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Cleavage fracture of ordered intermetallic alloys
Materials Science and Engineering, A153 (1992) 470-478
470
Cleavage fracture of ordered intermetallic alloys M. H. Yoo and C. L. Fu Oak Ridge National Laboratory, Metals and Ceramics Division, Oak Ridge, TN 37831-6115 (USA)
Abstract Fundamental aspects of cleavage fracture behavior of ordered intermetallic alloys are analyzed on the basis of firstprinciples total energy calculations of cleavage strength (and energy) and anisotropic elasticity modeling of crack tip slip (and twinning). Intrinsic brittleness of A13Sc and NiAI is rationalized from the comparative analyses with Ni3AI and FeAI, respectively. The cleavage habit planes of (110) for NiAI and (100) for FeAI are correctly predicted in terms of the anisotropy of cleavage energies and elastic properties of dislocations. Boron strengthening in NiaAI and hydrogen embrittlement in FeAI are discussed in view of the calculated results.
1. Introduction
One of the unique aspects of mechanical behavior of ordered intermetallic alloys is the propensity for planar slip and localized slip band formation which leads to brittle cleavage fracture, without necking, even after a significant amount of tensile elongation. Once nucleated, whether a typical crack of mode I type will propagate or be stopped depends on the competition between cleavage decohesion and crack-tip blunting and/or shielding by localized plastic deformation. In the case of ordered intermetallic alloys with the strong tendency for chemical ordering, both decohesion and shear processes are governed by the intrinsic atomic bonding strength and directionality [1], which may be altered by extrinsic factors such as alloying addition and/or environmental conditions [2]. This paper is concerned with the mechanistic understanding of intrinsic fracture behavior of ordered transition metal aluminides on the basis of first-principles total energy calculations and linear elasticity analyses of pre-existing cleavage cracks. First, using the calculated elastic constants and surface energies [1], ductile vs. brittle behavior of the aluminides will be estimated according to the reported fracture criteria [3-6]. The available fracture toughness data are compared with the calculated (ideal) values and the relative role of crack tip plasticity is assessed. The important effects of elastic anisotropy on Peach-Koehler force and Peierls stress for slip and twinning dislocations near crack tips will be established next. Lastly, extrinsic effects of solute strengthening and environmental embrittlement will be discussed, followed by recommendations for future needs for atomistic simulation studies of the crack tip behavior and micromechanics modeling 0921-5093/92/$5.00
studies of the strain rate effect on brittle-ductile transition temperature.
2. Brittle
vs.
ductile behavior
2.1. Bulk properties Within the framework of the local density functional (LDF) theory, the full potential linearized augmented plane wave (FLAPW) method [7] has been used to obtain the ground state properties of the seven transition metal aluminides with high order-disorder transition temperatures, T~, as shown in Table 1. Our approach represents a major advance in applying LDF theory of solids in that the LDF equations are solved without any shape approximation to the electron charge density and potential. The calculated lattice parameters for the stable crystal structures (L1 z for TABLE 1. Calculated ratios of Pugh's and Rice-Thomson criteria for transition metal aluminides at 0 K Alloy
T¢ (K)
ao (nm)
B P
pb Ys
Ni3A! Pt3AI AI3Tia AlaSc NiAl FeAl TiAl
1668 1830 ~ 1623 1593 1911 ~ 1534 1733
0.349, 0.385 0.392 0.404 0.281 0.283 0.279 (c/a = 1.01 ) 0.378
2.0 2.3 1.6 1.3 2.9 1.5 1.8
7.0 --13.8 5.3 8.0 6.2
1.2
--
TiAl s
1623
(c/,, = 2.25) aThe L 12 structure assumed. © 1992--Elsevier Sequoia. All rights reserved
M. H. Yoo, C. L. Fu
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Cleavage of ordered intermetallic alloys
471
Ni3A1, Pt3AI, and AI3Sc; B2 for NiAI and FeAI; L10 for TiA1, and D022 for TiA13) are found in excellent agreement with the tabulated experimental data [8]. Pugh [3] introduced the quotient of elastic bulk modulus to shear modulus, BliP, as an indicator for the plastic range of a material, such that a high value of B/t~ is associated with ductility and a low value with brittleness. The quotients listed in Table 1 were obtained from the elastic stiffness constants calculated using the F L A P W method [1]. According to Pugh's criterion, Ni3A1 should be relatively more ductile than the L12 trialuminide A13Sc, and TiAI 3 should be more ductile in the LI~ structure than in the D022 structure. T h e s e predictions are consistent with the experimental observations [9]. In the B2 structure, however, the quotient for NiA1 is almost twice as large as that for FeAI, hence giving a prediction opposite to the fact that FeA1 is intrinsically more ductile and tougher than NiAI [2, 10]. Obviously, the simple criterion based on elastic moduli is not adequate for predicting brittle vs. ductile behavior of the aluminides having the relatively open B2 structure.
TABLE 2. Ideal cleavage energy (G~) and theoretical cleavage strength (Oma~)of transition metal aluminides at 0 K
2.2. Surface energies
than the ratio of the theoretical cleavage strength to the theoretical shear strength of a material, then a fully brittle fracture is possible. If the converse is the case, some type of crack tip plasticity must be always associated with crack growth. Simple ways of estimating these theoretical (ideal) cleavage and shear strengths are well summarized elsewhere [13].
The FLAPW method for thin films was used to calculate surface electronic structures [11]. The surface is modeled by a single (free standing) slab consisting of 5-15 atomic layers with two-dimensional translational symmetry parallel to the surface and vacuum above and below the slab. In Table 1 the surface energies, ~,~, used in the Rice and Thomson criterion [5] are the average values of those calculated results for low index habit planes [1, 12] (see Table 2). The magnitudes of Burgers vectors, b, for L12 alloys and f.c.c.-based tetragonal alloys are of the 1/21110] type, and Burgers vectors in NiA1 and FeA1 are (100) and 1/2(111), respectively. The Rice and Thomson model [5] states that materials with wide dislocation cores and small values of t~b/v~ (<7.5-10) are ductile, while materials with narrow cores and large values of /zb/v, are brittle. Although the correlation with the relative ductility cannot be made as extensively as in the previous case of Blip, because of the lack of Vs data, A13Sc stands out as the sole brittle intermetallic according to the Rice and Thomson criterion. The extension of the Rice and Thomson model made recently by Sun et al. [6] is described in Section 3.3.
Alloy
(hkl)
Ni~AI AI3Sc
FeA1 TiAI
O'ma x '
O'ma x
(GPa)
(GPa)
100 111 100 110
5.8 4.6 3.4 3.7
235 351 189 182
8.(1 3.1 5.8 6.1
32 60
=30
27 32 33
19 19
100 110 100
5.5 4.1 6.5
233 318 290
11.7 5.5 4.8
29 43
30 30
110
5.8
375
3.2
49 68
35 35
100
4.6
190
8.3
001 110
5.6 5.3
185 198
7.7 7.4
111
NiA1
Gc c'22 s'2 (Jm 2) (GPa) (10 t2 m 2 N- 1)
111
3.3
4.5
180
254
6.2
5.0
26 3(t 36 44
a r ' O ' m a x is the theoretical strength based on the linear elasticity approximation (see text).
