings of the MicroMat’97. Berlin; 1997. p. 412. [7] Hangen UD, Sauthoff G. Intermetallics 1999;7:501. [8] Schryvers D, Boullay P, Potapov PL, Kohn RV, Ball JM. Int J Solids Struct 2002;39:3543. [9] Boullay P, Schryvers D, Ball JM. Acta Mater 2003;51:1421. [10] Bain EC. Trans AIME 1924;70:25. [11] Elliott RS, Shaw JA, Triantafyllidis N. J Mech Phys Solids 2006;54: 193. [12] Balandraud X, Zanzotto G. J Mech Phys Solids 2007;55:194. [13] Lee ES, Ahn S. Acta Mater 1998;46:4357. [14] Artunc¸ E. Physica B 2005;366:138. [15] Burgers WG. Physica 1934;1:561. [16] Guo YF, Wang CY, Zhao DL. Mater Sci Eng A 2003;349:29. [17] Guo YF, Wang YS, Zhao DL. Acta Mater 2007;55:401.
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[30] Gue´nin R, Clapp PC, Zhao Y, Rifkin JA. Mater Sci Eng B 1996;37:193. [31] Becquart CS, Clapp PC, Rifkin JA. Phys Rev B 1993;48:6. [32] Clapp PC, Rubins MJ, Charpenay S, Rifkin JA, Yu ZZ, Voter AF. Proc Symp Mater Res Soc 1989;133:29. [33] Sih GC, Liebowitz H. Mathematical theories of brittle facture. In: Liebowitz H, editor. Fracture: an advanced treatise. New York: Academic Press; 1968. [34] Riffkin J, Center for simulation, University of Connecticut, CT.
Acta Materialia 55 (2007) 6634–6641 www.elsevier.com/locate/actamat
Mechanisms of martensitic phase transformations in body-centered cubic structural metals and alloys: Molecular dynamics simulations Ya-Fang Guo b
a,*
, Yue-Sheng Wang a, Dong-Liang Zhao b, Wen-Ping Wu
a
a Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China Functional Materials Division, Central Iron and Steel Research Institute, Beijing 100081, China
Received 15 June 2007; received in revised form 30 July 2007; accepted 16 August 2007 Available online 4 October 2007
Abstract Two different mechanisms of the stress-induced martensitic phase transformation at the crack tip in body-centered cubic (bcc) structural metals and alloys have been studied by molecular dynamics simulations. For cracks with Æ1 0 0æ crack fronts, the bcc (B2) to facecentered cubic (fcc) (L10) phase transformation along the Bain stretch occurs. Whereas for cracks with Æ1 1 0æ crack fronts, either the bcc (B2) to fcc (L10) or the bcc (B2) to hexagonal close-packed (hcp) transformation is the candidate. We have found that the combination of local stress and crystal orientation plays an important role in the mechanism of the martensitic transformation. Thus a simple way to determine the mechanism of the martensitic transformation is developed. The complicated deformation behaviors at the crack tip in bcc iron and B2 NiAl are discussed in terms of this method. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Molecular dynamics simulations; Fracture; Martensitic transformation
1. Introduction Martensitic phase transformations are diffusionless lattice-distorting phase transformations of the crystal, in which atoms move cooperatively and often via a shear-like mechanism. In the past decades, martensitic transformations have been the subject of extensive experimental and theoretical investigations because they have an important effect on the physical and deformation properties of a material. Usually, martensitic transformations are temperatureinduced and may occur over a wide range of temperatures. However, stress-induced martensitic transformations are also widely observed and have a notable influence on the crack propagation properties and toughness of a material. For example, pioneering experiments [1] discovered that the ground state crystal structure of body-centered cubic *
Corresponding author. Fax: +86 10 51682094. E-mail address: [email protected] (Y.-F. Guo).
