Deformation twinning at aluminum crack tips

Acta Materialia 51 (2003) 117–131 www.actamat-journals.com

Deformation twinning at aluminum crack tips S. Hai, E.B. Tadmor ∗ Department of Mechanical Engineering, Technion—Israel Institute of Technology, Technion City, 32000 Haifa, Israel Received 7 May 2002; received in revised form 16 August 2002; accepted 21 August 2002

Abstract Recent experimental evidence has shown that even fcc materials that are not normally associated with deformation twinning, such as aluminum, will twin given a sufficiently high stress concentration such as at a crack tip. In this paper we present a computational study of the atomic structures that form at the tips of atomically sharp cracks in aluminum single crystals under loading. The simulations were carried out using the quasicontinuum method—a mixed continuum and atomistic approach. A variety of loading modes and orientations were examined. It was found that for certain combinations of loading mode and orientation, deformation twinning does occur at aluminum crack tips in agreement with experimental observation. For other configurations, either dislocation emission or in one case the formation of an intrinsic–extrinsic fault pair was observed. It was also found that the response at the crack tip can depend on the cracktip morphology in addition to the applied loading and crystallographic orientation.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Computer simulation; Aluminum; Fracture and fracture toughness; Dislocations; Deformation twinning

1. Introduction Deformation twinning is a common and important mechanism for plastic deformation in hcp metals. In bcc metals, it appears at low temperature where it becomes more favorable than dislocationbased slip processes [1]. In fcc metals deformation twinning is more rare. It normally requires low temperatures and high stresses or strain rates, although some fcc alloys twin more readily [2]. Of the fcc metals, pure aluminum and lead were traditionally cited as examples of metals that do not exhibit

Corresponding author. Tel.: +97248293466; fax: +972248324533. E-mail address: [email protected] (E.B. Tadmor).

deformation twinning at all [3]. However, more recently, experimental work has indicated that deformation twinning does occur in aluminum under certain conditions. In 1981, Pond and Garcia-Garcia [4] found the first evidence of deformation twinning in highpurity aluminum. The twin was found at the tip of an edge crack in a transmission electron microscopy (TEM) foil. The authors believe that the crack was formed as a result of mode III type loading resulting from the sample preparation procedure. TEM investigation of the crack tip revealed a deformation twin approximately 1500 nm long oriented along the [1¯ 12] direction with a (11¯1) twin plane.1 The twin

∗

1 All crystallographic orientations are given relative to the parent matrix phase.

1359-6454/03/$22.00  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00367-1

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is thought to have formed by the emission of 1/6[121] partial dislocations on successive (11¯ 1) planes. The tip of the twin is blunted and this is attributed to a dislocation reaction involving the leading three twinning dislocations resulting in the formation of residual and emissary dislocations. A more recent demonstration of deformation twinning in aluminum was obtained by Chen et al. [5] in 1999. The objective of this work was to study failure modes in thin crystals after dislocation slip mechanisms were exhausted. Three different metals were investigated: copper, aluminum and iron. In the experiments, thin-foil specimens were loaded into a TEM chamber and pulled in tension in situ until failure. In both copper and aluminum, deformation begins by the emission of large numbers of dislocations from a pre-existing crack tip and from sources ahead of it. The dislocation activity results in the formation of a thinned region ahead of the crack. When the thickness of the thinned region drops below a critical value, dislocation processes cease, the stress increases and other mechanisms become active. In copper, it was found that deformation twinning becomes the main deformation mechanism at this point eventually leading to tensile cracking. In aluminum, deformation twinning was less common. It occurred only in one case when the loading orientation strongly favored twinning. The nucleated twin was approximately 100 nm long and it lay along the [1¯ 1¯ 2¯ ] direction with a (111¯ ) twin plane. Chen et al. [5] point out that although their study was carried out in very thin crystals (⬇100 nm) it has relevance to macroscopic phenomena occurring after the exhaustion of work hardening. This suggests that deformation twinning may play an important role in the final stages of failure even in materials like copper and aluminum, which are not normally associated with twinning. In addition to the experimental evidence discussed before, numerical evidence for deformation twinning in aluminum also exists. In a study of deformation mechanisms beneath a plane strain knife-edge nanoindenter, Tadmor et al. [6] found that for certain orientations, deformation twinning was observed. The authors attribute the phenomenon to low-temperature effects (the simulations were carried out at zero temperature), to the large