3.1. Ideal cleavage strength Figure 1 shows schematically how the interracial energy, u, and the normal stress, o = - d u / d y , may vary with the separation distance, y, from an equilibrium interplanar spacing, do. The asymptotic value of u(y) gives the ideal cleavage energy, G c = 27~, which can be alternately obtained from a = o(y) by ,6
G ~ = J o(y)dy=-27~
(1)
dll
The maximum value of a(y) is the ideal cleavage strength, areax which is often approximated (the dashed curve in Fig. 1) by a'max=(ETJdo) 1/2, where E is Young's modulus. Taking into account the anisotropy of the crystal, one finds
=[
2s22ao
l
(2)
3. Theoretical strength Kelly et al. [4] first proposed the criterion that, if the ratio of the largest tensile stress to the largest shear stress close to the tip of an equilibrium crack is greater
for the cleavage strength of a rod when its sides are not t constrained (c'22 and s22 are the transformed elastic stiffness and compliance constants in the direction normal to a given (hkl) cleavage habit plane). The
472
M. H. Yoo, C. L. Fu
Yt z
Cleavageof ordered intermetallic alloys
_-x
oo =====11111 (a)
/
÷
U
Z
(b)
do
o-j
(c)
dl
df
~c~2
"Y
Fig. 1. Schematic illustrations of brittle cleavage fracture. (a) Decohesion by interatomic bond breaking, (b) interracial energy variation approaching an asymptotic value of the ideal cleavage
energy and (c) cohesive stress variation showing a maximum value of the ideal cleavage strength. approximate results based on the calculated surface energies and the elastic constants are shown in Table 2. The ratio of O'maJC'22 is in the range of 0.12-0.18. The first-principle total energy approach for calculating the cleavage energy and strength using the FLAPW method was first described by Fu [ 1 1 ]. Since it is known that the non-linear effect and electronic structure factor play important roles in determining the cleavage energy (and strength), the degree of approximation involved in the calculation of a'ax (based on linear elasticity) can be checked by the comparison of O'max with the first-principles results (trm~x and Go). Typically, as listed in Table 2, the calculated data for Ni3Al, AlaSc [1 1], NiA1 and FeAl [14] have ama~ values somewhat lower than a'max values (see Fig. 1). Obviously, it is essential to perform first-principles full potential calculations in order to obtain the correct values of cleavage energy (and strength). From our first-principles calculations, it was found that, for AlaSc, NiAl, and FeAl, the calculated a,~ax is essentially independent of the cleavage habit planes, despite Gc showing anisotropic behavior with respect to the crystallographic planes. Clearly, no direct proportionality exists between trm~x and G~, since G~ depends not only on a(y) but also on the range of
interatomic interaction (which is beyond elastic limit). The brittle fracture behavior of A13Sc is associated with much lower cleavage energies and a cleavage strength as compared with ductile Ni3AI. FeA1 has higher cleavage energies and a cleavage strength than NiA1, as would be expected from the stronger hybridization between iron d electrons and aluminum p electrons. The relatively large difference between the (100) and (110) cleavage energies in NiAI indicates that the interatomic interaction is dominated by the nearestneighbor interaction. 3.2. Interstitial solute effects In this section we consider the electronic mechanism underlying the hydrogen-induced embrittlement in FeAI and boron-induced strengthening in Ni3AI. Recently, it has been discovered that quite a number of ordered intermetallics exhibit environmental embrittlement (EE) at ambient temperatures. The embrittlement involves the reaction of water vapor with aluminum in intermetallics to form atomic hydrogen, which drives into the metals and causes fracture. The work by Liu et al. [2] clearly shows that FeAI is intrinsically ductile--the low ductility observed commonly is actually a result of the extrinsic factor of EE. Thus, our prediction that FeAI has the highest cleavage energy (and strength) among aluminides listed in Table 1 is consistent with experimental observation of the absence of cleavage fracture for FeAI when tested in hydrogen-neutral environments. Since the effects of hydrogen on the bonding in FeAI do not show up as large changes of the macroscopic yield strength, we focus on the intrinsic cleavage strength of FeAI, where the effect of hydrogen is dramatic [15]. We model the absorbed hydrogen in a supercell geometry with the periodicity of Fe-AIH-AI-Fe-A1 layers stacking along the [100] direction. The effect of hydrogen concentration is considered in two cases: (1) low concentration, in which the hydrogen-occupied tetrahedral sites are surrounded by hydrogen-free sites (up to second nearest-neighbor sites); and (2) high concentration limit, in which all the tetrahedral sites are occupied by a layer of hydrogen. We find that the effect of absorbed hydrogen is to reduce the Fe-A1 cleavage strength (see Table 3). Our calculation, thus, indicates that FeA1 is highly sensitive to the local environment of the material, and that the observed brittle cleavage fracture in FeAI tested in air is owing to the reduction of cohesive strength between iron and aluminum layers in the presence of absorbed hydrogen. This weakening effect by hydrogen is also found in the atomic cluster model using the augmented gaussian method [ 15]. From the study of bonding charge density, we find that the bonding in FeA1 has strong directional d
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Cleavageof ordered intermetallic alloys
TABLE 3. Effects of interstitial atoms on the cleavage energy and strength Alloy
Interstitials
(hkl)
Ni3A1
B
FeAI
H (low concentration) H (high concentration)
Gc (J m -2)
areax (GPa)
100
5.4
--
111
6.0
--
100 100
5.2 1.7
28 11
bonding character. This bonding charge is, however, considerably weakened in the presence of absorbed hydrogen, owing to the charge transfer from iron to hydrogen (i.e. a significant ionic component in the Fe-H bond). In other words, hydrogen acts as a negatively charged H - in aluminides. This Fe-H ionic bond mode is in sharp contrast to the Ni-B covalent bonding found for boron in Ni3AI, in which boron enhances the cohesive strength locally about the boron site. The effect of absorbed boron on the cohesion between atomic planes is studied using a supercell geometry (i.e. similar to that of hydrogen in FeA1), in which the boron-occupied octahedral site is surrounded by boron-free sites. The effect of boron on the cleavage energy is presented in Table 3. A comparison with the intrinsic cleavage energies of Ni3A1 (see Table 2) shows that boron increases the (111) cleavage energy but decreases the (100) cleavage energy. The decrease of (100) cleavage does not imply that (100) cleavage can be induced by boron, because the (100) cleavage energy with boron is still higher than that of (111) without boron. It does suggest, however, that boron tends to be absorbed in a more open site (with a local atomic environment similar to that of the (100) surface) as compared with the bulk octahedral site where each boron has six nickel atoms as its nearest neighbor. Indeed, the study of binding behavior of boron in the bulk defect sites shows that, energetically, boron prefers to be absorbed in the nickel-deficient defect sites with nearest-neighbor nickel coordination number of four to five (and without having aluminum atoms as its nearest neighbor). This indicates that boron tends to segregate preferentially to nickel-rich defect sites and to enhance atomic cohesion there through the formation of localized Ni-B covalent bonds. 3.3. Ideal shear strength No first-principles total energy calculation for the rigid body shear of one part of a crystal relative to another has been performed as yet. By extending the Peierls [16] concept, Rice [17[ has recently introduced a physical quantity 7us called "unstable stacking energy"
473
which is the maximum energy encountered in the rigid body shear process, hence a measure of the resistance to dislocation nucleation at a crack tip. For the case of an edge dislocation on a slip plane that intersects the mode I crack tip line at an angle 0, a revised criterion for ductile vs. brittle behavior was derived [6, 17] as Gd_ Yu~ Gc 2r~f2(O)
(3)
where f ( O ) = cos2(0/2)sin(0/2) for the case of elastic isotropy. Calculating ~us by the embedded atom method (EAM) and using the Y~ values obtained also from the E A M [18, 19], Sun et al. [6] predicted that Ni3A1 is more brittle than nickel in single crystal form, i.e. G J G c was larger for Ni3A1 than for nickel in both (100) and (110) mode I cracks. The importance of elastic anisotropy on dislocation multiplication near a crack is discussed in Section 4.3.
4. Fracture toughness 4.1. Critical stress intensity factor Within the linear elasticity approximation, Irwins stress intensity factor, Ki, is related to the energy release rate, G~, according to the relationship G i = C~K~ 2, where Cg is the elastic compliance constant [31]. For a mode I crack (i=I) in an elastic-perfectly brittle solid, the critical stress intensity factor is defined, for the elastically isostropic case, as Kic = {2/~ Gc/( 1 - v)} ~/2. For the rectilinear anisotropic (orthotropic) case, the ideal Ktc factor should be given by
t &J
(4)
where Sij are the reduced elastic compliance constants [32] for the plane strain conditions with respect to mode I crack orientation (Fig. 1). (It should be mentioned here that our earlier evaluation of the ideal K~c values [1, 14] are too large by a numerical factor of 10/(n)1/2.) The ideal Km values obtained from eqn. (4) using the elastic constants calculated at 0 K are listed in Table 4, and range from 0.71 for Ni-37at.%Al to 1.42 MPa (m) 1/2 for FeAI. The largest anisotropy in the ideal Klc is found to be 22% in NiAI along (110) direction, i.e. 1.06 for {100} and 0.85 for {110}. In Fe-40at.%A1 at 300 K, the differences in the ideal Klc between {100} and {110} are 21% along (100) and 20% along (110). Experimental data on fracture toughness of ordered intermetallics are sparse, particularly from single crystals. Table 4 lists K m values for transition metal aluminides available in the literature. The reported data on Ni3AI, A13Sc, and TiA1 are all from poly-
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Cleavage of ordered intermetallic alloys
TABLE 4. Comparisons of the theoretical (ideal) and measured critical stress intensity factor for mode I crack. (Note that the theoretical calculation is based on the assumption that the crystal is elastic-perfectly brittle) Alloy
(hkl)
Ni3AI
(100)
AI3Sc
(100) (110)
NiAI
(I00) (110)
Yi-36at.%Al b
(100) (110)
FeAl
(100) (110)
Fe-40at.%AF
(100) (110)
TiA1
(100) (001) (110)
[UVW]
[001 ] [011] [001] [011] [001] [110] [001] [011] [001] [110] [001 ] [011] [001] [110] [001 ] [011] [001] [110] [001 ] [011] [001] [110] [001] [011] [0101 [110] [001] [110]
KIC (MPa (m) 1/2) Ideal (0 K)
Experimental ( i>300 K)
Ref.