(bcc) iron undergoes a stress-induced martensitic phase transformation to a hexagonal close-packed (hcp) structure. For a certain range of pressure and temperature, bcc iron was also found to transform into a reversible face-centered cubic (fcc) structure, and other observations showed nanograin formation associated with stress-induced phase transformation in iron [2–4]. Moreover, martensites at crack tips in NiAl have been observed by transmission electron microscopy (TEM) [5] and atomic force microscopy [6]. The fracture toughness of NiAl was found to be influenced by the presence of thermally induced and stress-induced martensites [7]. Further TEM observations indicated that the martensite in NiAl alloys has a fine periodical microtwin structure with a typical stacking period of close-packed layers. The atomic configurations at macrotwin interfaces between microtwinned martensite plates were investigated in Refs. [8,9]. In addition to experimental observations, a detailed understanding of the martensitic transformation
1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.08.018
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mechanisms and the related phenomena of twinning and microstructure formation is necessary for the control, improvement and design of the materials properties. It is known that the martensitic transformations are purely displacing transformations. During the transformation, the atoms of a highly ordered parent crystal rearrange in a coordinated manner, leading to a new product crystalline phase. From a geometrical point of view, the martensitic transformation from cubic to compact structures is easily represented by the classical Bain relation between the bcc and fcc structures [10]. The different close-packed configurations are then discussed and interpreted as the result of suitable piecewise-linear deformations involving the Bain stretch variants [11–14]. In addition, a stress-induced martensitic phase transformation from bcc to hcp structures is clearly observed in iron. It is proposed that this martensitic transformation obeys certain crystallographic relations described by the Burgers scheme [15], which involves the opposite displacement of the adjacent {1 1 0} plane in the Æ1 1 0æ direction, composed of a stretching and a compression along the Æ1 1 0æ and Æ0 0 1æ directions, respectively. In our previous works, both twinning and phase transformation were observed at the crack tip in bcc iron by molecular dynamics simulations, and the mechanisms of phase transformation from bcc to hcp structures were carefully discussed [16,17]. More recently, we investigated the deformation behavior of the crack tip in B2 NiAl alloys, and a clear picture of the layered structures of martensites with a typical stacking period of close-packed layers was presented by atomistic simulations [18]. It is known that the local stress and composition conditions can play an important role in the final structures appearing in a material. Therefore, this work will compare the mechanisms of stress-induced martensitic transformations at the crack tip in bcc iron and B2 NiAl, with the aim of investigating the effects of local stress on the deformation behavior at the crack tip in bcc structural metals and alloys. 2. Conditions of computer simulations In the simulations, five types of mode I (opening mode) cracks both in bcc iron and in ordered B2 NiAl are studied, including types of {1 0 0}Æ1 0 0æ, {1 1 0}Æ1 0 0æ, {1 0 0}Æ1 1 0æ,
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{1 1 0}Æ1 1 0æ and {1 1 1}Æ1 1 0æ as shown in Fig. 1. Herein {1 0 0}, {1 1 0} and {1 1 1} indicate the directions of the crack planes; Æ1 0 0æ and Æ1 1 0æ indicate the directions of the crack fronts. The plane-strain problem is considered and the periodic boundary condition is applied in the direction parallel to the crack front. The outmost two (or four) layers of atoms in the model are identified to the boundary region (except for the crack faces) in atomistic simulations. The thickness of the boundary region is larger than the cutoff distance of the potential. For bcc iron, the Johnson pair potential [19] was used in the early atomistic simulations. Subsequently, many types of the embedded atom method (EAM) interatomic potentials [20–23] have been developed to describe the atomic interactions in bcc iron. In this work, the Finnis–Sinclair N-body potential [24] was employed, which has been widely used to study the deformation behavior, e.g. dislocation nucleation and emission, twinning, fracture and plasticity behavior, in bcc crystal [16,25,26]. It has been proved to be efficient for the calculation of deformation behavior in bcc crystals. For ordered B2 NiAl alloys, many-body interatomic potentials were used based on the EAM developed by Farkas et al. [27], which were built on the basis of the Ni and Al potentials from Voter and Chen [28]. In previous works, the potentials of Voter and Chen were successfully applied to simulations of the B2 to L10 martensitic transformation in NiAl [29–31]. However, these potential did not predict the B2 phase as the stable structure at the equi-atomic composition [32]. The potentials developed by Farkas et al. reproduce the properties of B2 NiAl and predict B2 as the stable phase for the equi-atomic composition. Before atomistic simulations, the pre-existing crack in the model is obtained according to continuum theory by shifting the atoms from their positions in a perfect crystal to positions specified by the anisotropic elastic continuum mechanics equations [33] for the desired stress intensity KI. After the initial configurations of the crack at different loads have been obtained, the atoms at the crack tip are fully relaxed for 4000 steps of magnitude 5 · 1015 s (or 2 · 1015 s for NiAl) with the fixed-displacement boundary condition, and the structure evolution of the crack tip at different loading levels is investigated. The temperature of
Fig. 1. Lattice structure of bcc crystal: (a) for {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks; (b) for {1 0 0}Æ1 1 0æ, {1 1 0}Æ1 1 0æ and {1 1 1}Æ1 1 0æ cracks.