stress concentration generated beneath the nanoindenter and to the geometrical constraint imposed by the two-dimensional (2D) nature of the plane strain loading. More recently Farkas et al. [7] observed a combined deformation twinning and cleavage mechanism at the tip of atomically sharp cracks in aluminum. The cracks were seen to extend in a two-step process: first a small twinned region was formed by successive dislocation emission and then a small cleavage event occurred extending the crack by several lattice spacings. The experimental and numerical evidence presented before suggests that deformation twinning in aluminum is possible under certain conditions. In particular, it appears that deformation twinning becomes favorable when dislocation slip is inhibited for reasons such as low temperature, dimensionality constraints or surface effects in thin specimens. It is of interest to further clarify the conditions under which deformation twinning occurs in aluminum and to attempt to reproduce and quantify the experimental results discussed before. This is the objective of this paper which presents an atomic-scale numerical simulation of deformation mechanisms at the tip of an atomically sharp crack in single-crystal aluminum. The effect of loading mode and orientation on the response mechanism at the crack tip is investigated. In particular, the configurations studied by Pond and Garcia-Garcia [4] and Chen et al. [5] are simulated in order to to allow for direct comparison with experiment. The calculations in this paper were carried out using the quasicontinuum (QC) method [8,9], which is a mixed continuum and atomistic approach. The method retains atomistic resolution were necessary while seemlessly grading out to a continuum model in the far field. The method is well-suited to simulate atomic-scale fracture problems where atomistic resolution is required at the crack tip while linear elastic fracture mechanics (LEFM) boundary conditions may be applied in the far field. The advantage of using this approach is that the number of degrees of freedom that have to be tracked is significantly reduced relative to conventional atomistic calculations making the problem tractable on a desktop workstation. Also, since the simulations involve quasistatic energy

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minimization, the unrealistic loading rates common to MD simulations are avoided. A limitation is that as a result thermally activated process are precluded. The structure of the paper is as follows. Section 2 briefly describes the QC methodology and its application to fracture problems. Section 3 describes the crack orientations and loading modes that are investigated and the main results obtained in the simulations. Section 4 concludes with a discussion of the results and suggestions for future work. 2. Methodology The QC methodology used here is a mixed continuum and atomistic method developed to study problems in the mechanics of materials where multiple scales operate simultaneously. It was originally introduced [8] to study single crystal mechanics and later extended [9] to treat polycrystals and polyphase materials. The basic idea is that in a crystal undergoing mechanical deformation the majority of the lattice experiences, a slowly varying deformation on the atomic scale which is well characterized by the continuum approximation. It is only in the vicinity of defects or in the presence of mechanical manipulations on the order of the lattice spacing where discrete atomic effects generally become important. There is thus, no need to explicitly treat every single atom in the crystal as is done in standard lattice statics and molecular dynamics approaches. Within the QC method the solution is to select a small subset of the total collection of atoms to represent the energetics of the whole. The crystal is then divided into disjoint cells each containing one of these selected atoms whose energetics are assumed to represent those of all other atoms in its cell. Thus, if the exact energy of a collection of N atoms is given by,

冘 N

Eexact ⫽

Ei,

(1)

i⫽1

where Ei is the energy of atom i, then within the QC method a reduced energy potential is defined, such that,

冘

119

R

Ereduced ⫽

niEi,

(2)

i⫽1

where, RN is the number of representative atoms in the selected subset and ni is the number of atoms represented by atom i. Clearly when all atoms are selected to be representative atoms the exact description is recovered. This reduced atomic description is stored on a finite element mesh [10], whose nodes coincide with the representative atom positions. The degrees of freedom of the system are the displacements of the representative atom nodes. The positions of all other atoms in the crystal, which are not explicitly accounted for, can be obtained by finite element interpolation. This becomes necessary when computing the energies of the representative atoms which depend on the positions of their neighboring atoms. To compute the energies of the representative atoms the embedded atom method (EAM) [11] is employed. In this scheme the energy of an atom is computed from the relative positions of all other atoms that fall within some specified cutoff, using the relation, Ei ⫽

冘

1 2

f(rij) ⫹ U(ri),

(3)

j

where rij is the distance from atom i to neighbor j, f(r) is a pair potential characterizing the core– core repulsion of the atomic nuclei, ri is the electron density at atom i and U(r) is the embedding energy due to the attraction between the nucleus and ambient electron density. Within the EAM approximation the electron density is also taken to have a pairwise form, ri ⫽

冘

f(rij).

(4)

j

The EAM offers a computationally tractable description of the material response, which appears to describe many metals quite adequately. For aluminum, we use the EAM potentials of Ercolessi and Adams [12]. The potential was selected because of its generally good agreement with basic experimental values (Table 1). In particular, the stacking fault energy, which is important in a study

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Table 1 Comparison of the values calculated using the Ercolessi and Adams EAM potential for aluminium with experimental values for a number of relevant material parameters. In the table, a0 is the lattice constant, Cij are the elastic constants, γ111 is the {111} surface energy and γsf is the stacking fault energy. The experimental values are taken from [12] where references are given. Parameter ˚] a0 [A ˚ 3] C11 [eV/A C12 C44 ˚ 2] γ111 [meV/A γsf