1.11 1.17 0.77 0.77 0.79 0.80 0.98 1.06 0.99 0.85 0.82 0.98 0.96 0.71 1.40 1.41 1.42 1.30 0.93 1.08 1.15 0.88 0.94 0.89 1.04 1.02 1.03 0.86
19-34 a 60-175 a 3a
20 21 22
10, 8 12 4.8, 4 5.2
23, 24 23 23, 24 23
~ 33
25
-~ 54
25
8-10" 6-9"
26 27
10-30
28
aData from polycrystals with alloying additions and/or microstructural modifications. bWith the elastic constants at 300 K [29] and Gc of NiAI at 0 K. cWith the elastic constants at 300 K [30] and Gc of FeAl at 0 K.
crystalline materials containing various micro- and macro-alloying additions and/or involving various microstructural parameters (e.g. grain size and shape, texture, and/or second-phase lamellar morphology, etc.). These are associated with both interracial and transgranular fracture processes. It is not surprising that the measured Kic values are much larger than the ideal K~c values, since the crack tip plasticity makes a dominant contribution to the fracture toughness. It should be pointed out however, that, despite the large contribution of crack tip plasticity to the critical stress intensity factor, the cohesion term is often suggested to play an important role in controlling the amount of plasticity and thus the cleavage fracture behavior (i.e. the stronger the cleavage cohesion, the larger the plastic work is allowed).
In the remainder of this paper, the focus of our discussion is placed on the role of crack tip plasticity in fracture toughness of NiA1 and Fe-40at.%Al. 4.2. Localized deformation in B2 alloys The most comprehensive experimental data on NiAI single crystals have been summarized recently by Vehoff [23]. The two data sets of Chang et al. [24] and Reuss and Vehoff [33] are in close agreement in that the Kic values measured at room temperature are 8 - 1 2 MPa (m) 1/2 for (100) cleavage and 4 - 5 MPa (m) m for (110) cleavage plane. The available data on temperature dependences of Kic values [24, 34] indicate that both (100) and (110) Kxc values decrease with decreasing temperatures, but an estimate of the ground state K~c values by extrapolation is not feasible.
M. 11. Yoo, C. L. Fu
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Cleavage of ordered intermetallic alloys
The premise of the question as to what loading of the mode I crack suffices to initiate a dislocation source from the tip is that cleavage decohesion does not occur first. For NiA1, we find that (110) has a relatively low cleavage energy, that is consistent with the reported (110) cleavage habit plane. As was discussed earlier [14], the large resolved shear stress for the (011)[100] slip system in the case of (100) [011] mode I crack (Fig. 2) and its absence in the case of (110)[001] crack are also consistent with the (110) habit plane, as far as the role of (110) [001 ] slip dislocations blunting the crack tip is concerned. The experimental estimates of K~c values for Fe-40at.%A1 at room temperature [25] are much higher, 33-54 MPa (m) ~/2, than the ideal values at 0 K, approximately 1 MPa (m) 1/2. Obviously, the role of crack tip plasticity is much more pronounced in Fe-40at.%A1 (with the premise that FeAI has strong cleavage cohesion), since it deforms by {112}(111 ) and {1]0}(111) slip systems at low and intermediate temperatures [35]. Figure 2 shows the amount of elastic anisotropy correction on the shear stress, rr0, at a (100) crack in Fe-40at.%A1 at 300 K. By including this anisotropic refinement, it is concluded that the driving (Peach-Koehler) force, rro_b, for b = 1/2(111) of the crack blunting type of {112}(111) slip system is higher for a (110) mode I crack than for a (100) crack [14]. This is consistent with the experimental determination of (100) cleavage plane by Chang and Rosa [25]. Furthermore, the dislocation mobility of the {112}(111) slip system is found to be suppressed by the resolved normal stress components, more at (100) than at (110) crack tip, hence also predicting the (100) cleavage plane [14].
primary slip (or twinning) and the complementary slip (or anti-twinning) directions, respectively, with 01 = 35.3 °. Those dislocations directed inward (r< rl) are attracted to the crack owing to the image forces acting on them, and they may be incorporated into a blunted crack tip by creating an extra surface step on the (112) plane. Those dislocations directed outward (r > rj ) may make contributions to shielding the crack tip. In Fig. 3, the sources $3 and $4 operate on the (112) planes that do not intersect the crack tip, and hence are not playing a role in crack tip blunting, but generating crack tip shielding (or anti-shielding) dislocations. As was mentioned earlier [36], the elastic shear anisotropy, A=2C44/(C11-Cl2), is generally high in ordered intermetallic alloys, especially those offstoichiometric NiA1, FeA1, and CuZn, e.g. A = 2.1, 3.8, 7.5, and 9.1 for FeA1 (0 K), NiA1 (0 K), Fe-40at.%,~l (300K), and Ni-37at.%A1 (300K), respectively. Effects of the elastic anisotropy on the positive pressure p and the shear stress fro are shown in Fig. 2 for the case of (100) mode I crack in Fe-40at.%Al. The specific values of rr0 for the four sources are listed in Table 5. Comparing with the isotropic values, shown within the parentheses, one finds that the anisotropic correction can be quite substantial, e.g. + 16% for the (011)[100] slip and - 3 3 % for the (112)[111] slip in Fig. 2. Therefore, the elastic anisotropy correction would make a significant contribution to the revised Rice criterion for ductile vs. brittle behavior given by eqn. (3). For example, in the case of Fig. 3, the correction off(01) = 0.27 for the isotropic case to 0.18 for the source S~ increases the G j / G c value by a factor of 2.25, •hence the brittleness increases.
4.3. Dislocation sources a n d mobility
Consider four possible dislocation sources of the (112)[111] slip (or twin) system located at the equidistance, rl, from the tip of a (100)[011] mode I crack as shown in Fig. 3. The sources S~ and $2 are in the
475
[tt2]
Y
Boo] Iltt]
J St
10oo]
~ J
:
MODE-T
_ ,.~O,
[0tt] X
is2 (b} Fig. 2. Stress fields at a mode I crack tip of (100) type in Fe-40at.%Al at 300 K in units of Kl/(2ztr) ~/2 under plane strain condition. (a) Positive (dilatant) pressure, p, and (b) shear component, fro.
Fig. 3. Four possible sources of edge dislocations of + [ 111 ](112) type near a (100) mode I crack tip in cubic crystals.
476
M. H. Yoo, C L. Fu
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Cleavage of ordered intermetallic alloys
TABLE 5. Ease-of-gliding enhancement factor, a, for :l: [ 11 i ](112) edge dislocations near a (100) mode I crack in Fe-40at.%Al at 300 K Source
0 (deg)
Kl/(2~r) 1/2 ¢7r
(70
a TrO
$1
35.3
1.37 (1.04)
0.91 (0.87)
0.18 (0.27)
-0.62
S,
- 144.7
0.84 (0.57)
0.03 (0.03)
-0.11 (-0.09)
-0.28
S~
- 54.7
0.97
0.79
- 0.27
0.54
125.3
(1.08) 1.19 (0.83)
(0.70) 0.13 (0.10)
( - 0.36) 0.27 (0.19)
0.84
$4
The stress values within the parentheses are for elastic isotropy.
The relative mobility of edge dislocations situated around a crack tip can be estimated by including the anisotropic coupling effect of the normal stress components, o r and ao, on an effective dislocation core (planar) width, ¢, such that the modified Peierls stress may be given as [14, 35], rp = 2Ke exp[-2ar(1 + a)~/b], where K~ is the energy factor of an edge dislocation and b is the magnitude of Burgers vector. The ease of gliding parameter, ¢/b, is modified by (1 + a)¢/b, where the enhancement factor is given by a =
$16or + $26o0
(5)
$66 TrO
The numerical values of the reduced compliance constants for Fe-40at.%A1 at 300 K are $16 = +0.10, S26 = -T-0.43, and $ 6 6 = 2 . 2 8 in units of 10 -11 m 2 N -1 for the + [ 111 ] (112) system. Table 5 lists the calculated enhancement factors for the four cases of dislocation sources. As was discussed previously [14], both the primary and complementary dislocations are retarded by the anisotropic coupling effect (a = - 0 . 6 2 and -0.28). In addition, we find that the relative mobility of these dislocations originating from the sources S3 and $4 is greatly enhanced (a = 0.54 and 0.84).