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the system is kept invariant at 5 K throughout the simulation, which is achieved by scaling the instantaneous velocities of all atoms with the appropriate Maxwell–Boltzmann distribution at a specified temperature. The XMD program is used for the atomistic simulations [34]. According to anisotropic linear elastic continuum theory, the displacement and stress fields of a straight crack for the plane-strain condition [33] are given by fug ¼ K I ðrÞ
1=2
ff ðhÞg;
ð1=2Þ
frg ¼ K I ðrÞ
ð1Þ
fgðhÞg;
ð2Þ
where fug ¼ fux ; uy gT ;
ff g ¼ ffx ; fy gT ;
frg ¼ frx ; ry ; rxy gT ;
fgg ¼ fgx ; gy ; gxy gT ;
ð3Þ
and K I ¼ rðapÞ
1=2
:
ð4Þ
In Eqs. (1)–(4), KI is the stress intensity factor; r and h are the polar coordinates of the atom for which the displacement is calculated; r is the distance from the atom to the crack tip; g is a function of the angle h between r and the crack plane; r is the applied load; and a is the length of the crack. In the present study, the stress intensity factor KI is used to express the corresponding external load r, which can be defined as nKIC, where KIC is the critical stress intensity factor. Several systems of different sizes were tested beforehand, and it was verified that the basic results of the structure evolution at the crack tip in the simulation are neither affected by the presence of the fixed boundary nor insensitive to the change in the system size if the simulated atomistic region is sufficiently large. For bcc iron, we select systems with dimensions of 35 · 35 · 0.8 (or 1.1) nm (for the Æ0 0 1æ or Æ1 1 0æ crack front, respectively), which contain 97,200–138,528 atoms in total. The initial crack length in ˚ . For B2 NiAl, the choour calculation model is about 37 A sen systems had dimensions of 33 · 33 · 1.4 (or 1.6) nm, which contain about 128,000–143,360 atoms for different ˚. crack systems. The initial crack length is about 33 A 3. Stress-induced martensitic phase transformation in bcc iron
Fig. 2 shows the atomistic configuration of the crack tip at 5 K under a load of 1.9KIC for the {1 0 0}Æ1 0 0æ, {1 1 0}Æ1 0 0æ and {1 0 0}Æ1 1 0æ cracks; two long twin strips form at the stressed crack tip for these three types of cracks. In Fig. 2a and b for the {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks, we find that the cubic structure unit transforms to a tetragonal unit at the twinned region, and a bcc-to-fcc phase transformation occurs. A sharp habit plane is observed between the parent and transformed structures, and the twinned martensite strips at the crack tip grow on the habit planes of ð1 1 0Þ and (1 1 0) along the [1 1 0] and ½1 1 0 directions separately. The resulting twin boundary is a prior {1 1 0}bcc plane. The (0 1 1)fcc plane of the twinned martensite unit originates from the former (0 0 1)bcc plane of the bcc crystal. We observe that this martensitic transformation occurs along the Bain path. As illustrated in Fig. 3, by varying the ratio c/a, the bcc unit is continuously transformed into an fcc unit along the Bain path. When a bcc lattice unit undergoes a stretching along the Æ0 1 0æ direction, the bcc-to-fcc phase transformation p ffiffiffi occurs, and the perfect ratio c/a of the Bain stretch is 2. However, owing to the volume compatibility conditions between the parent and product phases, we notice that in our present work, the atomic distance in the [0 1 0] direction ˚ to approximately 3.60 A ˚ after the increases from 2.8665 A phase transformation, while the atomic distance in the ˚ to approximately [1 0 0] direction decreases from 2.8665 A ˚ . Thus the ratio c/a of the formed fcc unit is about 2.69 A 1.38, pffiffiffi a little smaller than the