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Effects of lattice misfit on plastic deformation behavior of single-crystalline micropillars of Ni-based superalloys
Accepted Manuscript Effects of lattice misfit on plastic deformation behavior of single-crystalline micropillars of Ni-based superalloys
Kabir Arora, Kyosuke Kishida, Katsushi Tanaka, Haruyuki Inui PII:
S1359-6454(17)30614-6
DOI:
10.1016/j.actamat.2017.07.044
Reference:
AM 13943
To appear in:
Acta Materialia
Received Date:
04 December 2016
Revised Date:
15 July 2017
Accepted Date:
21 July 2017
Please cite this article as: Kabir Arora, Kyosuke Kishida, Katsushi Tanaka, Haruyuki Inui, Effects of lattice misfit on plastic deformation behavior of single-crystalline micropillars of Ni-based superalloys, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.07.044
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Effects of lattice misfit on plastic deformation behavior of single-crystalline micropillars of Ni-based superalloys
Kabir Arora1, Kyosuke Kishida1,2*, Katsushi Tanaka3, and Haruyuki Inui1,2 1
Department of Materials Science and Engineering, Kyoto University, Sakyo-ku, Kyoto
606-8501, Japan 2
Center for Elements Strategy Initiative for Structural Materials (ESISM), Kyoto
University, Sakyo-ku, Kyoto 606-8501, Japan 3
Department of Mechanical Engineering, Kobe University, 1-1 Rokkoudai-cho, Nada-
ku, Kobe 657-8501, Japan
Abstract The deformation behaviors of rectangular-parallelepiped micropillars fabricated from [001]-oriented Ni-based superalloys with negative and positive lattice misfits, respectively, were investigated with respect to the specimen size. The yield stresses of the larger specimens (side length L > ~2 μm) of both considered alloy types were found to be nearly independent of the size, while the smaller specimens (L < ~2 μm) exhibited “smaller is stronger” and “smaller is (slightly) weaker” trends for negative and positive lattice misfits, respectively. The internal stresses calculated based on the micro-elasticity theory revealed that the internal shear stress fields of the micropillars varied with their sizes. The size dependence of the calculated internal shear stress distribution agreed fairly well with the experimentally determined size dependence of the yield stress, indicating that the distribution of the internal misfit stress significantly contributed to the sizedependent yield stress of the smaller micropillars. 1
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Keywords: Ni-based superalloys; Yield strength; Size effects; Lattice misfit; Internal stress *Corresponding author. Tel./Fax: +81-75-753-5461 E-mail: [email protected]
1. Introduction The microcompression test method introduced by Uchic et al. has been recognized as a new technique to investigate the deformation behavior of various materials at sizescales of tens of micrometers or less [1, 2]. Extensive studies have been conducted on the deformation behaviors of single crystals of various materials such as pure metals, intermetallic compounds, and semiconductors, and most of them have been found to exhibit “smaller is stronger” size-dependent deformation [1–15]. Various possible mechanisms have been used to explain the size-dependent strengthening of single crystals of metallic materials, such as dislocation starvation (surface nucleation) [16, 17] and truncation of the single-arm dislocation sources (SASs), in which the length of the SASs is considered to be a controlling factor [18–20]. A similar size-dependent strengthening has also been observed in single crystals of Ni-based superalloys, which have a two-phase microstructure consisting of cuboidal precipitates of Ni3Al (γ’) coherently embedded in an FCC nickel (γ) matrix [10, 21, 22]. However, the underlying mechanism of the size-dependent strengthening is yet to be clarified, which is because the mechanical properties of Ni-based superalloys are affected by several factors, including the (i) flow stress of the γ matrix, (ii) size, morphology, and volume fraction of the γ′ precipitates, (iii) APB energy, (iv) misfit stress, and (v) pre2
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existing dislocations [22, 23]. The morphology of the γ′ precipitates is known to be mainly controlled by the lattice misfit δ between the two constituent phases, given by
2(a ' a ) /(a ' a ) , where aγ′ and aγ are the lattice parameters of the γ′ precipitate and γ matrix, respectively [23, 24]. An important determinant of the lattice misfit is the internal misfit stress, which is the primary reason for the development of the so-called raft structure comprising γ and γ′ lamellae alternately stacked in the <001> loading direction, as observed during the tensile creep testing of Ni-based superalloys with a negative lattice misfit [23, 25, 26].
Fig. 1 shows schematic illustrations of the
distributions of the internal misfit stresses in Ni-based superalloys with negative and positive lattice misfits, respectively. Ni-based superalloys with a negative lattice misfit typically have biaxial compressive stress in the 𝛾 channels, while triaxial tensile stress act on the 𝛾' precipitates [23]. The reverse is the case in such superalloys with a positive lattice misfit; i.e., biaxial tensile and triaxial compressive stress act on the 𝛾 channels and the 𝛾' precipitates, respectively. On a scale comparable to the size of the γ′ precipitates (≤ 1 μm), the effects of the internal misfit stress on the deformation behavior may be significant. Previous investigations of the deformation behaviors of two-phase Ni-based superalloys by microcompression using cylindrical micropillar specimens of diameters > 2 µm have revealed a size dependence of the flow stress at 1%–3% plastic strain, similar to that exhibited by single-phase materials [21, 22]. The present authors also recently observed a similar size-dependent flow stress at 1% plastic strain during micropillar compression testing of the [001]-oriented superalloy CMSX-4 [10, 21, 22]. However, these previous works only considered the effects of the specimen size without explicitly
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examining the effects of the lattice misfit on the deformation behavior of small specimens with sizes comparable to that of the γ′ precipitates. In the present study, we focused on the compression behavior of Ni-based superalloy micropillars measuring ≤ 8 µm and the effects of lattice misfit on their deformation characteristics at room temperature. We specifically considered the commercial alloy CMSX-4 with a negative misfit [23] and a newly developed alloy, hereafter referred to as alloy-A, with a positive misfit. The two alloys were used to compare the effects of the misfit type on the deformation characteristics. The effects of the lattice misfit δ on the plastic deformation behavior of single-crystalline micropillars of the Ni-based superalloys were examined with respect to the internal elastic stress fields calculated based on the micro-elasticity theory [27].