4.4. Transformation toughening In general, fracture toughness of a crystalline solid can be enhanced not only by the energy dissipation of dilatational transformation within the positive pressure field (Fig. 2(a)), but also by that of shear transformation within the shear stress field (Fig. 2(b)). The important role of twinning in cleavage fracture of titanium aluminides has been the subject of recent overview papers [37, 38]. Recently, Deve and Evans [39] reported an experimental confirmation of "twin toughening" in y-
TiA1. Two-fold increase in fracture resistance was observed when a high density of deformation twinning occurred in a process zone as the crack propagates. This may be the first experimental observation of crack shielding type toughening by twinning. No experimental evidence of crack blunting by twinning has been reported. However, there is atomistic simulation evidence by Kohlhoff and Schmauder [40] that a mode I crack in a-Fe is blunted by twin nucleation at the crack tip. The discussion of the previous section (Section 4.3) on the driving force on a dislocation and the dislocation mobility is equally applicable to [112](111) deformation twinning and pseudo-twinning in b.c.c, and B2-type crystals, respectively. Correspondingly, by transposing the plane and the direction, one can extend the foregoing discussion to the analogous case of {111 }(112) deformation twinning and pseudo-twinning in f.c.c.- and L12-type crystals, respectively. Possible roles of microtwinning and extended stacking faults in fracture toughness of L12 alloys, and of premartensitic displacive transformation in B2 alloys, deserve further studies especially in response to test temperature and applied strain rate.
5. Discussion
The major problem of ordered intermetallic alloys in general, and especially in B2 alloys, is the brittleness at room temperature. The brittle-ductile transition temperature (BDTT) in NiAI single crystal along the (110) orientation was reported to be around 200 °C [23, 34]. Along the (100) hard orientations, the plastic strain to failure measured in NiAI exhibits a very sharp increase at about 350 °C, and this transition temperature shifts up about 175 °C as the strain rate is raised from 8.3 x 10 -5 s -l by two orders of magnitude [34]. In the directionally-solidified Fe-40at.%Al, Chang [41] has recently observed that while the tensile elongation increased gradually, the Charpy impact energy decreased with increasing temperature. At high strain rates in the range of impact loading most intermetallic alloys, including NiAI [34] and TiAI [27], can fail in a brittle manner even at above the BDTT. In order to understand the temperature and strain rate dependencies of cleavage fracture in ordered intermetallics, it is of utmost importance to understand the mechanism of dislocation multiplication and the mobility of superdislocations beyond the simple relative analyses based on the Peierl's concept (Section 4.3). This can be studied using both dislocation micromechanics models and atomistic simulation techniques (with realistic interatomic potentials) by determining the "saddle point" configuration not only for kink pairs
M. H. }1oo, C. L. Fu
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Cleavage of ordered intermetallic alloys
on a screw dislocation, but also for jog pairs on an edge dislocation. According to Figs. 2 and 3, the positive pressure field, p, in units of K~/(2:rr) 1/2 gives relatively high values 1.09 (0.89) and 0.96 (0.83) for the edge dislocation pairs from the sources S~ and $3, respectively. This means for a mode I (100)[011] crack in Fe-40at.%A1 that while the elastic anisotropy reduces the glide mobility of ( 112)[111] edge dislocations, it increases their climb mobility. Molecular dynamic simulations would reveal crack tip processes in atomic scale, such as anharmonicity effect [42], elasto-softening [43] and micro-twinning [40] in a-Fe, hydrogeninduced decohesion in nickel [44], and localized structural phase transformation in NiAI [45]. As the current effort on constructing interatomic potentials for ordered intermetallics continues, the future outlook for detailed understanding of cleavage fracture from the combined atomistic and dislocation micromechanics viewpoint is very promising.
6. Summary The intrinsic and extrinsic cleavage behavior of transition metal aluminides has been analyzed based on first-principles total energy calculations of cleavage strength and energy and continuum modeling of cracktip slip and twinning. The following predictions are made. (1) As compared with Ni3A1 , A13Sc (i.e. trialuminides) is intrinsically more brittle because of the low cleavage strengths (and energies) and Kic values. (2) The low cleavage energy, owing to the predominantly nearest-neighbor interaction, and the difficulty of (110) [001 ] slip lead to the (110) cleavage in NiAI. (3) For the (112)[111] slip in elastically anisotropic Fe-40at.%A1, the crack tip shear stress is lower and the modified Peierls stress is higher for the (100) mode I crack than for (110), hence the (100) cleavage habit plane. (4) In Ni3AI, boron tends to segregate preferentially to nickel-rich defect site with a nearest-neighbor nickel coordination number of four to five. Boron is also found to enhance the ( 111 ) cleavage energy. (5) In the (100) cleavage fracture of FeA1, absorbed hydrogen reduces the cleavage strength (and energy) between the Fe-A1 interlayers.
Acknowledgments We thank J. H. Schneibel for helpful discussion and Connie Dowker for manuscript preparation. This research was sponsored by the Division of Materials Sciences, US Department of Energy, under contract
477
DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.
References 1 M.H. YooandC. L. Fu, IS1Jlnt., 31 (1991) 1049. 2 C. T. Liu, C. L. Fu, E. R George and G. S. Painter, 1SH Int., 31 (1991) 1191. 3 S.F. Pugh, Philos. Mag., 45 (1954) 823. 4 A. Kelly, W. R. Tyson and A. H. Cottrell, Philos. Mag., 15 (J967) 567. 5 J. R. Rice and R. Thomson, Philos. Mag., 29 (1973) 73. 6 Y. Sun, J. R. Rice and L. Truskinovsky, Mater. Res. Soc. Symp. Proc., 213 (1991) 243. 7 E. Wimmer, H. Krakauer, M. Weinert and A. J. Freeman, Phys. Rev. B, 24 (1981) 864. 8 P. Villars and L. D. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases, ASM, Metals Park, OH, 1985. 9 E. E George, D. P. Pope, C. L. Fu and J. H. Schneibel, ISIJ Int., 31 (1991) 1063. 10 K. M. Chang, R. Darolia and H. A. Lipsitt, Mater. Res. Soc. Symp. Proc., 213 (1991) 529. 11 C.L. Fu, J. Mater. Res., 5 (1990) 971. 12 C. L. Fn and M. H. Yoo, in C. T. Liu, R. W. Cahn and G. Sauthoff (eds.), Ordered Intermetallics--Physical Metallurgy and Mechanical Behavior, Kluwer, Boston, in the press. 13 A. Kelly and N. H. Macmillan, Strong Solids, Clarendon, Oxford, 3rd edn., 1986. 14 M.H. Yoo and C. L. Fu, Scr. Metall., 25 ( 1991 ) 2345. 15 C.L. Fu and G. S. Painter, J. Mater. Res., 6 (1991) 719. 16 R.E. Peierls, Proc. Phys. Soc., 52 (1940) 34. 17 J.R. Rice, J. Mech. Phys. Solids, 40 (1992) 239. 18 S. M. Foiles, M. I. Baskes and M. S. Daw, Phys. Rev. B, 33 (1986) 7983. 19 S.M. Foiles and M. S. Daw, J. Mater. Res., 2 (1987) 5. 20 J. D. Rigney, R S. Khadkikar, J. J. Lewandowski and K. Vedula, Mater. Res. Soc. Symp. Proc., 133 (1989) 603. 21 D.J. Alexander, Scr. Metall., 24 (1990) 845. 22 J. H. Schneibel and E. P. George, Scr. MetalL, 24 (1990) 1069. 23 H. Vehoff, in C. T. Liu, R. W. Cahn and G. Sauthoff (eds.), Ordered lntermetallics--Physical Metallurgy and Mechanical Behavior, Kluwer, Boston, in the press. 24 K. M. Chang, R. Darolia and H. A. Lipsitt, Mater. Res. Soc. Symp. Proc., 213(1991)597. 25 K. M. Chang and R. A. Rosa, Cleavage behavior of intermetallic Fe-40Ai single crystals, GE Report, 1990. 26 K. J. Venkateswara Rao, G. R. Odetta and R. O. Ritchie, Acta Metall., 40(1992) 353. 27 Y.W. Kim, Mater. Res. Soc. Syrup. Proc., 213 (1991)777. 28 S. Tsuyama, S. Mitao and K. Minakawa, in Y. W. Kim and R, R. Boyer (eds.), Microstructure/Property Relationships in Titanium Aluminides and Alloys, TMS, Warrendale, PA, 1991, p. 213. 29 N. Rusovic and H. Warlimont, Phys. Status Solidi, A, 44 (1977) 609. 30 H.N. Leamy, E. D. Gibson and F. X. Kayser, Acta Metall., 15 (1967) 1827. 31 J. E Knott, Fundamentals of Fracture Mechanics, Wiley, New York, 1973, p. 106. 32 G. C. Shih and H. Liebowitz, in H. Liebowitz (ed.), Fracture-An Advanced Treatise, Vol. II, Academic Press, New York, 1968, p. 68.