EAM 4.032 0.737 0.388 0.229 54.3 6.62

Experiment 4.032 0.738 0.390 0.203 71–75 7.5–9.0

of deformation twinning, is predicted reasonably well, in contrast to many EAM potentials which tend to grossly underestimate this value. Two separate methodologies are employed to obtain the positions of the atomic neighbors of the representative atoms that are necessary for the evaluation of (3). The energy of representative atoms experiencing a slowly varying deformation in their vicinity is computed in a local fashion where it is assumed that the nearby environment of the atom is well-characterized by the deformation gradient at its position. This is essentially the continuum approximation and this limit of the formulation corresponds to a nonlinear anisotropic elastic description of the material. At the other extreme are atoms experiencing large variations of deformation in their vicinity. These atoms are computed nonlocally in the sense that the positions of the neighboring atoms are independent of the deformation at the representative atom position. This corresponds to the lattice statics or atomistics limit of the formulation [8]. The total energy of the system can now be computed from (2) and (3) and equilibrium configurations are identified by minimizing this energy with respect to the representative atom positions. The minimization is carried out by a quasi-Newton solver with a conjugate gradient backup when the initial guess is outside the basin of attraction of the Newton solver (See [9] for details). This approach corresponds to a zero temperature quasi-static solution.

Due to the 2D nature of the investigated crack configuration, a pseudo-2D implementation of QC was used in the current study. Although all atomistic calculations were made in three-dimensions (i.e. each representative atom is surrounded by a sphere of atoms for the purpose of calculating its energy), the displacement fields are constrained to have no variation in the out-of-plane z-direction, thus, ux ⫽ ux(x,y), uy ⫽ uy(x,y), uz ⫽ uz(x,y),

(5)

where, ux, uy and uz are the displacements in the respective directions. This is a form of generalized plane strain. The implication of this is that only dislocations with lines normal to the simulation plane may be nucleated. This is a reasonable approximation given the 2D nature of the crack configuration studied here. This assumption is supported by the work of Xu et al. [13], who found using a Peierls model approach that dislocation nucleation on planes that are oblique to the crack surface was unfavorable. The authors do note, however, that heterogeneous dislocation nucleation at ledges along the crack front can be significant. This effect is left out of the present analysis. Application of this approach to fracture problems is straightforward. A brief explanation is given here. For a more detailed discussion of modeling fracture using the QC method, see Miller et al. [14]. A crack is introduced into the crystal by removing five atomic layers. An initial set of representative atoms is selected such that the crack tip region is fully refined (i.e. all atoms are represented) and the remainder of the model is more coarsely represented (Fig. 1). The following procedure is adopted for loading the crack: (1) All atoms are displaced according to the anisotropic LEFM solution [15] for a small initial stress intensity factor (SIF) K0, u(X) ⫽ uLEFM(X,K0),

(6)

where u is the displacement field, X is any point in the model and uLEFM is the anisotripic LEFM solution. (2) The displacements at the model boundaries are fixed (except for the crack faces) and the positions of all other nodes/atoms are obtained by energy minimization as explained before.

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3. Results A schematic of the geometry used in the simulations is presented in Fig. 2. Three different crystallographic orientations and a variety of loading modes were considered as detailed in Table 2. The motivation for selecting these models was based on existing experimental and numerical results from previous studies, as follows:

Fig. 1. An initial QC mesh used for the fracture simulations. The inset shows a closeup of the crack tip region which is fully refined down to atomic resolution.

(3) The mesh is automatically adapted (i.e. representative atoms are added and removed according to an automatic criterion based on the variation of deformation gradients about the atoms) [9] and the local/nonlocal status of all atoms is re-evaluated. (4) The SIF is incremented by a small amount ⌬K and all nodes are displaced by the associated incremental anisotropic LEFM solution, u(X) ⫽ u(X) ⫹ uLEFM(X,⌬K),

앫 Model A—An orientation studied by Hoagland et al. [16], where perfect dislocations were nucleated under mode I loading. 앫 Model B—The Pond and Garcia-Garcia [4] experimental crack orientation. 앫 Model C—The Chen et al. [5] experimental crack orientation. The models will be described in more detail in the following sections. 3.1. Model A Model A corresponds to an orientation studied by Hoagland et al. [16], where perfect dislocation emission was observed under mode I loading. We include this orientation as a check to ensure that

(7)

The procedure now resumes at step (2) and continues until a critical event occurs at the crack tip, i.e. dislocation emission, deformation twinning or brittle cleavage is observed. In the current simula˚ 2.5 tions values of K0 ⫽ 0 and ⌬K ⫽ 0.01eV / A were used. In certain cases where higher resolution ˚ 2.5. was required, ⌬K was reduced to 0.001 eV/A It is important to point out that by adopting this procedure we are in fact modeling a generic crack of arbitrary configuration. Thus, while the actual numerical model configuration is of an edge crack (as explained in the next section), in practice this can be thought of as the tip region of a larger crack with arbitrary geometry and loading, the specifics of which are a matter of interpretation of the SIF.

Fig. 2. A schematic diagram of the simulated crack configuration along with the coordinate system and relevant dimensions.