2. Experimental procedures Two materials with lattice misfits of opposite signs, namely, CMSX-4 (δ = ˗0.001–˗0.023 at room temperature [28]) and alloy-A (δ = ~+0.004 at room temperature) were used in this study. The nominal composition of alloy-A was determined by decreasing Mo, W, Al, Ti, Hf, Re and increasing Ta from that of CMSX-4 [23] so as to obtain a cuboidal microstructure with a positive lattice misfit. A single crystal of alloy-A was grown by the Bridgman technique at a growth rate of 5 mm/h. The as-grown single crystal was homogenized at 1300°C for 1 h and at 1100°C for 4 h, and finally aged at 870°C for 4 h. The typically observed {001}-section microstructures of the as-received CMSX-4 and heat-treated alloy-A single crystals are shown in Fig. 2. The two materials had similar cuboidal microstructures, with the average edge length of the cuboidal γ′
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precipitates being 400–500 nm. The volume fractions of the γ′ phase in CMSX-4 and alloy-A were estimated to be about 67% and 70%, respectively. Rectangular-parallelepiped bulk specimens with their surfaces parallel to the {001} plane (hereafter (001) surfaces) were cut from the single crystals by electrodischarge machining. The (001) surfaces of the bulk specimens were mechanically polished and then electropolished in a solution of 60% perchloric acid, n-butyl alcohol, and methanol (1:6:12 volume ratio) at 18 V and -55°C. [001]-oriented micropillars with a square cross section (side length L values of 0.5–8 μm) were fabricated from the electropolished (001) surface by the focused ion beam (FIB) technique using a JEOL JIB4000 FIB system with a Ga ion source operated at an acceleration voltage of 30 kV. Two orthogonal side surfaces of the micropillars were set parallel to the (100) and (010) planes, respectively, to align all the edges of the micropillar with those of the cuboidal precipitates. The height-to-length (L) ratio of the micropillars was set to be 2 - 4. Compression tests were conducted on the micropillars using a Shimadzu DUH211 microhardness testing machine equipped with a flat diamond tip of diameter 20 μm. All the micropillars were tested at room temperature in a load control mode (corresponding nominal strain rates of 10-3–10-4 s-1). The microstructures of the micropillars before and after compression testing were examined by scanning electron microscopy (SEM) using a JEOL JSM-7001FA electron microscope equipped with a field emission gun.
3. Modeling
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If a material that contains coherent inclusions is assumed to be an elastically homogeneous medium, an elastic displacement in the material can be solved in the Fourier space [27]. The elastic displacement uk(r) is given by the following equation:
uk (r ) where
1 iGik g g j ijpq *pq r g eigr d 3g 3 2
(1),
Gik g , ijpq , and *pq r g denote the Green’s function, elastic constants, and
Fourier transformation of the eigenstrain (stress-free strain), respectively. In the present case of the Ni-based superalloys with cuboidal microstructures, the eigenstrain is simply given by
*pq r ( r ' precipitates and p = q) = 0 (otherwise)
(2).
To simulate the surface relaxation effect, we introduced a (virtual) buffer layer in directions perpendicular to the side surfaces of a micropillar. The buffer layer was assumed to have the same elastic constants as the micropillar. Plastic and volumetric deformations of the buffer layer was allowed to eliminate the elastic constraint of the buffer layer on the micropillar. With these assumptions, the eigenstrain of the buffer layer could be determined using the following iterative calculation: *pq r
i 1
r , *pq r elastic pq i
i
r buffer
layer
(3),
where elastic r is the elastic strain determined at the i-th iteration, and is a dumping i factor for stabilizing the iterative calculation. Two types of cubic unit blocks (configurations A and B, as illustrated in Fig. 3) were considered in this study, each of which contained the same volume fraction of the γ′ phase (0.67) embedded in the γ matrix. Each cubic unit block, with edges set parallel to x 6
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= [100], y = [010], and z = [001], contained 96 × 96 × 96 grid points. For each configuration, five models comprising n × n × 1 cubic unit blocks (n = 1, 2, 3, 4, and 8) were used for the calculation, as illustrated in Fig. 3(c). The unit blocks were surrounded by a buffer layer of thickness 16n grid points in the x and y directions. The five models are hereafter referred to as model n (n = 1, 2, 3, 4, and 8). Because the average edge length of the cuboidal precipitates for both considered materials is 400–500 nm, the edge length of a cubic unit block was about 460–570 nm. Thus, the five models approximately represented micropillars with side length L values of 0.5, 1.0, 1.5, 2.0, and 4.0 μm, respectively. In addition, the internal stress distributions of the bulk specimens (infinite L) were simulated using an additional model with configuration A without the surrounding buffer layer. The model was designated as the bulk model. Lattice misfit values of -0.002 (negative lattice misfit) and +0.004 (positive lattice misfit) were used to calculate the stress distributions corresponding to the CMSX-4 and alloy-A micropillars, respectively. In the calculations, all models were assumed to be elastically homogeneous with average elastic constants C11 = 205 GPa, C12 = 121 GPa, and C44 = 118 GPa for both the γ and γ′ phases, the values of which were estimated from those reported for the γ and γ′ phases [29]. In order to clarify the influences of the lattice misfit only, it is sufficient to use the average elastic constants for the calculations of the internal stress distribution. No external applied stress was considered in the calculations.