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33 S. Reuss and H. Vehoff, Proc. 2nd European Conf. Advanced Materials and Processing, EUROMAT '91, Cambridge, UK, in the press. 34 R. Darolia, D. E Lahrman, R. D. Field, J. R. Dobbs, K. M. Chang, E. H. Goldman and D. G. Konitzer, in C. T. Liu, R. W. Cahn and G. Sauthoff (eds.), Ordered Intermetallics-Physical Metallurgy and Mechanical Behavior, Kluwer, Boston, in the press. 35 M. H. Yoo, T. Takasugi, S. Hanada and O. Izumi, Mater. Trans. JIM, 31 (1990) 435. 36 M. H. Yoo, in O. Izumi (ed.), Proc. Int. Symp. Intermetallic Compounds--Structure and Mechanical Properties (JIMIS6), Sendai, Japan, 1991, p. 11. 37 M. H. Yoo, C. L. Fu and J. K. Lee, Mater. Res. Soc. Syrup. Proc., 213 (1991) 545.
38 M. H. Yoo, C. L. Fu and J. K. Lee, J. Phys. Ill, 1 (1991) 1065. 39 H. Deve and A. G. Evans, Acta Metall., 39 ( 1991 ) 1171. 40 S. Kohlhoff and S. Schmauder, in V. Vitek and D. J. Srolovitz (eds.), Atomistic Modelling in Materials--Beyond Pair Potentials, Plenum, New York, 1989, p. 411. 41 K.M. Chang, Metall. Trans. A, 21 (1990) 3028. 42 K. Nishioka and J. K. Lee, Philos. Mag. A, 44 ( 1981 ) 779. 43 K S. Cheug, A. S. Argon and S. Yip, J. Appl. Phys., 69 (1991 ) 2088. 44 M. S. Daw, M. I. Baskes and W. G. Wolfer, Proc. Spec. Symp. Modelling Environmental Effects on Crack Initiation and Propagation, TMS-AIME, Warrendale, PA, 1986. 45 D. Kim, P. C. Clapp and J. A. Rifkin, Mater. Res. Soc. Symp. Proc., 213 (1991) 249.
470
Cleavage fracture of ordered intermetallic alloys M. H. Yoo and C. L. Fu Oak Ridge National Laboratory, Metals and Ceramics Division, Oak Ridge, TN 37831-6115 (USA)
Abstract Fundamental aspects of cleavage fracture behavior of ordered intermetallic alloys are analyzed on the basis of firstprinciples total energy calculations of cleavage strength (and energy) and anisotropic elasticity modeling of crack tip slip (and twinning). Intrinsic brittleness of A13Sc and NiAI is rationalized from the comparative analyses with Ni3AI and FeAI, respectively. The cleavage habit planes of (110) for NiAI and (100) for FeAI are correctly predicted in terms of the anisotropy of cleavage energies and elastic properties of dislocations. Boron strengthening in NiaAI and hydrogen embrittlement in FeAI are discussed in view of the calculated results.
1. Introduction
One of the unique aspects of mechanical behavior of ordered intermetallic alloys is the propensity for planar slip and localized slip band formation which leads to brittle cleavage fracture, without necking, even after a significant amount of tensile elongation. Once nucleated, whether a typical crack of mode I type will propagate or be stopped depends on the competition between cleavage decohesion and crack-tip blunting and/or shielding by localized plastic deformation. In the case of ordered intermetallic alloys with the strong tendency for chemical ordering, both decohesion and shear processes are governed by the intrinsic atomic bonding strength and directionality [1], which may be altered by extrinsic factors such as alloying addition and/or environmental conditions [2]. This paper is concerned with the mechanistic understanding of intrinsic fracture behavior of ordered transition metal aluminides on the basis of first-principles total energy calculations and linear elasticity analyses of pre-existing cleavage cracks. First, using the calculated elastic constants and surface energies [1], ductile vs. brittle behavior of the aluminides will be estimated according to the reported fracture criteria [3-6]. The available fracture toughness data are compared with the calculated (ideal) values and the relative role of crack tip plasticity is assessed. The important effects of elastic anisotropy on Peach-Koehler force and Peierls stress for slip and twinning dislocations near crack tips will be established next. Lastly, extrinsic effects of solute strengthening and environmental embrittlement will be discussed, followed by recommendations for future needs for atomistic simulation studies of the crack tip behavior and micromechanics modeling 0921-5093/92/$5.00
studies of the strain rate effect on brittle-ductile transition temperature.
2. Brittle
vs.
ductile behavior
2.1. Bulk properties Within the framework of the local density functional (LDF) theory, the full potential linearized augmented plane wave (FLAPW) method [7] has been used to obtain the ground state properties of the seven transition metal aluminides with high order-disorder transition temperatures, T~, as shown in Table 1. Our approach represents a major advance in applying LDF theory of solids in that the LDF equations are solved without any shape approximation to the electron charge density and potential. The calculated lattice parameters for the stable crystal structures (L1 z for TABLE 1. Calculated ratios of Pugh's and Rice-Thomson criteria for transition metal aluminides at 0 K Alloy
T¢ (K)
ao (nm)
B P
pb Ys
Ni3A! Pt3AI AI3Tia AlaSc NiAl FeAl TiAl
1668 1830 ~ 1623 1593 1911 ~ 1534 1733
0.349, 0.385 0.392 0.404 0.281 0.283 0.279 (c/a = 1.01 ) 0.378
2.0 2.3 1.6 1.3 2.9 1.5 1.8
7.0 --13.8 5.3 8.0 6.2
1.2
--
TiAl s
1623
(c/,, = 2.25) aThe L 12 structure assumed. © 1992--Elsevier Sequoia. All rights reserved
M. H. Yoo, C. L. Fu
/
Cleavage of ordered intermetallic alloys
471
Ni3A1, Pt3AI, and AI3Sc; B2 for NiAI and FeAI; L10 for TiA1, and D022 for TiA13) are found in excellent agreement with the tabulated experimental data [8]. Pugh [3] introduced the quotient of elastic bulk modulus to shear modulus, BliP, as an indicator for the plastic range of a material, such that a high value of B/t~ is associated with ductility and a low value with brittleness. The quotients listed in Table 1 were obtained from the elastic stiffness constants calculated using the F L A P W method [1]. According to Pugh's criterion, Ni3A1 should be relatively more ductile than the L12 trialuminide A13Sc, and TiAI 3 should be more ductile in the LI~ structure than in the D022 structure. T h e s e predictions are consistent with the experimental observations [9]. In the B2 structure, however, the quotient for NiA1 is almost twice as large as that for FeAI, hence giving a prediction opposite to the fact that FeA1 is intrinsically more ductile and tougher than NiAI [2, 10]. Obviously, the simple criterion based on elastic moduli is not adequate for predicting brittle vs. ductile behavior of the aluminides having the relatively open B2 structure.
TABLE 2. Ideal cleavage energy (G~) and theoretical cleavage strength (Oma~)of transition metal aluminides at 0 K
2.2. Surface energies
than the ratio of the theoretical cleavage strength to the theoretical shear strength of a material, then a fully brittle fracture is possible. If the converse is the case, some type of crack tip plasticity must be always associated with crack growth. Simple ways of estimating these theoretical (ideal) cleavage and shear strengths are well summarized elsewhere [13].