perfect Bain stretch value of 2. In Fig. 2c for the {1 0 0}Æ1 1 0æ crack, although the representative structure unit in the {1 1 0} plane is different from that in the {1 0 0} plane, the same bcc-to-fcc transformation occurring at the twinned martensite strips at the crack tip is observed. However, the twin boundary is a prior {1 2 1}bcc plane, instead of the {1 1 0}bcc plane as shown in Fig. 2a and b. The twinned martensite strips at the crack tip grow on the habit planes of f1 2 1g and f1 2 1g along the ½1 1 1 and ½1 1 1 directions, respectively. The (0 0 1)fcc plane of the fcc unit originates from the former (0 1 1)bcc plane, thus the (0 1 1)bcc plane of the parent bcc crystal is identified as the (0 01·)fcc plane of the formed fcc lattice. The ratio c/a of the formed fcc unit is about 1.33, a little smaller than the perfect Bain stretch value. Moreover, crack blunting is also observed due to new grain nucleation at the crack tip in Fig. 2c.
3.1. Phase transformation from bcc to fcc structures 3.2. Mechanisms of martensitic transformation in bcc iron In our previous work [17], phase transformation from bcc to hcp structures has been observed at the crack tip, and the mechanism of the martensitic transformation has been discussed. Moreover, previous works [16] have shown that the twinning and the cleavage fracture at low temperatures or at high strain rates are cooperating processes in bcc iron. However, the twinning at the crack tip is found to be accompanied by a bcc-to-fcc phase transformation in the present work.
We have found two different mechanisms of the stressinduced martensitic phase transformation at the crack tip in bcc iron for different crack orientations: (i) a bcc-to-fcc transformation caused by a stretch in the [0 1 0] direction along the classical Bain path; (ii) a bcc-to-hcp transformation through a shear in the {1 1 0} plane and relative displacement of atoms in different atomic layers [17], as previously proposed by Mao et al. in 1967 [35]. For cracks
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Fig. 2. Atomistic configuration of crack at 5 K under a load of 1.9 KIC for: (a) {1 0 0}Æ1 0 0æ crack; (b) {1 1 0}Æ1 0 0æ crack; and (c) {1 0 0}Æ1 1 0æ crack. Solid and open circles indicate atoms in A and B layers; small squares indicate the representative structure units.
Fig. 3. Bain relation between bcc and fcc structures.
with Æ1 0 0æ crack fronts, such as the {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks, only the bcc-to-fcc transformation along the Bain path occurs. For cracks with Æ1 1 0æ crack fronts, two types of martensitic transformation can occur. For example, both the bcc-to-fcc and bcc-to-hcp transformations occur at the crack tip for the {1 1 1}Æ1 1 0æ crack. As shown in Fig. 4, a new grain nucleates above the crack plane, accompanying a bcc-to-hcp phase transformation. The (0 1 1)bcc plane of the parent bcc crystal is identified to the (0 0 0 1)hcp plane of the formed hcp crystal. Below the crack plane, a twinned martensite strip is observed lying along the ½1 1 1 direction with a ð1 2 1Þ twin plane, and a bcc-to-fcc transformation occurs. The (0 1 1)bcc plane of the parent bcc crystal is identified as the (0 0 1)fcc plane, this is exactly consistent with the results in Fig. 2c for the {1 0 0}Æ1 1 0æ crack. Experimental results have revealed that for a certain combination of pressure and temperature, both the bccto-fcc and bcc-to-hcp phase transformations can be
Fig. 4. Details of the crack tip region at 5 K under a load of 2.0 KIC: (a) {1 1 0}Æ1 1 0æ crack after 1000 steps of relaxation; (b) {1 1 1}Æ1 1 0æ crack after 400 steps of relaxation. Solid and open circles indicate atoms in A and B layers.