4. Results 4.1 Microcompression experiments
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Representative stress-strain curves of the [001]-oriented single-crystalline CMSX-4 (negative lattice misfit) and alloy-A (positive lattice misfit) micropillars are shown in Fig. 4(a) and 4(b), respectively. Stress-strain curves of bulk-size specimens of the [001]-oriented single crystals tested in compression at a constant strain rate of 10-4 /s are also indicated in Fig. 4 for comparison. The stress-strain curves of the relatively large CMSX-4 micropillars (L ≥ 2.0 μm) (Fig. 3(a)) are characterized by a transition region between elastic and plastic flows with continuously decreasing work hardening rate, followed by a region with a large strain burst or low work hardening. The low work hardening behavior of the larger micropillars is similar to that observed in the stress-strain curve of the bulk-size specimen of CMSX-4. In the case of the smaller CMSX-4 micropillars (L < 2.0 μm), the stress-strain curves generally exhibit a staircase-like pattern, with alternations between (almost) elastic loading and microstrain bursts. This results in a higher flow stress at a plastic strain above 1%. These observed characteristics of the CMSX-4 micropillars agree well with previous reports regarding other Ni-based superalloys, namely, UM-F19 and René N5 [21, 22]. The characteristics of the stressstrain curves of the alloy-A micropillars with a positive lattice misfit are generally similar to those of the CMSX-4 micropillars, except for the relatively high work hardening behavior observed for the larger micropillars (L ≥ 2.0 μm) as well as the bulk-size specimen of alloy-A. Fig. 5 shows the side length L dependence of the yield stress (flow stress at 0.2% plastic strain). For both alloys, the yield stress of the larger micropillars (L > ~2 μm) is almost independent of the specimen size. However, opposite trends are observed for the yield stress of the smaller micropillars (L ≤ ~2 μm), namely, “smaller is stronger” for CMSX-4 (Fig. 5(a)) and “smaller is (slightly) weaker” for alloy-A (Fig. 5(b)). A nearly 8
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size-independent 0.2% flow stress has been similarly observed for the compression of large René N5 single-crystalline micropillars oriented for single-slip [22]. Because the microstructural characteristics of the two materials, such as the volume fractions and morphologies of their γ′ precipitates, are quite similar, the difference between the yield stress of their smaller micropillars (L < ~2 μm) is believed to be due to the differing internal stress distributions induced by the opposite signs of their lattice misfits.
4.2 SEM observations Figs. 6 and 7 show the SEM secondary electron images of some CMSX-4 and alloy-A micropillars after compression, respectively. Multiple slip traces parallel to more than two {111} planes are observed to have developed rather homogeneously during a relatively early stage of the plastic deformation (about 0.25% compressive strain (Fig. 6(b)). Most of the slip traces extend continuously from one edge to another, indicating that, not only the γ channels, but also the γ′ precipitates were sheared during a very early stage of the plastic deformation. Similar trends are also observed for the micropillars of alloy-A (Fig. 7). No apparent differences are observed between the characteristics of the slip traces of the two alloys.
4.3. Internal stress calculation 4.3.1 Bulk model Typical results of the internal stress calculations for the bulk models are presented in Fig. 8, which shows the two-dimensional contour maps of the resolved shear stress of the (111)[011] slip system, generated by the lattice misfits of -0.002 (CMSX-4) and +0.004 (alloy-A). The resolved shear stress generated by a lattice misfit is hereafter 9
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designated the internal shear stress
int . The (111)[011] slip system is one of the eight
operative slip systems with the highest Schmid factor of 0.408 under uniaxial compression in the [001] direction. Three contour plots with mutually orthogonal cross sections passing through the center of the cubic unit block of model 1 are presented for each model. Figs. 8(a)–8(c) and 8(g)–8(i) show the contour maps of the values of
int
calculated using the bulk model for the negative and positive lattice misfits, respectively. In the contour maps, the positive maximum (+600 MPa), zero, and negative maximum (600 MPa) values of
int are indicated in green (8-bit RGB code (0, 255, 0)), white (255,
255, 255), and red (255, 0, 0), respectively. The sign of the resolved shear stress is assigned such that the application of a positive shear stress on the slip system yields a compressive strain in the [001] direction. Consequently, the external applied stress in the [001] direction also yields a positive resolved shear stress on the (111)[011] slip system (hereafter designated the external shear stress ext ). The effective (or total) shear stress
eff acting on the slip system is simply given by eff int ext
(4).
Figs. 9(a,b) schematically illustrate the distributions of
int on the (111) slip plane for
the bulk models with the negative and positive lattice misfits, respectively. In the figures, red, white and green colors qualitatively indicate negative, zero and positive values of
int , respectively and blue circles indicate the positions exhibiting the highest int values. For a negative lattice misfit (Figs. 8(a) and 9(a)), only the vertical γ channels perpendicular to the [010] axis (hereafter referred to as the Vy channel) exhibit relatively 10
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high positive values (green regions). The (111)[011] slip is thus expected to be initially activated and propagate inside the Vy channels, because
eff is able to reach the critical
value for slip activation under a relatively low external compressive load in the [001] direction, according to eq. (4). In contrast, for a positive lattice misfit (Figs. 8(c) and 9(b)), the regions favorable for the (111)[011] slip activation (green regions) under a [001] compressive load are initially limited in the horizontal γ channels (H channels) perpendicular to the [001] axis. The distributions of the shear stress can be more quantitatively appreciated from Fig. 10, which shows histograms of
int for the γ and γ′ phases as obtained from the entire
grid points inside the model micropillars. In addition to the total distributions shown as bar charts, the
int distributions inside the various γ channels (Vx (vertical channel
perpendicular to the [100] axis), Vy, H, and their intersections (designated mixed channels)) are also indicated by the line plots in Figs. 10(a) and 10(e). The
int data from
the mixed channels are excluded from the plots for the Vx, Vy, and H channels. As can be clearly seen from Figs. 10(a) and 10(e), the three geometrically different major γ channels, namely, the Vx, Vy, and H channels with relatively large volumes, have different
int
distribution profiles. In the case of the negative lattice misfit, the average values of
int
in the major Vx, Vy and H channels are zero, positive, and negative, respectively, whereas they are zero, negative, and positive for the positive lattice misfit, respectively. It is important to note that the average value of
int for the (111)[011] slip inside the γ′
precipitate of the bulk model is zero with a relatively small standard deviation irrespective of the sign of the lattice misfit. This is quite reasonable considering that the internal stress 11
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field inside the γ′ precipitates is virtually isotropic for an ideal bulk model. These results for the present bulk models are consistent with those in a recent report [30].