The FLAPW method for thin films was used to calculate surface electronic structures [11]. The surface is modeled by a single (free standing) slab consisting of 5-15 atomic layers with two-dimensional translational symmetry parallel to the surface and vacuum above and below the slab. In Table 1 the surface energies, ~,~, used in the Rice and Thomson criterion [5] are the average values of those calculated results for low index habit planes [1, 12] (see Table 2). The magnitudes of Burgers vectors, b, for L12 alloys and f.c.c.-based tetragonal alloys are of the 1/21110] type, and Burgers vectors in NiA1 and FeA1 are (100) and 1/2(111), respectively. The Rice and Thomson model [5] states that materials with wide dislocation cores and small values of t~b/v~ (<7.5-10) are ductile, while materials with narrow cores and large values of /zb/v, are brittle. Although the correlation with the relative ductility cannot be made as extensively as in the previous case of Blip, because of the lack of Vs data, A13Sc stands out as the sole brittle intermetallic according to the Rice and Thomson criterion. The extension of the Rice and Thomson model made recently by Sun et al. [6] is described in Section 3.3.
Alloy
(hkl)
Ni~AI AI3Sc
FeA1 TiAI
O'ma x '
O'ma x
(GPa)
(GPa)
100 111 100 110
5.8 4.6 3.4 3.7
235 351 189 182
8.(1 3.1 5.8 6.1
32 60
=30
27 32 33
19 19
100 110 100
5.5 4.1 6.5
233 318 290
11.7 5.5 4.8
29 43
30 30
110
5.8
375
3.2
49 68
35 35
100
4.6
190
8.3
001 110
5.6 5.3
185 198
7.7 7.4
111
NiA1
Gc c'22 s'2 (Jm 2) (GPa) (10 t2 m 2 N- 1)
111
3.3
4.5
180
254
6.2
5.0
26 3(t 36 44
a r ' O ' m a x is the theoretical strength based on the linear elasticity approximation (see text).
3.1. Ideal cleavage strength Figure 1 shows schematically how the interracial energy, u, and the normal stress, o = - d u / d y , may vary with the separation distance, y, from an equilibrium interplanar spacing, do. The asymptotic value of u(y) gives the ideal cleavage energy, G c = 27~, which can be alternately obtained from a = o(y) by ,6
G ~ = J o(y)dy=-27~
(1)
dll
The maximum value of a(y) is the ideal cleavage strength, areax which is often approximated (the dashed curve in Fig. 1) by a'max=(ETJdo) 1/2, where E is Young's modulus. Taking into account the anisotropy of the crystal, one finds
=[
2s22ao
l
(2)
3. Theoretical strength Kelly et al. [4] first proposed the criterion that, if the ratio of the largest tensile stress to the largest shear stress close to the tip of an equilibrium crack is greater
for the cleavage strength of a rod when its sides are not t constrained (c'22 and s22 are the transformed elastic stiffness and compliance constants in the direction normal to a given (hkl) cleavage habit plane). The
472
M. H. Yoo, C. L. Fu
Yt z
Cleavageof ordered intermetallic alloys
_-x
oo =====11111 (a)
/
÷
U
Z
(b)
do
o-j
(c)
dl
df
~c~2
"Y
Fig. 1. Schematic illustrations of brittle cleavage fracture. (a) Decohesion by interatomic bond breaking, (b) interracial energy variation approaching an asymptotic value of the ideal cleavage
energy and (c) cohesive stress variation showing a maximum value of the ideal cleavage strength. approximate results based on the calculated surface energies and the elastic constants are shown in Table 2. The ratio of O'maJC'22 is in the range of 0.12-0.18. The first-principle total energy approach for calculating the cleavage energy and strength using the FLAPW method was first described by Fu [ 1 1 ]. Since it is known that the non-linear effect and electronic structure factor play important roles in determining the cleavage energy (and strength), the degree of approximation involved in the calculation of a'ax (based on linear elasticity) can be checked by the comparison of O'max with the first-principles results (trm~x and Go). Typically, as listed in Table 2, the calculated data for Ni3Al, AlaSc [1 1], NiA1 and FeAl [14] have ama~ values somewhat lower than a'max values (see Fig. 1). Obviously, it is essential to perform first-principles full potential calculations in order to obtain the correct values of cleavage energy (and strength). From our first-principles calculations, it was found that, for AlaSc, NiAl, and FeAl, the calculated a,~ax is essentially independent of the cleavage habit planes, despite Gc showing anisotropic behavior with respect to the crystallographic planes. Clearly, no direct proportionality exists between trm~x and G~, since G~ depends not only on a(y) but also on the range of
interatomic interaction (which is beyond elastic limit). The brittle fracture behavior of A13Sc is associated with much lower cleavage energies and a cleavage strength as compared with ductile Ni3AI. FeA1 has higher cleavage energies and a cleavage strength than NiA1, as would be expected from the stronger hybridization between iron d electrons and aluminum p electrons. The relatively large difference between the (100) and (110) cleavage energies in NiAI indicates that the interatomic interaction is dominated by the nearestneighbor interaction. 3.2. Interstitial solute effects In this section we consider the electronic mechanism underlying the hydrogen-induced embrittlement in FeAI and boron-induced strengthening in Ni3AI. Recently, it has been discovered that quite a number of ordered intermetallics exhibit environmental embrittlement (EE) at ambient temperatures. The embrittlement involves the reaction of water vapor with aluminum in intermetallics to form atomic hydrogen, which drives into the metals and causes fracture. The work by Liu et al. [2] clearly shows that FeAI is intrinsically ductile--the low ductility observed commonly is actually a result of the extrinsic factor of EE. Thus, our prediction that FeAI has the highest cleavage energy (and strength) among aluminides listed in Table 1 is consistent with experimental observation of the absence of cleavage fracture for FeAI when tested in hydrogen-neutral environments. Since the effects of hydrogen on the bonding in FeAI do not show up as large changes of the macroscopic yield strength, we focus on the intrinsic cleavage strength of FeAI, where the effect of hydrogen is dramatic [15]. We model the absorbed hydrogen in a supercell geometry with the periodicity of Fe-AIH-AI-Fe-A1 layers stacking along the [100] direction. The effect of hydrogen concentration is considered in two cases: (1) low concentration, in which the hydrogen-occupied tetrahedral sites are surrounded by hydrogen-free sites (up to second nearest-neighbor sites); and (2) high concentration limit, in which all the tetrahedral sites are occupied by a layer of hydrogen. We find that the effect of absorbed hydrogen is to reduce the Fe-A1 cleavage strength (see Table 3). Our calculation, thus, indicates that FeA1 is highly sensitive to the local environment of the material, and that the observed brittle cleavage fracture in FeAI tested in air is owing to the reduction of cohesive strength between iron and aluminum layers in the presence of absorbed hydrogen. This weakening effect by hydrogen is also found in the atomic cluster model using the augmented gaussian method [ 15]. From the study of bonding charge density, we find that the bonding in FeA1 has strong directional d
M. 14. Yoo, C L. Fu /
Cleavageof ordered intermetallic alloys
TABLE 3. Effects of interstitial atoms on the cleavage energy and strength Alloy
Interstitials
(hkl)
Ni3A1
B
FeAI
H (low concentration) H (high concentration)
Gc (J m -2)
areax (GPa)
100
5.4
--
111
6.0
--
100 100
5.2 1.7
28 11
bonding character. This bonding charge is, however, considerably weakened in the presence of absorbed hydrogen, owing to the charge transfer from iron to hydrogen (i.e. a significant ionic component in the Fe-H bond). In other words, hydrogen acts as a negatively charged H - in aluminides. This Fe-H ionic bond mode is in sharp contrast to the Ni-B covalent bonding found for boron in Ni3AI, in which boron enhances the cohesive strength locally about the boron site. The effect of absorbed boron on the cohesion between atomic planes is studied using a supercell geometry (i.e. similar to that of hydrogen in FeA1), in which the boron-occupied octahedral site is surrounded by boron-free sites. The effect of boron on the cleavage energy is presented in Table 3. A comparison with the intrinsic cleavage energies of Ni3A1 (see Table 2) shows that boron increases the (111) cleavage energy but decreases the (100) cleavage energy. The decrease of (100) cleavage does not imply that (100) cleavage can be induced by boron, because the (100) cleavage energy with boron is still higher than that of (111) without boron. It does suggest, however, that boron tends to be absorbed in a more open site (with a local atomic environment similar to that of the (100) surface) as compared with the bulk octahedral site where each boron has six nickel atoms as its nearest neighbor. Indeed, the study of binding behavior of boron in the bulk defect sites shows that, energetically, boron prefers to be absorbed in the nickel-deficient defect sites with nearest-neighbor nickel coordination number of four to five (and without having aluminum atoms as its nearest neighbor). This indicates that boron tends to segregate preferentially to nickel-rich defect sites and to enhance atomic cohesion there through the formation of localized Ni-B covalent bonds. 3.3. Ideal shear strength No first-principles total energy calculation for the rigid body shear of one part of a crystal relative to another has been performed as yet. By extending the Peierls [16] concept, Rice [17[ has recently introduced a physical quantity 7us called "unstable stacking energy"
473
which is the maximum energy encountered in the rigid body shear process, hence a measure of the resistance to dislocation nucleation at a crack tip. For the case of an edge dislocation on a slip plane that intersects the mode I crack tip line at an angle 0, a revised criterion for ductile vs. brittle behavior was derived [6, 17] as Gd_ Yu~ Gc 2r~f2(O)
(3)
where f ( O ) = cos2(0/2)sin(0/2) for the case of elastic isotropy. Calculating ~us by the embedded atom method (EAM) and using the Y~ values obtained also from the E A M [18, 19], Sun et al. [6] predicted that Ni3A1 is more brittle than nickel in single crystal form, i.e. G J G c was larger for Ni3A1 than for nickel in both (100) and (110) mode I cracks. The importance of elastic anisotropy on dislocation multiplication near a crack is discussed in Section 4.3.