observed in bcc iron [1–4]. Moreover, multimillion-atom molecular dynamics simulations [36] were used to investigate the shock-induced phase transformation of solid iron, and both the bcc-to-fcc and bcc-to-hcp phase transformations were observed under shock loading. The results indi-
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cated that the dynamics and orientation of the developing close-packed grains depend on the shock strength and the crystallographic shock direction. More recently, some experimental works have focused on the role of shear stresses in the critical pressure for phase transformation [37,38]. The results indicated that the combination of the high strain-rate sensitivity and dominant shear loading conditions seem to trigger the phase transition [39]. Atomistic calculations have also shown the significant contribution of shear strain to the bcc-to-hcp transition [40]. In our results, we have found that both the twinning and phase transformation are formed by the emission of 16 h1 1 1i partial dislocations along the {1 1 2} slip plane, which is actually caused by the shear stress at the crack tip region [17]. Thus we surmise that the combination of the local shear stress and the crystal orientation plays an important role on the mechanism of the martensitic transformation. Next we will develop a simple way to determine the deformation mechanism at the crack tip based on the analyses of the crystal structure and the local stress. 3.3. Method to determine the mechanisms of martensitic transformation First we have to select the representative structure units in the {1 0 0} plane and {1 1 0} plane, respectively. As shown in Fig. 2a and b, a cubic unit in the {1 0 0}bcc plane is selected for cracks with Æ1 0 0æ crack fronts. For cracks with Æ1 1 0æ crack fronts, a tetragonal unit in the {1 1 0}bcc plane is selected as shown in Figs. 2c and 4. Secondarily, the local stress applied on the selected unit which causes the structural transformation is simplified to a simple two-dimensional shear stress. As shown in Fig. 5, for cracks with Æ1 0 0æ crack fronts, a (1 0 0) lattice unit deforms under a simple two-dimensional shear of ½1 1 0ð 1 1 0Þ. The direction of the horizontal axis is parallel to the shear direction of [1 1 0], and the vertical axis is perpendicular to the ð 1 1 0Þ habit plane. Fig. 6a shows that for cracks with Æ1 1 0æ crack fronts, a (1 1 0) tetragonal lattice unit deforms under a simple two-dimensional shear of ½1 1 1ð 1 2 1Þ. In Fig. 6b, an opposite shear of ½ 1 1 1ð 1 2 1Þ is applied. The direction of the horizontal axis is parallel to the shear direction of ½1 1 1 or ½ 1 1 1, respectively. The vertical axis is perpendicular to the ð 1 2 1Þ habit plane.
Fig. 6. Lattice deformation of a (1 0 1) unit under simple two-dimensional shears of (a) ½1 1 1ð1 2 1Þ; and (b) ½1 1 1ð1 2 1Þ. Dot lines and solid lines indicates representative units before and after deformation, respectively.