4.3.2 Effects of micropillar size and in-plane configuration of γ′ precipitates on internal shear stress The determined distributions of
int for model 1 (n = 1) with surface relaxation
are significantly different from those for the bulk models. The trends of the contour maps and distributions of
int for model 1 for negative (Figs. 8(b), 9(c), 10(c), and 10(d)) and
positive (Figs. 8(d), 9(d), 10(g), and 10(h)) lattice misfits are clearly opposite. Specifically, the values of
int tend to be negative (red region in Figs. 8(b) and 9(c)) and
positive (green region in Figs. 8(d) and 9(d)) for negative and positive lattice misfits, respectively. Similar trends were observed for all the micropillar models n (n = 1, 2, 3, 4, and 8). The average values of
int for the various γ channels and γ’ precipitates for the
two in-plane configurations of the γ′ precipitates (configurations A and B) are plotted as a function of the number of unit blocks (n) in the x and y directions in Fig. 11, in which the horizontal axis is plotted on a logarithmic scale. The error bars in Fig. 11 indicate the standard deviations. In the case of the negative lattice misfit (Figs. 11(a) and 11(b)), only the Vy channels has a positive average value of
int ; the other γ channels and the γ′
precipitates have negative or nearly zero average values of
int for all the models (n = 1,
2, 3, 4, and 8 approximately represented micropillars with side length L of 0.5, 1, 1.5, 2, and 4.0 μm, respectively) irrespective of the in-plane configuration. This suggests that the activation and propagation of [011] dislocations on (111) are suppressed in most of the γ 12
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channels (except for the Vx channels) as well as the γ′ precipitates by the internal stress
int . The situation is completely opposite for the positive misfit. In this case, most of the γ channels and γ′ precipitates, with the exception of the Vy channels and the mixed channels between the Vx and Vy channels, have positive
int values. This indicates that,
for a positive lattice misfit, slip activation and propagation in most of the γ channels are strongly assisted by the internal stress
int . The varying int distributions of the model
1 micropillars are thus considered to significantly affect their yield behavior. With regard to the size dependence, the values of
int for all the γ channels and
γ′ precipitates vary with the micropillar size exhibiting different size dependences depending on the channel. Among the major γ channels, i.e., the Vx, Vy, and H channels, the size dependences of
int are greater for the latter two. The int values for the Vy, and
H channels gradually decrease and increase with decreasing n for the negative and positive lattice misfits, respectively. In addition, the size dependence of
int for the H
channels appears stronger for the in-plane configuration A, while that in the Vy channels is slightly stronger for the in-plane configuration B. These results indicate that the surface relaxation effects are greater in the H channels when the γ channels are located on the side surfaces of the smaller micropillars. The
int values for the most of the mixed γ
channels, with the exception of the mixed channel between the Vx and Vy, channels, exhibit greater and opposite size dependences compared to those for the major γ channels. However, the internal stresses
int for the mixed γ channels are expected to give minor
contributions to the macroscopic yielding because of relatively small volume fractions 13
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and higher
int of the mixed γ channels compared to those of the major γ channels with
the lowest average
int . The internal stress int for the γ′ precipitates does not exhibit
apparent size dependence in the size range of n = 1 ~ 8.
5. Discussion In the present study, we experimentally determined that the yield stress (0.2% flow stress) in smaller micropillars (L < ~2 μm) exhibited different trends depending on the sign of the lattice misfit. Moreover, our calculation results clearly indicated that the effects of surface relaxation on the internal stress distributions were significant for micropillars. Here, we discuss the possible influences of the internal stress distributions on macroscopic yielding of micropillar specimens. Figs. 9(c) and 9(d) schematically illustrate the distributions of
int on the (111)
slip plane for model 1 (configuration A) with the negative and positive lattice misfits, respectively. In the case of the negative lattice misfit (Fig. 9(c)), the positions with the highest
int (marked with a blue circle) was found to be located inside the Vy channel
near the edge of the γ’ precipitate. Thus, it is reasonable to assume that dislocations inside the Vy channel near the position exhibiting the highest
int are initially activated when
eff exceeds the critical value for the (111)[011] slip activation. In addition, considering the fact that γ’ precipitates were sheared during a very early stage of the plastic deformation of the micropillars of CMSX-4 (Fig. 6), dislocations are required to glide through the entire slip plane including the H channels with the lowest average well as the γ’ precipitates with a negative average 14
int as
int at the 0.2 % flow stress level. To
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achieve this, the effective stresses eff for dislocation activation in both the H channels and the γ’ precipitates needs to exceed certain critical values under external compressive loading, which indicates that a dominant controlling factor for macroscopic yielding of micropillars with the negative lattice misfit is considered to be the internal stress either the H channels or the γ’ precipitates. Although the critical values of
int for
eff required
for dislocation activation in the γ channels and the γ’ precipitates are unknown, it is reasonable to assume that these two
eff values does not differ much as a first
approximation, considering the fact that dislocation activation in both phases at early stages of plastic deformation is generally observed in bulk-size single crystals of Ni-based superalloys tested in tension at room temperature [31-32]. Under this assumption, the H channels with the lowest average
int are considered to play a key role for macroscopic
yielding. As seen in Figs. 11(a) and 11(b), the
int values for the H channels gradually
decrease with decreasing n for the negative lattice misfit. According to eq. (4), a decrease in internal stress
int should increase the minimum external stress ext to make eff
exceed the critical values for macroscopic yielding with the decrease in the specimen size. This is consistent qualitatively with the experimentally observed size dependence of the yield stress. Thus, the assumption for
eff is considered to be reasonable. If the similar
argument is applied for the cases of the positive lattice misfit, the
int value of the Vy
channels is expected to be the controlling factor for macroscopic yielding (Fig. 9(d)). Because the
int values for the Vy channels slightly increase with decreasing n for the
positive lattice misfit (Fig. 11(c) and 11(d)), the minimum external stress 15
ext is
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considered to decrease with the specimen size, which is also consistent fairly well with the experimentally observed tendency of the size dependent yield stress. Thus, the variations of the
int distributions in the major γ channels with the lowest average int
is considered to predominantly affect the size dependence of the yield stress of the Nibased superalloy micropillars. However, the above discussions may only be valid for the earliest stage of the plastic deformation. Indeed, a previous study on the compression of René N5 singlecrystalline micropillars oriented for single-slip activation revealed that the power law factor of the size-dependent flow stress increased with increasing plastic strain [22]. A similar observation has also been made for CMSX-4 micropillars [10]. With the progression of plastic deformation, the distribution of the internal stress
int is expected
to vary gradually because the lattice misfit can be partly relaxed by the interaction between activated dislocations and interfaces between the γ matrix and γ′ precipitates. Because the distribution of
int affects various dislocation-related factors such as the
activation of dislocation sources in the γ matrix, the dislocation velocity, the occurrence of cross-slip and shearing of the γ′ precipitates, the quantitative estimation of the effects of
int on the observed flow behavior is very complex. Further study involving, for
example, simulation of the dislocation dynamics and modeling based on dislocation kinetics is required to obtain a full understanding of the observed variations of the sizedependent flow stress with respect to the plastic strain.