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Table 2 Crystallographic orientations, loading modes and dimensions of the different crack configurations investigated. All dimensions are ˚. in A Model

A B C

Loading Modes

x-axis

y-axis

z-axis

a

h

w

I I,II,III I

[111] [1¯ 12] [111¯ ]

[1¯ 10] [11¯ 1] [112]

[1¯ 1¯ 2] [110] [11¯ 0]

2980 2528 772

5840 5960 1586

5960 5056 1563

d

8.8 14.13 8.66

t

4.94 2.85 2.85

the simulation reproduces previous numerical results. In this orientation, the only available slip plane is the (111) plane. All other {111}-type planes lie obliquely to the simulation plane and are thus, precluded by the imposed plane strain conditions as explained in Section 2. ˚ 2.5 a dissociated At a SIF of KI ⫽ 0.142 eV / A ¯ 1 / 2[110] (BA) edge dislocation is nucleated on the (111) plane below the crack tip. The dislocation has dissociated according to the following reaction, 1 1 1 ¯ [110](BA)→ [21¯ 1¯ ](Bd) ⫹ [12¯ 1](dA), 2 6 6

(8)

where both the Miller index and Thompson vector notation [17] is given. Almost immediately after the first emission, at a SIF of KI ⫽ ˚ 2.5, an identical dissociated dislocation 0.144 eV / A is nucleated above the crack tip resulting in the nearly symmetrical configuration presented in Fig. 3. The contour shading corresponds to out-of-plane displacement, which indicates the presence of a stacking fault between the two partials. The split˚ in agreement with preting distance is about 15 A vious results for this potential [6]. This value is large compared to the experimental splitting dis˚ measured by Mills and tance in aluminum of 5.5 A Stadelmann [18]. The discrepancy between the simulated value and the experimental value may be attributed to the still too low stacking fault energy of the Ercolessi and Adams [12] potential. The above results qualitatively agree with the findings of Hoagland et al. [16], where a different EAM potential for aluminum was used. For the same orientation and loading, the same dislocation structure is observed. In the Hoagland et al. simul˚ 2.5, ation, nucleation occurs at a KI ⫽ 0.313 eV / A which is about double the value observed in the

Fig. 3. The model A orientation loaded in mode I with KI ⫽ ˚ 2.5. Two dissociated edge dislocations have been 0.144 eV / A emitted. The shading corresponds to levels of out-of-plane dis˚ (lightest) to 0.34 A ˚ (darkest). placement, ranging from ⫺0.36 A The relevant face of the Thompson tetrahedron for the activated (111) slip plane is presented along side the contour plot in a schematic 3D diagram of the crack tip and slip plane.

current simulation. The difference is most likely related to the small model size used in the Hoagland et al., calculation (only a 5 nm region about the crack tip was allowed to relax) and to the different potential used.

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The symmetric nucleation of dislocations above and below the crack tip may be understood by considering the schematic diagram in Fig. 3. It is clear that the leading and trailing partials on both sides of the crack tip are oriented in the same manner relative to the loading direction. This is not the case in the model C configuration, considered later where an asymmetric response is observed. 3.2. Model B Model B reproduces the crack orientation of the Pond and Garcia-Garcia [4] experiment. In the experiment, a deformation twin with a (11¯ 1) twinning plane and a [121] shear direction was found at the tip of an edge crack in an aluminum TEM foil. The exact loading in the experiment is not known, but the authors surmise that both the crack and the deformation twin were formed as a result of mode III type loading inherent to the foil preparation process. In the simulations, all three modes (I, II and III) were investigated to see which, if any, produce the deformation observed in the experiment. The model B configuration contains two {111}-type planes on which dislocations may nucleate and glide (Fig. 4). These are the (11¯ 1) planes running parallel to the crack plane and the (11¯ 1¯ ) planes lying at an angle of 70.5° to the crack plane. Under mode I loading, a twin begins nucleating ˚ 2.5 from the crack tip at a SIF of KI ⫽ 0.13 eV / A 1/2 ( ⫽ 0.21 MPa(m) ). The twinning direction is [11¯ 2] and the twinning plane is (11¯ 1¯ ). The SIF at nucleation is about half of the critical SIF for cleavage computed from the Griffith criterion for this orientation [15], which is KIc ⫽ ˚ 2.5. Due to symmetry, KIc is the same 0.235 eV / A for all orientations studied in this paper. The twin is formed by the emission of 1 / 6[1¯ 12¯ ] (Ab) partial dislocations on successive (11¯ 1¯ ) planes. Fig. 5 presents a series of snapshots of the crack tip region immediately prior to each emission event. The finite element mesh and the atomic positions are displayed. The twinned region in each frame is marked by dark shading and the layer due to twin next is marked by light shading. Frame (a) displays the crack tip immediately prior to the nucleation of the first partial at KI ⫽

Fig. 4. A schematic diagram of the model B configuration. The available slip planes have been indicated along with the relevant faces of the Thompson tetrahedron.