We define the shorter axis of the tetragonal unit as the c axis in Fig. 6, the angle between the c axis and the shear direction is a. We find in Fig. 6a that, when the angle a is smaller than 90, shear deformation occurs easily, and the tetragonal unit transforms to a square unit along the Bain stretch. However, if the angle a is larger than 90, shear deformation becomes very difficult due to the higher deformation resistance, thus the Bain stretch is not the preferred path. Therefore, for cracks with Æ1 1 0æ crack fronts, two types of martensitic transformation mechanisms are optional. One is the bcc-to-fcc transformation along the Bain path as we have mentioned in Section 3.1; the other is the bcc-to-hcp transformation. For the {1 0 0}Æ1 1 0æ crack in Fig. 2c, the angle a is smaller than 90, and a bcc-to-fcc transformation occurs along the Bain path. Whereas for the {1 1 0}Æ1 1 0æ crack, the angle a is larger than 90, thus the bcc-to-hcp transformation is preferred. For the {1 1 1}Æ1 1 0æ crack in Fig. 4, both types of martensitic transformation occur. Above the crack plane, the angle a is larger than 90, and a new grain nucleates accompanying the bcc-to-hcp transformation; below the crack plane, the angle a is smaller than 90, and a twinned martensite strip forms along the Bain path. In Fig. 5 for cracks with Æ1 0 0æ crack fronts, a bcc-to-fcc transformation can occur along the Bain path either under the shear of [1 1 0] or the opposite ½1 1 0. Thus in Fig. 2a and b for the {1 0 0}Æ1 0 0æ and {1 1 0}Æ1 0 0æ cracks, a bccto-fcc transformation occurs along the Bain path accompanying long, twinned martensite strips formed at the crack tip. From the above analyses, we conclude that this method can be used to determine the mechanisms of the martensitic transformation in bcc iron. In the following section, we will apply this method to analyze martensitic phase transformations in ordered B2 NiAl alloys. 4. Discussion on the deformation behavior in B2 NiAl 4.1. For cracks with Æ1 0 0æ crack fronts
Fig. 5. Lattice deformation of a (0 0 1) unit under a simple twodimensional shear of ½1 1 0ð1 1 0Þ. Dot lines and solid lines indicate representative units before and after deformation, respectively.
The deformation behavior at the crack tip in B2 NiAl has been investigated previously [18]. In this section, based on the analysis of the martensitic transformation mechanisms in bcc iron, the deformation mechanisms at the crack
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tip in B2 NiAl will be discussed. As shown in Fig. 7 for cracks with Æ1 0 0æ crack fronts in B2 NiAl, we have observed a similar bcc (B2)-to-fcc (L10) structure transformation at the crack tip. After a stretching along the Æ1 0 0æ direction, the transformation of the B2 phase into the L10 martensite phase occurs along the Bain path. The habit plane is also the {1 1 0} plane. The ratio c/a of the transformed L10 martensite phase is about 1.25, showing pffiffiaffi clear deviation from the perfect Bain stretch value of 2. It is also a little smaller than the ratio c/a of the formed fcc unit in iron. Moreover, there is a great difference between bcc iron and B2 NiAl in terms of the bcc (B2)-to-fcc (L10) structure transformations. The transformed martensites in B2 NiAl are found with layered structures. The faulted lines lying on the {1 0 0} plane can be seen clearly in Fig. 7. These internally faulted martensites formed at the crack tip in NiAl alloys are different from the twinned martensite strips in pure metallic iron. It is known that the primary slip plane in NiAl is the {1 0 0} plane, which is consistent with the faulted martensites plates in Fig. 7. Thus we assume that the formation of this internally faulted structure is closely related with the atoms bonding in different atoms layers, as well as the alloy composition. The relatively close-packed {1 1 0} plane is the primary cleavage plane in NiAl alloys, which is different from the {1 0 0} cleavage plane in bcc iron. Thus we observe the dislocation emission perpendicular to the crack surface in Fig. 7a, which causes the crack blunting. Moreover, due to the dislocation emission along the ½0 1 0 direction, we find that the shear direction in the left side of the dislocations is different to that in the right side. Thus in the left region of Fig. 7a, just below the crack plane, we can observe that the Bain stretch occurs along the ½ 1 0 0 direction. A faulted line appears on the (1 0 0) plane, which is perpendicular to those (0 1 0) faulted planes. A crossing type of martensites units is observed just below the crack tip region.