6. Conclusions
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We conducted systematic micropillar compression tests on [001]-oriented Ni-based superalloys, namely, CMSX-4 (with negative misfit) and alloy-A (with positive misfit), and used the micro-elasticity theory to calculate the internal stresses with respect to the specimen size. Following is a summary of the main findings: 1.
The yield stress of the [001]-oriented CMSX-4 and alloy-A micropillars are nearly independent on the specimen size, for side length L values of 2–8 μm. In contrast, smaller (L ≤ ~2 μm) CMSX-4 (negative lattice misfit) and alloy-A (positive lattice misfit) micropillars exhibit “smaller is stronger” and “smaller is (slightly) weaker” trends, respectively.
2.
The calculated internal stress fields of the Ni-based superalloy micropillars vary with the specimen size because of the surface relaxation effects on the lattice misfit strain. The overall distributions of the internal shear stress
int resolved for a slip system
favorable to uniaxial compression shift toward the opposing and assistive sides for slip activation in micropillars with negative and positive lattice misfits, respectively. The numerically determined trends of the
int distributions are consistent with
experimental results, which indicate an increase and decrease in the yield stress with decreasing micropillar size for negative and positive lattice misfits, respectively. These results clearly indicate that the variation of the
int distributions is a
determinant of the size-dependent yield behavior of Ni-based superalloy micropillars under compression.
Acknowledgements 17
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This work was supported by the Elements Strategy Initiative for Structural Materials (ESISM) and Grants-in-Aid for Scientific Research (Nos. 26289258, 26630346, 15H02300 and 15K14162), both funded by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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[10] K. Arora, K. Kishida, H. Inui. Micro-pillar compression of Ni-base superalloy single crystals. MRS Symp. Proc. 2013; 1516: 209–214. [11] N. L. Okamoto, D. Kashioka, M. Inomoto, H. Inui, H. Takabayashi, S. Yamaguchi. Compression deformability of Γ and ζ Fe-Zn intermetallics to mitigate detachment of brittle intermetallic coating of galvannealed steels. Scripta Mater. 2013; 69: 307– 310. [12] N. L. Okamoto, M. Inomoto, H. Adachi, H. Takebayashi, H. Inui. Micropillar compression deformation of single crystals of the intermetallic compound ζ-FeZn13. Acta Mater. 2014; 65: 229–239. [13] S. Nakatsuka, K. Kishida, H. Inui. Micropillar compression of MoSi2 single crystals. MRS Online Proc. Lib. 2015; 1760: mrsf14-1760-yy05-09. [14] Z. M. T. Chen, N. L. Okamoto, M. Demura, H. Inui. Micropillar compression deformation of single crystals of Co3(Al,W) with the L12 structure. Scripta Mater. 2016; 121: 28–31. [15] N. L. Okamoto, S. Fujimoto, Y. Kambara, M. Kawamura, Z. M. T. Chen, H. Matsunoshita, K. Tanaka, H. Inui, E. P. George. Size effect, critical resolved shear stress, stacking fault energy, and solid solution strengthening in the CrFeMnCoNi high-entropy alloy. Sci. Rep. 2016; 6: 35863. [16] J. R. Greer, W. D. Nix. Nanoscale gold pillars strengthened through dislocation starvation. Phys. Rev. B 2006; 73: 245410–245416. [17] W. D. Nix, S. W. Lee. Micro-pillar plasticity controlled by dislocation nucleation 19
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at surfaces. Philos. Mag. 2011; 91: 1084–1096. [18] T. A. Parthasarathy, S. I. Rao, D. M. Dimiduk, M. D. Uchic, D. R. Trinkle. Contribution to size effect of yield strength from the stochastics of dislocation source lengths in finite samples. Scripta Mater. 2007; 56: 313–316. [19] S. I. Rao, D. M. Dimiduk, T. A. Parthasarathy, M. D. Uchic, M. Tang, C. Woodward. Athermal mechanisms of size-dependent crystal flow gleaned from three-dimensional discrete dislocation simulations. Acta Mater. 2008; 56: 3245– 3259. [20] S. W. Lee, W. D. Nix. Size dependence of the yield strength of fcc and bcc metallic micropillars with diameters of a few micrometers. Philos. Mag. 2012; 92: 1238– 1260. [21] M. D. Uchic, D. M. Dimiduk. A methodology to investigate size scale effects in crystalline plasticity using uniaxial compression testing. Mater. Sci. Eng. A 2005; 400–401: 268–278. [22] P. A. Shade, M. D. Uchic, D. M. Dimiduk, G. B. Viswanathan, R. Wheeler, H. L. Fraser. Size-affected single-slip behavior of René N5 microcrystals. Sci. Eng. A 2012; 535: 53–61. [23] R. C. Reed. The superalloys Fundamentals and Applications. Cambridge University Press, Cambridge, 2006. [24] R. A. Ricks, A. J. Porter, R. C. Ecob. The growth of γ′ precipitates in nickel-base superalloys. Acta Metall. 1983; 31: 43–53. [25] T. M. Pollock, A. S. Argon. Directional coarsening in nickel-base single crystals with high volume fractions of coherent precipitates. Acta Metal. Mater. 1994; 42: 1859–1874. 20
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[26] T. Ichitsubo, D. Koumoto, M. Hirano, K. Tanaka, M. Osawa, T. Yokokawa, H. Harada. Rafting mechanism for Ni-base superalloy under external stress: Elastic or elastic-plastic phenomena? Acta Mater. 2003; 51: 4033–4044. [27] T. Mura. Micromechanics of Defects in Solids. 2nd revised ed. Kluwer Academic Publishers, Dordrecht, 1987. [28] R. Völkl, U. Glatzel, M. Feller-Kniepmeier. Measurement of the unconstrained misfit in the nickel-base superalloy CMSX-4 with CBED. Scripta Mater. 1998; 38: 893–900. [29] A. Hazotte, A. Racine, S. Denis. Internal Mismatch Stresses in Nickel-Based Superalloys: A Finite Element Approach. J. de Phys. IV Colloq. 1996; 6: C1-119128. [30] S. Gao, M. Fivel, A. Ma, A. Hartmaier. Influence of misfit stresses on dislocation glide in single crystal superalloys: A three-dimensional discrete dislocation dynamics study. J. Mech. Phys. Sol. 2015; 76: 276–290. [31] W. W. Milligan, S.D. Antolovich. Yielding and Deformation Behavior of the Single Crystal Superalloy PWA 1480. Metall. Trans. A 1987; 85-95. [32] A. Sengupta, S.K. Putatunda, L. Bartosiewicz, J. Hangas, P.J. Nailos, M. Peputapeck, F.E. Alberts. Tensile Behavior of a New Single Crystal Nickel-Based Superalloy (CMSX-4) at Room and Elevated Temperatures. J. Mater. Eng. Perf. 1994; 3: 664-672.