4. Fracture toughness 4.1. Critical stress intensity factor Within the linear elasticity approximation, Irwins stress intensity factor, Ki, is related to the energy release rate, G~, according to the relationship G i = C~K~ 2, where Cg is the elastic compliance constant [31]. For a mode I crack (i=I) in an elastic-perfectly brittle solid, the critical stress intensity factor is defined, for the elastically isostropic case, as Kic = {2/~ Gc/( 1 - v)} ~/2. For the rectilinear anisotropic (orthotropic) case, the ideal Ktc factor should be given by
t &J
(4)
where Sij are the reduced elastic compliance constants [32] for the plane strain conditions with respect to mode I crack orientation (Fig. 1). (It should be mentioned here that our earlier evaluation of the ideal K~c values [1, 14] are too large by a numerical factor of 10/(n)1/2.) The ideal Km values obtained from eqn. (4) using the elastic constants calculated at 0 K are listed in Table 4, and range from 0.71 for Ni-37at.%Al to 1.42 MPa (m) 1/2 for FeAI. The largest anisotropy in the ideal Klc is found to be 22% in NiAI along (110) direction, i.e. 1.06 for {100} and 0.85 for {110}. In Fe-40at.%A1 at 300 K, the differences in the ideal Klc between {100} and {110} are 21% along (100) and 20% along (110). Experimental data on fracture toughness of ordered intermetallics are sparse, particularly from single crystals. Table 4 lists K m values for transition metal aluminides available in the literature. The reported data on Ni3AI, A13Sc, and TiA1 are all from poly-
474
M. H. Yoo, C. L. Fu /
Cleavage of ordered intermetallic alloys
TABLE 4. Comparisons of the theoretical (ideal) and measured critical stress intensity factor for mode I crack. (Note that the theoretical calculation is based on the assumption that the crystal is elastic-perfectly brittle) Alloy
(hkl)
Ni3AI
(100)
AI3Sc
(100) (110)
NiAI
(I00) (110)
Yi-36at.%Al b
(100) (110)
FeAl
(100) (110)
Fe-40at.%AF
(100) (110)
TiA1
(100) (001) (110)
[UVW]
[001 ] [011] [001] [011] [001] [110] [001] [011] [001] [110] [001 ] [011] [001] [110] [001 ] [011] [001] [110] [001 ] [011] [001] [110] [001] [011] [0101 [110] [001] [110]
KIC (MPa (m) 1/2) Ideal (0 K)
Experimental ( i>300 K)
Ref.
1.11 1.17 0.77 0.77 0.79 0.80 0.98 1.06 0.99 0.85 0.82 0.98 0.96 0.71 1.40 1.41 1.42 1.30 0.93 1.08 1.15 0.88 0.94 0.89 1.04 1.02 1.03 0.86
19-34 a 60-175 a 3a
20 21 22
10, 8 12 4.8, 4 5.2
23, 24 23 23, 24 23
~ 33
25
-~ 54
25
8-10" 6-9"
26 27
10-30
28
aData from polycrystals with alloying additions and/or microstructural modifications. bWith the elastic constants at 300 K [29] and Gc of NiAI at 0 K. cWith the elastic constants at 300 K [30] and Gc of FeAl at 0 K.
crystalline materials containing various micro- and macro-alloying additions and/or involving various microstructural parameters (e.g. grain size and shape, texture, and/or second-phase lamellar morphology, etc.). These are associated with both interracial and transgranular fracture processes. It is not surprising that the measured Kic values are much larger than the ideal K~c values, since the crack tip plasticity makes a dominant contribution to the fracture toughness. It should be pointed out however, that, despite the large contribution of crack tip plasticity to the critical stress intensity factor, the cohesion term is often suggested to play an important role in controlling the amount of plasticity and thus the cleavage fracture behavior (i.e. the stronger the cleavage cohesion, the larger the plastic work is allowed).
In the remainder of this paper, the focus of our discussion is placed on the role of crack tip plasticity in fracture toughness of NiA1 and Fe-40at.%Al. 4.2. Localized deformation in B2 alloys The most comprehensive experimental data on NiAI single crystals have been summarized recently by Vehoff [23]. The two data sets of Chang et al. [24] and Reuss and Vehoff [33] are in close agreement in that the Kic values measured at room temperature are 8 - 1 2 MPa (m) 1/2 for (100) cleavage and 4 - 5 MPa (m) m for (110) cleavage plane. The available data on temperature dependences of Kic values [24, 34] indicate that both (100) and (110) Kxc values decrease with decreasing temperatures, but an estimate of the ground state K~c values by extrapolation is not feasible.
M. 11. Yoo, C. L. Fu
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Cleavage of ordered intermetallic alloys
The premise of the question as to what loading of the mode I crack suffices to initiate a dislocation source from the tip is that cleavage decohesion does not occur first. For NiA1, we find that (110) has a relatively low cleavage energy, that is consistent with the reported (110) cleavage habit plane. As was discussed earlier [14], the large resolved shear stress for the (011)[100] slip system in the case of (100) [011] mode I crack (Fig. 2) and its absence in the case of (110)[001] crack are also consistent with the (110) habit plane, as far as the role of (110) [001 ] slip dislocations blunting the crack tip is concerned. The experimental estimates of K~c values for Fe-40at.%A1 at room temperature [25] are much higher, 33-54 MPa (m) ~/2, than the ideal values at 0 K, approximately 1 MPa (m) 1/2. Obviously, the role of crack tip plasticity is much more pronounced in Fe-40at.%A1 (with the premise that FeAI has strong cleavage cohesion), since it deforms by {112}(111 ) and {1]0}(111) slip systems at low and intermediate temperatures [35]. Figure 2 shows the amount of elastic anisotropy correction on the shear stress, rr0, at a (100) crack in Fe-40at.%A1 at 300 K. By including this anisotropic refinement, it is concluded that the driving (Peach-Koehler) force, rro_b, for b = 1/2(111) of the crack blunting type of {112}(111) slip system is higher for a (110) mode I crack than for a (100) crack [14]. This is consistent with the experimental determination of (100) cleavage plane by Chang and Rosa [25]. Furthermore, the dislocation mobility of the {112}(111) slip system is found to be suppressed by the resolved normal stress components, more at (100) than at (110) crack tip, hence also predicting the (100) cleavage plane [14].
primary slip (or twinning) and the complementary slip (or anti-twinning) directions, respectively, with 01 = 35.3 °. Those dislocations directed inward (r< rl) are attracted to the crack owing to the image forces acting on them, and they may be incorporated into a blunted crack tip by creating an extra surface step on the (112) plane. Those dislocations directed outward (r > rj ) may make contributions to shielding the crack tip. In Fig. 3, the sources $3 and $4 operate on the (112) planes that do not intersect the crack tip, and hence are not playing a role in crack tip blunting, but generating crack tip shielding (or anti-shielding) dislocations. As was mentioned earlier [36], the elastic shear anisotropy, A=2C44/(C11-Cl2), is generally high in ordered intermetallic alloys, especially those offstoichiometric NiA1, FeA1, and CuZn, e.g. A = 2.1, 3.8, 7.5, and 9.1 for FeA1 (0 K), NiA1 (0 K), Fe-40at.%,~l (300K), and Ni-37at.%A1 (300K), respectively. Effects of the elastic anisotropy on the positive pressure p and the shear stress fro are shown in Fig. 2 for the case of (100) mode I crack in Fe-40at.%Al. The specific values of rr0 for the four sources are listed in Table 5. Comparing with the isotropic values, shown within the parentheses, one finds that the anisotropic correction can be quite substantial, e.g. + 16% for the (011)[100] slip and - 3 3 % for the (112)[111] slip in Fig. 2. Therefore, the elastic anisotropy correction would make a significant contribution to the revised Rice criterion for ductile vs. brittle behavior given by eqn. (3). For example, in the case of Fig. 3, the correction off(01) = 0.27 for the isotropic case to 0.18 for the source S~ increases the G j / G c value by a factor of 2.25, •hence the brittleness increases.