˚ 2.5. 0.12 eV / A In frame (b) at KI ⫽ ˚ 2.5, the nucleation begins with the emis0.13 eV / A sion of the first partial dislocation on the plane intersecting the crack tip. The partial travels a dis˚ leaving an intrinsic stacking fault in tance of 60 A its wake (the partial is not visible in the figure). In ˚ 2.5, a second partial is frame (c) at KI ⫽ 0.14 eV / A ¯ ¯ emitted on the (111) plane adjacent and behind the previous emission plane laying down an extrinsic stacking fault. A microtwin2 two layers thick has formed. At this stage, the first partial is located at ˚ from the crack tip and the a distance of 230 A ˚ behind second partial is at a distance of 190, 40 A the first. It is clear from the figure that the formation of the microtwin is blunting the crack tip. ˚ 2.5, a third Finally, in frame (d) at KI ⫽ 0.15 eV / A partial is emitted, increasing the thickness of the microtwin to three layers and thereby forming a true twin. The distances of the first, second and 2

The term microtwin refers to the two-layer twin nucleus formed by the emission of the two partials. A microtwin cannot be considered as a true twin due to the interaction of the twin boundaries across the extrinsic stacking fault. These boundaries will generally have a higher energy than true twin boundaries. Despite this, we refer to the defect as a microtwin for two reasons: (1) the morphology of the defect which is bound by twin planes and terminated by an incoherent boundary is characteristic of twins; and (2) in all the simulations presented here, once a microtwin nucleates, increased load always causes it to thicken into a true twin.

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Fig. 5. The sequence of nucleation steps leading to the formation of the twin at the tip of the model B crack loaded in mode I ˚ 2.5; (b) KI ⫽ 0.13 eV/ A ˚ 2.5; (c) KI ⫽ 0.14 eV/ A ˚ 2.5; and (d) KI ⫽ 0.15 eV / A ˚ 2.5. The dark shading indicates with: (a) KI ⫽ 0.12 eV/ A twinned layers and the light shading indicates layers that will twin in the next load step.

third partials from the crack tip are now 480, 440 ˚ . The 40 A ˚ spacing between the leading and 360 A partials is preserved, while the third partial is ˚ . It is very interfurther back at a distance of 80 A esting that at this stage the twin continues to grow from the opposite (right) end. The fourth layer to twin lies ahead of the crack tip and does not intersect the crack surface. This is more difficult to do because the twinning process now involves the creation of new surface area at the bottom of the twinned layer. The fact that this occurs, suggests that the left end of the twin has grown out of the region where the crack-tip stress field is sufficient to cause a nucleation event to occur. The twin continues to grow with increasing load

˚ 2.5 it is eight layers thick at and at KI ⫽ 0.17 eV / A ˚ long. Fig. 6 shows the full the base and is 550 A twin at this stage with detail of the crack tip and some of the partial dislocations along the twin. Note the cleavage at the crack tip and corresponding generation of new surface area as a result of the nucleation of the last four twin layers. The distances of the partials from the crack tip are 170, ˚ . The typical 270, 360, 430, 480, 520 and 540 A twin needle-like morphology is clear. In the simulation, the leading partials do not react to form residual and emissary dislocations as in the experiment [4]. The reason for this is most-likely tied to limitations of the EAM potential used in the simulation as explained below.

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Fig. 6. The model B orientation loaded in mode I with KI ⫽ ˚ 2.5. The deformation twin appears shaded. Details of 0.17 eV/ A the atomic structure at the crack tip and at locations of partial dislocations along the twin are given.

The driving force for the dislocation reaction at the tip of twin is the higher energy of the stacking faults relative to the twin structure. This energy difference tends to pull the partials back towards the twin. In aluminum this attractive force is sufficiently large to overcome the mutual repulsion between the partials [4] and thus, spontaneous blunting of the tip of the twin is expected as observed in the experiment. The Ercolessi and Adams [12] EAM potential used in the simulation can distinguish between intrinsic and extrinsic stacking fault and twin boundary energies, however the ordering between them is incorrect. The ˚ 2, gesf ⫽ potential predicts gisf ⫽ 6.62 meV / A 2 ˚ 2, so that ˚ and gt ⫽ 3.63 meV / A 7.28 meV / A gisf ⬍ gesf⬇2gt, whereas ab initio calculations [19] find gisf ⬎ gesf ⬎ 2gt. This difference is significant because the incorrect ordering of the energies predicted by the EAM potential means that in the simulation there is no driving force for the dislocation reaction to occur. Under mode II loading a very different result is ˚ 2.5, two observed. At a SIF of KII ⫽ 0.093 eV / A dissociated dislocations nucleate one above the other on adjacent slip planes and react to form an intrinsic–extrinsic fault pair [20]. Fig. 7 shows the dislocated structure near the crack tip and at the end of the slipped region. One atomic layer has

Fig. 7. The model B orientation loaded in mode II with ˚ 2.5. An intrinsic–extrinsic fault pair has been KII ⫽ 0.093 eV /A formed as explained in the text. Frame (a) shows the crack tip region and frame (b) shows the intrinsic and extrinsic stacking faults at the end of the slipped region. The shading corresponds ˚ (lightest) to to out-of-plane displacement, ranging from 0 A ˚ (darkest). The slanted lines help to highlight the change 1.42 A of stacking across the faulted regions.