Figure captions
21
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Fig. 1. Schematic illustrations of the internal misfit stresses for (a) negative and (b) positive lattice misfits. Fig. 2. Backscattered electron images of (a) CMSX-4 and (b) Alloy-A. Fig. 3. Schematic illustrations of the models used for the internal stress calculations. (a,b) Two types of cubic unit blocks with different geometrical configurations of the γ′ phase in the γ matrix, and (c) structure of a model comprising n × n × 1 cubic unit blocks surrounded by a buffer layer. Fig. 4. Stress-strain curves of the [001]-oriented (a) CMSX-4 and (b) alloy-A micropillars. Fig. 5. Side length L dependences of the yield stress (0.2% flow stresses) of the [001]oriented (a) CMSX-4 and (b) Alloy-A micropillars. The dashed lines indicate the yield stresses determined for the bulk specimens tested in compression. Fig. 6. Deformation microstructures of the [001]-oriented CMSX-4 micropillars for (a) L = 0.7 μm and 1.2% plastic strain, (b) L = 0.8 μm and 0.25% plastic strain, (c) L = 1.5 μm and 1.1% plastic strain, and (d) L = 4.0 μm and 2.7% plastic strain. Fig. 7. Deformation microstructures of the [001]-oriented Alloy-A micropillars for (a) L = 0.5 μm and 1.5% plastic strain, (b) L = 0.8 μm and 0.8% plastic strain, (c) L = 2.0 μm and 1.5% plastic strain, and (d) L = 4.0 μm and 3.9% plastic strain. Fig. 8. Contour maps of the internal shear stress
int values resolved on the (111)[011]
slip system, induced by lattice misfits of (a,b) -0.002 and (c,d) +0.004. (a) and (c) were
22
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obtained using the bulk model, while (b) and (d) were obtained using model 1 with configuration A of the cubic unit blocks (see Fig. 3). Fig. 9. Schematic illustrations of the distributions of the internal shear stress
int resolved
on the (111)[011] slip system. Blue circles indicate the positions exhibiting the highest
int values. Red, white and green colors qualitatively indicate negative, zero and positive values of
int , respectively.
Fig. 10. Histograms of the γ- and γ′-phase internal shear stress
int values for (a–d)
negative (δ = -0.002) and (e–h) positive (δ = +0.004) lattice misfits. (a), (b), (e), and (f) were obtained using the bulk model, while (c), (d), (g), and (h) were obtained using model 1 with configuration A of the cubic unit blocks (see Fig. 3). Fig. 11. Variations of the average internal shear stress
int values in the various γ
channels and γ′ precipitates with respect to the number (n) of unit blocks in the x and y directions, for (a,b) negative (δ = -0.002) and (c,d) positive (δ = +0.004) lattice misfits. (a) and (c) were obtained using configuration A of the cubic unit blocks, while (b) and (d) were obtained using configuration B.