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4.2. For cracks with Æ1 1 0æ crack fronts As discussed in Section 3, two types of martensitic transformation mechanisms are possible for cracks with Æ1 1 0æ crack fronts in bcc iron. In Fig. 8 for the Æ1 1 0æ cracks in ordered B2 NiAl alloys, we have only observed a bcc (B2)-to-hcp transformation at the crack tip. In the top left of Fig. 8b for the {1 1 0}Æ1 1 0æ crack, a representative unit with the ½1 1 1 shear direction is marked out. The angle a is larger than 90, and thus the phase transformation from B2 to hcp occurs. This martensitic transformation is caused by atomic relative displacement in different atomic layers and a shear in the {1 1 0} plane, which is consistent with the mechanism in bcc iron. Moreover, two long martensite strips appear at the crack tip accompanying crack cleavage along the ½1 0 1 plane, which is completely different from the result in bcc iron in which a new martensite grain nucleates at the crack tip accompanying crack branching. In Fig. 8a for the {1 0 0}Æ1 1 0æ crack, crack blunting occurs due to the dislocations emission. At the blunted crack tip region, some changes that are not distinguishable are observed. The stress concentration at the crack tip is relaxed by dislocation emission and crack blunting, thus no martensite strip exists at the crack tip. In the left and right sides of the dislocation, we present two representative units, respectively. The shear directions for these two representative units are opposite due to the dislocation emission. For the left side unit, the angle a is larger than 90. Thus just below the crack surface, the martensitic phase transformation from the B2 structure to a hcp structure occurs, which is different from the result of the {1 0 0}Æ1 1 0æ crack in bcc iron that a bcc-to-fcc transition occurs along the Bain path. It can be concluded from this result that the deformation mechanisms at the crack tip depend strongly on the condition of the local stress. It should be mentioned that the cases of phase transformations below the crack surface, separately shown in Figs. 7a and 8a, may not agree well with the experimental results
Fig. 7. Details of the crack tip region at 5 K under a load of (a) 2.0 KIC for a {1 0 0}Æ1 0 0æ crack and 3.0 KIC a {1 1 0}Æ1 0 0æ crack. Open circles represent Ni atoms, while solid circles represent Al atoms.
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Fig. 8. Details of the crack tip region at 5 K under a load of (a) 1.8 KIC for a {1 0 0}Æ1 1 0æ crack and (b) 2.0 KIC for a {1 1 0}Æ1 1 0æ crack. Big circles represent Ni atoms, while small circles represent Al atoms. Solid and open circles indicate atoms in A and B layers, respectively.
in an actual material. These phase transformations are caused by the combined effects of dislocation emission and the restriction of the fixed boundary due to the simulation condition. However, we can conclude from above simulation results that the deformation mechanisms at the local positions are finally determined by the local stress and composition condition. In an actual material under the influences of the external load and temperature, the local stress and composition condition are very complicated. Therefore, very complicated deformation behaviors have been observed in the experimental results. For example, the high-resolution TEM observations have revealed a crossing-type macrotwin with different orientations of microtwin invariants in NiAl martensite [8]. Although our simulated results, such as the formation the crossing type of martensite units as shown in Fig. 7a, are too simple compared with this experimental phenomenon, it provides a useful comparison with the experimental results. 5. Conclusions We have investigated the low-temperature deformation at the crack tip in bcc iron by molecular dynamics simulations. Two different mechanisms of martensitic phase transformations for different types of cracks are discussed in particular. One is the bcc-to-fcc transformation caused by a stretch in the Æ0 1 0æ direction along the Bain path. The other is the bcc-to-hcp transformation through a shear in the {1 1 0} plane and atomic relative displacement in different atomic layers. The choice of the mechanisms of the martensitic transformations depends on the local stress at the crack tip and the crystal orientation. A simple way to determine the deformation behavior at the crack tip is presented. We apply this method to analyze the martensitic phase transformation in the ordered B2 NiAl alloys. The complicated deformation behavior at the crack tip is discussed.
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