˚ displaced in the out-of-plane direction by 1.42 A as indicated by the shading. Slanted lines have been drawn to help visualize the change of stacking across the faulted regions. To understand the nature of the emitted dislocations we consider the slip discontinuities ⌬1 and ⌬2 across the slip planes marked 1 and 2 in the figure. In the crack-tip ˚ which correregion ⌬1 ⫽ (2.64,0.01,⫺1.42)A sponds to the passage of a 1 / 2[101¯ ] (BC) dislo˚ which correcation and ⌬2 ⫽ (2.64,0.01,1.42)A

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sponds to the passage of 1 / 2[01¯ 1¯ ] (BD) dislocation. The passage of these two dislocations has restored the perfect crystal structure in the crack tip region except for some elastic straining. The dissociated dislocations BC and BD react to form the intrinsic–extrinsic fault pair visible in Fig. 7(b). A schematic view of the fault pair structure is given in Fig. 8(a). The fault pair is bounded by three partial dislocations of the same Burgers vector Ba, where we note that 1 1 1 ¯ [211](aC) ⫹ [1¯ 2¯ 1¯ ](aD)→ [11¯ 2¯ ](Ba). 6 6 6

(9)

The slip in the x-direction associated with this fault is shown in Fig. 8(b). The dashed and solid lines correspond to slip across planes 1 and 2, respectively, for idealized Volterra dislocations.

Fig. 8. Intrinsic–extrinsic fault pair structure: (a) schematic diagram of the dislocation structure of a BC/BD fault pair; (b) slip distribution corresponding to the fault pair in (a); (c) slip distribution ahead of the crack tip in model B/II at KII ⫽ ˚ 2.5. The dashed and solid lines in frames (b) and 0.093 eV/ A (c) correspond to slip across planes 1 and 2, respectively, as indicated in frame (a) and Fig. 7(a).

The dotted lines show schematically the slip profiles for the case where the partial dislocation cores are spread out (as occurs in the simulation). Compare the dotted line profiles with the actual slip distribution observed in the simulation (Fig. 8(c)). The distribution is consistent with the fault pair structure (note that for aluminum a0(6)1 / 2 / 4 ⫽ ˚ and a0(6)1 / 2 / 12 ⫽ 0.82 A ˚ ). Similarly, the 2.47 A out-of-plane slip across planes 1 and 2 observed in ˚ ) is consistent with the the simulation ( ± 1.42 A out-of-plane components of the partial dislocations bounding the fault pair. A possible mechanism for the formation of an intrinsic–extrinsic fault pair from two co-planar 具110典 dislocations was suggested by Mahajan [21]. The mechanism involves the recombination of one of the dislocations and its subsequent redissociation on an adjacent plane. In the simulation, the fault pair forms in a single minimization step, thus, the details of the interaction of dislocations leading to its formation were not observed and the Mahajan model could not be verified. Mahajan and Chin [22] have postulated that intrinsic–extrinsic fault pairs form an important step in the formation of three-layer twin nuclei. This was not observed in the simulation for the range of loads investigated. After the formation of the fault pair, increased load caused the defect to propagate further away from the crack tip into the bulk. It would be of interest to continue increasing the load in this simulation to see whether the Mahajan–Chin mechanism for deformation twinning appears at higher loads. This was not done at this stage due to the computational intensity of the calculation. Under mode III loading, a perfect screw dislocation with Burgers vector 1 / 2[1¯ 1¯ 0] (CD) is emitted from the crack tip at a SIF of KIII ⫽ ˚ 2.5. Because of the plane strain con0.05 eV / A straints, dissociation of the screw dislocation into partials does not occur. It is noteworthy that the critical SIF for dislocation emission in mode III is significantly lower (about half) of the critical SIF’s in modes I and II. Although deformation twinning is observed for mode I, none of the pure loading modes discussed above reproduced the experimentally observed twin (11¯ 1)[121]. The twinning direction in the

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experiment lies in the (11¯ 1) (xz) plane at a 60° angle relative to the x-axis. This suggests that a mixed mode II and III loading is necessary to activate this deformation mechanism. To test this, the model B orientation was loaded in mixed mode along the [121] direction with KII ⫽ Kcos60° and KIII ⫽ Ksin60° where K is the magnitude of the applied SIF. Two different crack tip morphologies are studied: one is referred to as the ‘blunt’ configuration (Fig. 9(a)) and the other is referred to as the ‘sharp’ configuration (Fig. 9(b)). The blunt and sharp terminology refers to the angle of the acute corner at the crack tip. Both cracks have the same crystallographic orientation and are subjected to the same loading. The difference between them is only in the surface delimiting the end of the crack. In the blunt configuration the crack tip is delimited by a (11¯ 1¯ ) plane and the crack tip angle is 70.5°.