4.3. Dislocation sources a n d mobility
Consider four possible dislocation sources of the (112)[111] slip (or twin) system located at the equidistance, rl, from the tip of a (100)[011] mode I crack as shown in Fig. 3. The sources S~ and $2 are in the
475
[tt2]
Y
Boo] Iltt]
J St
10oo]
~ J
:
MODE-T
_ ,.~O,
[0tt] X
is2 (b} Fig. 2. Stress fields at a mode I crack tip of (100) type in Fe-40at.%Al at 300 K in units of Kl/(2ztr) ~/2 under plane strain condition. (a) Positive (dilatant) pressure, p, and (b) shear component, fro.
Fig. 3. Four possible sources of edge dislocations of + [ 111 ](112) type near a (100) mode I crack tip in cubic crystals.
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TABLE 5. Ease-of-gliding enhancement factor, a, for :l: [ 11 i ](112) edge dislocations near a (100) mode I crack in Fe-40at.%Al at 300 K Source
0 (deg)
Kl/(2~r) 1/2 ¢7r
(70
a TrO
$1
35.3
1.37 (1.04)
0.91 (0.87)
0.18 (0.27)
-0.62
S,
- 144.7
0.84 (0.57)
0.03 (0.03)
-0.11 (-0.09)
-0.28
S~
- 54.7
0.97
0.79
- 0.27
0.54
125.3
(1.08) 1.19 (0.83)
(0.70) 0.13 (0.10)
( - 0.36) 0.27 (0.19)
0.84
$4
The stress values within the parentheses are for elastic isotropy.
The relative mobility of edge dislocations situated around a crack tip can be estimated by including the anisotropic coupling effect of the normal stress components, o r and ao, on an effective dislocation core (planar) width, ¢, such that the modified Peierls stress may be given as [14, 35], rp = 2Ke exp[-2ar(1 + a)~/b], where K~ is the energy factor of an edge dislocation and b is the magnitude of Burgers vector. The ease of gliding parameter, ¢/b, is modified by (1 + a)¢/b, where the enhancement factor is given by a =
$16or + $26o0
(5)
$66 TrO
The numerical values of the reduced compliance constants for Fe-40at.%A1 at 300 K are $16 = +0.10, S26 = -T-0.43, and $ 6 6 = 2 . 2 8 in units of 10 -11 m 2 N -1 for the + [ 111 ] (112) system. Table 5 lists the calculated enhancement factors for the four cases of dislocation sources. As was discussed previously [14], both the primary and complementary dislocations are retarded by the anisotropic coupling effect (a = - 0 . 6 2 and -0.28). In addition, we find that the relative mobility of these dislocations originating from the sources S3 and $4 is greatly enhanced (a = 0.54 and 0.84).
4.4. Transformation toughening In general, fracture toughness of a crystalline solid can be enhanced not only by the energy dissipation of dilatational transformation within the positive pressure field (Fig. 2(a)), but also by that of shear transformation within the shear stress field (Fig. 2(b)). The important role of twinning in cleavage fracture of titanium aluminides has been the subject of recent overview papers [37, 38]. Recently, Deve and Evans [39] reported an experimental confirmation of "twin toughening" in y-
TiA1. Two-fold increase in fracture resistance was observed when a high density of deformation twinning occurred in a process zone as the crack propagates. This may be the first experimental observation of crack shielding type toughening by twinning. No experimental evidence of crack blunting by twinning has been reported. However, there is atomistic simulation evidence by Kohlhoff and Schmauder [40] that a mode I crack in a-Fe is blunted by twin nucleation at the crack tip. The discussion of the previous section (Section 4.3) on the driving force on a dislocation and the dislocation mobility is equally applicable to [112](111) deformation twinning and pseudo-twinning in b.c.c, and B2-type crystals, respectively. Correspondingly, by transposing the plane and the direction, one can extend the foregoing discussion to the analogous case of {111 }(112) deformation twinning and pseudo-twinning in f.c.c.- and L12-type crystals, respectively. Possible roles of microtwinning and extended stacking faults in fracture toughness of L12 alloys, and of premartensitic displacive transformation in B2 alloys, deserve further studies especially in response to test temperature and applied strain rate.
5. Discussion
The major problem of ordered intermetallic alloys in general, and especially in B2 alloys, is the brittleness at room temperature. The brittle-ductile transition temperature (BDTT) in NiAI single crystal along the (110) orientation was reported to be around 200 °C [23, 34]. Along the (100) hard orientations, the plastic strain to failure measured in NiAI exhibits a very sharp increase at about 350 °C, and this transition temperature shifts up about 175 °C as the strain rate is raised from 8.3 x 10 -5 s -l by two orders of magnitude [34]. In the directionally-solidified Fe-40at.%Al, Chang [41] has recently observed that while the tensile elongation increased gradually, the Charpy impact energy decreased with increasing temperature. At high strain rates in the range of impact loading most intermetallic alloys, including NiAI [34] and TiAI [27], can fail in a brittle manner even at above the BDTT. In order to understand the temperature and strain rate dependencies of cleavage fracture in ordered intermetallics, it is of utmost importance to understand the mechanism of dislocation multiplication and the mobility of superdislocations beyond the simple relative analyses based on the Peierl's concept (Section 4.3). This can be studied using both dislocation micromechanics models and atomistic simulation techniques (with realistic interatomic potentials) by determining the "saddle point" configuration not only for kink pairs
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Cleavage of ordered intermetallic alloys
on a screw dislocation, but also for jog pairs on an edge dislocation. According to Figs. 2 and 3, the positive pressure field, p, in units of K~/(2:rr) 1/2 gives relatively high values 1.09 (0.89) and 0.96 (0.83) for the edge dislocation pairs from the sources S~ and $3, respectively. This means for a mode I (100)[011] crack in Fe-40at.%A1 that while the elastic anisotropy reduces the glide mobility of ( 112)[111] edge dislocations, it increases their climb mobility. Molecular dynamic simulations would reveal crack tip processes in atomic scale, such as anharmonicity effect [42], elasto-softening [43] and micro-twinning [40] in a-Fe, hydrogeninduced decohesion in nickel [44], and localized structural phase transformation in NiAI [45]. As the current effort on constructing interatomic potentials for ordered intermetallics continues, the future outlook for detailed understanding of cleavage fracture from the combined atomistic and dislocation micromechanics viewpoint is very promising.
6. Summary The intrinsic and extrinsic cleavage behavior of transition metal aluminides has been analyzed based on first-principles total energy calculations of cleavage strength and energy and continuum modeling of cracktip slip and twinning. The following predictions are made. (1) As compared with Ni3A1 , A13Sc (i.e. trialuminides) is intrinsically more brittle because of the low cleavage strengths (and energies) and Kic values. (2) The low cleavage energy, owing to the predominantly nearest-neighbor interaction, and the difficulty of (110) [001 ] slip lead to the (110) cleavage in NiAI. (3) For the (112)[111] slip in elastically anisotropic Fe-40at.%A1, the crack tip shear stress is lower and the modified Peierls stress is higher for the (100) mode I crack than for (110), hence the (100) cleavage habit plane. (4) In Ni3AI, boron tends to segregate preferentially to nickel-rich defect site with a nearest-neighbor nickel coordination number of four to five. Boron is also found to enhance the ( 111 ) cleavage energy. (5) In the (100) cleavage fracture of FeA1, absorbed hydrogen reduces the cleavage strength (and energy) between the Fe-A1 interlayers.
Acknowledgments We thank J. H. Schneibel for helpful discussion and Connie Dowker for manuscript preparation. This research was sponsored by the Division of Materials Sciences, US Department of Energy, under contract
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DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.
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