23
Kabir Arora, Kyosuke Kishida, Katsushi Tanaka, Haruyuki Inui PII:
S1359-6454(17)30614-6
DOI:
10.1016/j.actamat.2017.07.044
Reference:
AM 13943
To appear in:
Acta Materialia
Received Date:
04 December 2016
Revised Date:
15 July 2017
Accepted Date:
21 July 2017
Please cite this article as: Kabir Arora, Kyosuke Kishida, Katsushi Tanaka, Haruyuki Inui, Effects of lattice misfit on plastic deformation behavior of single-crystalline micropillars of Ni-based superalloys, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.07.044
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Effects of lattice misfit on plastic deformation behavior of single-crystalline micropillars of Ni-based superalloys
Kabir Arora1, Kyosuke Kishida1,2*, Katsushi Tanaka3, and Haruyuki Inui1,2 1
Department of Materials Science and Engineering, Kyoto University, Sakyo-ku, Kyoto
606-8501, Japan 2
Center for Elements Strategy Initiative for Structural Materials (ESISM), Kyoto
University, Sakyo-ku, Kyoto 606-8501, Japan 3
Department of Mechanical Engineering, Kobe University, 1-1 Rokkoudai-cho, Nada-
ku, Kobe 657-8501, Japan
Abstract The deformation behaviors of rectangular-parallelepiped micropillars fabricated from [001]-oriented Ni-based superalloys with negative and positive lattice misfits, respectively, were investigated with respect to the specimen size. The yield stresses of the larger specimens (side length L > ~2 μm) of both considered alloy types were found to be nearly independent of the size, while the smaller specimens (L < ~2 μm) exhibited “smaller is stronger” and “smaller is (slightly) weaker” trends for negative and positive lattice misfits, respectively. The internal stresses calculated based on the micro-elasticity theory revealed that the internal shear stress fields of the micropillars varied with their sizes. The size dependence of the calculated internal shear stress distribution agreed fairly well with the experimentally determined size dependence of the yield stress, indicating that the distribution of the internal misfit stress significantly contributed to the sizedependent yield stress of the smaller micropillars. 1
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Keywords: Ni-based superalloys; Yield strength; Size effects; Lattice misfit; Internal stress *Corresponding author. Tel./Fax: +81-75-753-5461 E-mail: [email protected]
1. Introduction The microcompression test method introduced by Uchic et al. has been recognized as a new technique to investigate the deformation behavior of various materials at sizescales of tens of micrometers or less [1, 2]. Extensive studies have been conducted on the deformation behaviors of single crystals of various materials such as pure metals, intermetallic compounds, and semiconductors, and most of them have been found to exhibit “smaller is stronger” size-dependent deformation [1–15]. Various possible mechanisms have been used to explain the size-dependent strengthening of single crystals of metallic materials, such as dislocation starvation (surface nucleation) [16, 17] and truncation of the single-arm dislocation sources (SASs), in which the length of the SASs is considered to be a controlling factor [18–20]. A similar size-dependent strengthening has also been observed in single crystals of Ni-based superalloys, which have a two-phase microstructure consisting of cuboidal precipitates of Ni3Al (γ’) coherently embedded in an FCC nickel (γ) matrix [10, 21, 22]. However, the underlying mechanism of the size-dependent strengthening is yet to be clarified, which is because the mechanical properties of Ni-based superalloys are affected by several factors, including the (i) flow stress of the γ matrix, (ii) size, morphology, and volume fraction of the γ′ precipitates, (iii) APB energy, (iv) misfit stress, and (v) pre2
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existing dislocations [22, 23]. The morphology of the γ′ precipitates is known to be mainly controlled by the lattice misfit δ between the two constituent phases, given by
2(a ' a ) /(a ' a ) , where aγ′ and aγ are the lattice parameters of the γ′ precipitate and γ matrix, respectively [23, 24]. An important determinant of the lattice misfit is the internal misfit stress, which is the primary reason for the development of the so-called raft structure comprising γ and γ′ lamellae alternately stacked in the <001> loading direction, as observed during the tensile creep testing of Ni-based superalloys with a negative lattice misfit [23, 25, 26].
Fig. 1 shows schematic illustrations of the
distributions of the internal misfit stresses in Ni-based superalloys with negative and positive lattice misfits, respectively. Ni-based superalloys with a negative lattice misfit typically have biaxial compressive stress in the 𝛾 channels, while triaxial tensile stress act on the 𝛾' precipitates [23]. The reverse is the case in such superalloys with a positive lattice misfit; i.e., biaxial tensile and triaxial compressive stress act on the 𝛾 channels and the 𝛾' precipitates, respectively. On a scale comparable to the size of the γ′ precipitates (≤ 1 μm), the effects of the internal misfit stress on the deformation behavior may be significant. Previous investigations of the deformation behaviors of two-phase Ni-based superalloys by microcompression using cylindrical micropillar specimens of diameters > 2 µm have revealed a size dependence of the flow stress at 1%–3% plastic strain, similar to that exhibited by single-phase materials [21, 22]. The present authors also recently observed a similar size-dependent flow stress at 1% plastic strain during micropillar compression testing of the [001]-oriented superalloy CMSX-4 [10, 21, 22]. However, these previous works only considered the effects of the specimen size without explicitly
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examining the effects of the lattice misfit on the deformation behavior of small specimens with sizes comparable to that of the γ′ precipitates. In the present study, we focused on the compression behavior of Ni-based superalloy micropillars measuring ≤ 8 µm and the effects of lattice misfit on their deformation characteristics at room temperature. We specifically considered the commercial alloy CMSX-4 with a negative misfit [23] and a newly developed alloy, hereafter referred to as alloy-A, with a positive misfit. The two alloys were used to compare the effects of the misfit type on the deformation characteristics. The effects of the lattice misfit δ on the plastic deformation behavior of single-crystalline micropillars of the Ni-based superalloys were examined with respect to the internal elastic stress fields calculated based on the micro-elasticity theory [27].
2. Experimental procedures Two materials with lattice misfits of opposite signs, namely, CMSX-4 (δ = ˗0.001–˗0.023 at room temperature [28]) and alloy-A (δ = ~+0.004 at room temperature) were used in this study. The nominal composition of alloy-A was determined by decreasing Mo, W, Al, Ti, Hf, Re and increasing Ta from that of CMSX-4 [23] so as to obtain a cuboidal microstructure with a positive lattice misfit. A single crystal of alloy-A was grown by the Bridgman technique at a growth rate of 5 mm/h. The as-grown single crystal was homogenized at 1300°C for 1 h and at 1100°C for 4 h, and finally aged at 870°C for 4 h. The typically observed {001}-section microstructures of the as-received CMSX-4 and heat-treated alloy-A single crystals are shown in Fig. 2. The two materials had similar cuboidal microstructures, with the average edge length of the cuboidal γ′
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precipitates being 400–500 nm. The volume fractions of the γ′ phase in CMSX-4 and alloy-A were estimated to be about 67% and 70%, respectively. Rectangular-parallelepiped bulk specimens with their surfaces parallel to the {001} plane (hereafter (001) surfaces) were cut from the single crystals by electrodischarge machining. The (001) surfaces of the bulk specimens were mechanically polished and then electropolished in a solution of 60% perchloric acid, n-butyl alcohol, and methanol (1:6:12 volume ratio) at 18 V and -55°C. [001]-oriented micropillars with a square cross section (side length L values of 0.5–8 μm) were fabricated from the electropolished (001) surface by the focused ion beam (FIB) technique using a JEOL JIB4000 FIB system with a Ga ion source operated at an acceleration voltage of 30 kV. Two orthogonal side surfaces of the micropillars were set parallel to the (100) and (010) planes, respectively, to align all the edges of the micropillar with those of the cuboidal precipitates. The height-to-length (L) ratio of the micropillars was set to be 2 - 4. Compression tests were conducted on the micropillars using a Shimadzu DUH211 microhardness testing machine equipped with a flat diamond tip of diameter 20 μm. All the micropillars were tested at room temperature in a load control mode (corresponding nominal strain rates of 10-3–10-4 s-1). The microstructures of the micropillars before and after compression testing were examined by scanning electron microscopy (SEM) using a JEOL JSM-7001FA electron microscope equipped with a field emission gun.