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In the sharp configuration the crack tip is delimited by a (001) plane and the crack tip angle is 54.7°. The results for the blunt and sharp cracks after loading are seen in Fig. 9(c) and (d), respectively. The difference is striking. In the blunt configuration a dissociated dislocation is nucleated at the crack tip. The emission occurs in two stages. At a ˚ 2.5 the leading 1 / 6[1¯ 2¯ 1¯ ] SIF of K ⫽ 0.041 eV / A (aD) partial dislocation is emitted, followed at ˚ 2.5 by the trailing partial K ⫽ 0.056 eV / A 1 / 6[11¯ 2¯ ] (Ba) to complete the 1 / 2[01¯ 1¯ ] (BD) dislocation. The dislocation moves away from the crack tip and is not in the figure where only the slip left in its wake is visible through the shearing of the finite element mesh. The top half of the crystal above the crack has displaced along the x-axis by an amount equal to the projection of the Burgers vector onto the x-axis (a0(6)1 / 2 / 4). The atoms in

Fig. 9. The model B orientation loaded in mixed mode II and III with two different crack tip morphologies. Frame (a) shows the blunt crack tip morphology prior to loading and frame (b) shows the sharp crack configuration prior to loading. Frame (c) shows the ˚ 2.5. A perfect dislocation has nucleated and moved out of the frame to the right. Frame blunt crack when loaded to K ⫽ 0.056 eV / A ˚ 2.5. A (11¯ 1)[121] microtwin has nucleated from the crack tip. The twin (d) shows the blunt crack when loaded to K ⫽ 0.045 eV/ A ˚ (lightest) to 3A ˚ (darkest). boundaries are indicated by TB. The shading corresponds to out-of-plane displacement, ranging from ⫺3A

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the figure have been shaded to show the out-ofplane displacement. The darker shading corresponds to motion out of the the plane and the lighter shading corresponds to motion into the plane. In the sharp crack configuration deformation twinning is observed. A microtwin two layers thick ˚ 2.5. The is nucleated at a SIF of K ⫽ 0.045 eV / A twinning direction is [121] and the twinning plane is (11¯ 1) as in the experiment. The microtwin is formed near the sharp tip of the crack at the bottom by the emission of two 1 / 6[1¯ 2¯ 1¯ ] (aD) partial dislocations on adjacent (11¯ 1) planes. The twinning planes are indicated in the figure along with a dark line as a guide to the eye to show the change in the stacking direction across the twin. The twinned region continues to grow and thicken with increasing load as more partial dislocations are emitted on the planes above the microtwin. This is exactly the mode of deformation observed in the experiment of Pond and Garcia-Garcia [4]. The simulation thus reproduces the experimental behavior. It is noteworthy that the SIF for the formation of the twin in the sharp configuration is lower than that for nucleation of the dislocation in the blunt configuration. Since it is reasonable to assume that both configurations existed in the experimental system, the fact that twinning was observed in the experiment is consistent with the predictions of the simulation. 3.3. Model C The model C configuration corresponds to the orientation of the Chen et al. [5] experiment. This experiment was more controlled than that of Pond and Garcia-Garcia [4] in the sense that it was specifically set up to study crack tip deformation. A thin foil sample with a pre-existing crack was loaded in mode I in situ under TEM. In the experiment, a deformation twin was formed at an angle of 90° to the crack face by the emission of 1/6[112] (gD) partial dislocations on successive (111¯ ) planes. Due to the uncontrolled sample geometry in the experiment it was not possible to extract the critical SIF for the nucleation event [23]. In the simulation, the same qualitative result is obtained as in the experiment. At a SIF of KI ⫽

˚ 2.5, a deformation twin nucleates from 0.17 eV / A the crack tip at an angle of 90° downwards. The twinning plane and direction are the same as in the experiment. Fig. 10 shows a series of snapshots of the crack tip region during the nucleation and growth of the twin. The twin nucleates on the plane intersecting the crack tip and grows to the right with increasing load. At one load step (going from ˚ 2.5 to KI ⫽ 0.21 eV / A ˚ 2.5) two KI ⫽ 0.20 eV / A twin layers nucleate simultaneously. This probably indicates that the load step was too large. As a result of the twinning the crack tip is advancing and the crack is becoming sharper. It is interesting that this apparent crack extension is occurring at a SIF which is approaching the critical SIF for Grif˚ 2.5). fith cleavage (KIc ⫽ 0.235 eV / A Fig. 11 presents the entire twin at a SIF of ˚ 2.5. The twin is five layers thick at KI ⫽ 0.26 eV / A ˚ long. However, this length the base and and 185 A has no physical significance, because in the model C simulation, adaptive meshing was not activated and the size of the twin was limited by the size of the initial fully refined mesh at the crack tip. This may also be the reason why cleavage is not observed even though KIc has been exceeded. Both in the experiment and the simulation, the twin nucleates on one side of the crack in the [1¯ 1¯ 2¯ ] direction. The reason for this is clear if one considers the available slip directions in the schematic in Fig. 11. Below the crack tip, the most favorable slip system is oriented in such a way that the leading partial (Dg) is aligned with the direction of loading. Above the crack, the leading partial (Ag) is inclined with respect to the loading direction and is thus, less favorable.

4. Conclusions The objective of this work was to computationally study the atomic-scale processes at an aluminum crack tip with special emphasis on deformation twinning. The calculations were carried out using the QC method which has been shown to reproduce lattice statics results at a greatly reduced computational effort. It was found that the crack tip behavior is quite complex. The resulting deformation mechanism strongly depends

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Fig. 10. The sequence of nucleation steps leading to the formation of the twin at the tip of the model C crack loaded in mode I ˚ 2.5; (b) KI ⫽ 0.20 eV / A ˚ 2.5; (c) KI ⫽ 0.22 eV/ A ˚ 2.5; and (d) KI ⫽ 0.23 eV / A ˚ 2.5. The dark shading indicates with (a) KI ⫽ 0.16 eV/ A twinned layers and the light shading indicates layers that will twin in the next load step.

on the crystallographic orientation, loading mode and the specifics of the crack tip morphology. Table 3 summarizes the simulation results for all orientations and loadings studied. Of particular interest are the last two results in the table which reproduce experimental observation. The model B orientation loaded in mixed mode II and III, with a sharp crack tip morphology, reproduces the twinning observed in the experiment of Pond and Garcia-Garcia [4] and the model C orientation loaded

in mode I reproduces the twinning observed in the experiment of Chen et al. [5]. The complexity manifested in Table 3 raises a basic question: what is the criterion determining whether the crack will twin or deform by slip for a given loading? Investigation of the crystallography and loading in each of the simulated systems leads to the qualitative observation that the mode of deformation that occurs is determined by the direction of the maximum resolved shear stress in the

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Fig. 11. The model C orientation loaded in mode I with ˚ 2.5. A deformation twin (shown in black) has KI ⫽ 0.26 eV/ A nucleated downward from the crack tip. The relevant face of the Thompson tetrahedron for the activated (111¯ ) slip plane is presented along side the mesh.

slip plane. Dislocation emission will occur when this direction coincides with the Burgers vector of a perfect dislocation. Twinning will occur when the direction coincides with the Burgers vector of a

partial dislocation. This is clear, for example, when comparing models A/I and C/I (Figs. 3 and 11). In both cases the slip plane is the yz plane and the maximum resolved shear stress is in the y-direction. In model A, the y-direction corresponds to a perfect dislocation direction [1¯ 10] and a dislocation is emitted. In model C, the y-direction corresponds to a partial dislocation direction [112] and twinning occurs. This simple rule has several limitations: (1) The rule does not apply to more complex loading situations where the direction of the maximum resolved shear stress does not correspond to a Burgers vector direction. In that case, a transition from slip to twinning may be expected as the maximum resolved shear stress becomes more closely aligned with a partial dislocation Burgers vector relative to a perfect dislocation direction. (2) The rule also does not address the issue of competing mechanisms on alternate planes. For example, in model B/II there is a competition between the (11¯ 1¯ ) and (11¯ 1) planes, where in both cases the maximum resolved shear stress direction does not coincide with a Burgers vector direction. (3) The rule does not take into account the morphology of the crack tip, which in model B/II–III profoundly affected the resulting deformation. (4) The rule does not address temperature and strain rate effects. These effects are not relevant in the present simulations, but may be important in experimental systems. To address some of these issues a quantitative model for twin and dislocation nucleation from a crack tip, based on the Peierls model, has been

Table 3 Summary of the simulation results. For each model/loading combination the event that occurred is listed along with the critical SIF at nucleation (Kc). The SIF values are given for the second partial dislocation emission which definitively determines whether a twin or a dislocation is nucleating. Multiply Kc by 1.6 to convert the units to MPa m0.5. The model orientations are defined in Table 2. Model/Loading

Critical Event

˚ 2.5] Kc[eV/A

A/I B/I B/II B/III B/II-III (blunt) B/II-III (sharp) C/I

edge Ⲛ (111)[11¯ 0] twin (11¯ 1¯ )[1¯ 12¯ ] fault pair (11¯ 1)[01¯ 1¯ ]&[101¯ ] screw Ⲛ (11¯ 1)[1¯ 1¯ 0] mixed Ⲛ (11¯ 1)[01¯ 1¯ ] twin (11¯ 1)[121] twin (111¯ )[112]

0.142 0.140 0.093 0.050 0.056 0.045 0.210

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developed by the authors. The model is an extension of the criterion developed by Rice [24] for dislocation emission from a crack tip. The predictions of the model are in good agreement with the results in Table 3. The model will appear in a forthcoming publication.

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