3. Modeling
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If a material that contains coherent inclusions is assumed to be an elastically homogeneous medium, an elastic displacement in the material can be solved in the Fourier space [27]. The elastic displacement uk(r) is given by the following equation:
uk (r ) where
1 iGik g g j ijpq *pq r g eigr d 3g 3 2
(1),
Gik g , ijpq , and *pq r g denote the Green’s function, elastic constants, and
Fourier transformation of the eigenstrain (stress-free strain), respectively. In the present case of the Ni-based superalloys with cuboidal microstructures, the eigenstrain is simply given by
*pq r ( r ' precipitates and p = q) = 0 (otherwise)
(2).
To simulate the surface relaxation effect, we introduced a (virtual) buffer layer in directions perpendicular to the side surfaces of a micropillar. The buffer layer was assumed to have the same elastic constants as the micropillar. Plastic and volumetric deformations of the buffer layer was allowed to eliminate the elastic constraint of the buffer layer on the micropillar. With these assumptions, the eigenstrain of the buffer layer could be determined using the following iterative calculation: *pq r
i 1
r , *pq r elastic pq i
i
r buffer
layer
(3),
where elastic r is the elastic strain determined at the i-th iteration, and is a dumping i factor for stabilizing the iterative calculation. Two types of cubic unit blocks (configurations A and B, as illustrated in Fig. 3) were considered in this study, each of which contained the same volume fraction of the γ′ phase (0.67) embedded in the γ matrix. Each cubic unit block, with edges set parallel to x 6
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= [100], y = [010], and z = [001], contained 96 × 96 × 96 grid points. For each configuration, five models comprising n × n × 1 cubic unit blocks (n = 1, 2, 3, 4, and 8) were used for the calculation, as illustrated in Fig. 3(c). The unit blocks were surrounded by a buffer layer of thickness 16n grid points in the x and y directions. The five models are hereafter referred to as model n (n = 1, 2, 3, 4, and 8). Because the average edge length of the cuboidal precipitates for both considered materials is 400–500 nm, the edge length of a cubic unit block was about 460–570 nm. Thus, the five models approximately represented micropillars with side length L values of 0.5, 1.0, 1.5, 2.0, and 4.0 μm, respectively. In addition, the internal stress distributions of the bulk specimens (infinite L) were simulated using an additional model with configuration A without the surrounding buffer layer. The model was designated as the bulk model. Lattice misfit values of -0.002 (negative lattice misfit) and +0.004 (positive lattice misfit) were used to calculate the stress distributions corresponding to the CMSX-4 and alloy-A micropillars, respectively. In the calculations, all models were assumed to be elastically homogeneous with average elastic constants C11 = 205 GPa, C12 = 121 GPa, and C44 = 118 GPa for both the γ and γ′ phases, the values of which were estimated from those reported for the γ and γ′ phases [29]. In order to clarify the influences of the lattice misfit only, it is sufficient to use the average elastic constants for the calculations of the internal stress distribution. No external applied stress was considered in the calculations.
4. Results 4.1 Microcompression experiments
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Representative stress-strain curves of the [001]-oriented single-crystalline CMSX-4 (negative lattice misfit) and alloy-A (positive lattice misfit) micropillars are shown in Fig. 4(a) and 4(b), respectively. Stress-strain curves of bulk-size specimens of the [001]-oriented single crystals tested in compression at a constant strain rate of 10-4 /s are also indicated in Fig. 4 for comparison. The stress-strain curves of the relatively large CMSX-4 micropillars (L ≥ 2.0 μm) (Fig. 3(a)) are characterized by a transition region between elastic and plastic flows with continuously decreasing work hardening rate, followed by a region with a large strain burst or low work hardening. The low work hardening behavior of the larger micropillars is similar to that observed in the stress-strain curve of the bulk-size specimen of CMSX-4. In the case of the smaller CMSX-4 micropillars (L < 2.0 μm), the stress-strain curves generally exhibit a staircase-like pattern, with alternations between (almost) elastic loading and microstrain bursts. This results in a higher flow stress at a plastic strain above 1%. These observed characteristics of the CMSX-4 micropillars agree well with previous reports regarding other Ni-based superalloys, namely, UM-F19 and René N5 [21, 22]. The characteristics of the stressstrain curves of the alloy-A micropillars with a positive lattice misfit are generally similar to those of the CMSX-4 micropillars, except for the relatively high work hardening behavior observed for the larger micropillars (L ≥ 2.0 μm) as well as the bulk-size specimen of alloy-A. Fig. 5 shows the side length L dependence of the yield stress (flow stress at 0.2% plastic strain). For both alloys, the yield stress of the larger micropillars (L > ~2 μm) is almost independent of the specimen size. However, opposite trends are observed for the yield stress of the smaller micropillars (L ≤ ~2 μm), namely, “smaller is stronger” for CMSX-4 (Fig. 5(a)) and “smaller is (slightly) weaker” for alloy-A (Fig. 5(b)). A nearly 8
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size-independent 0.2% flow stress has been similarly observed for the compression of large René N5 single-crystalline micropillars oriented for single-slip [22]. Because the microstructural characteristics of the two materials, such as the volume fractions and morphologies of their γ′ precipitates, are quite similar, the difference between the yield stress of their smaller micropillars (L < ~2 μm) is believed to be due to the differing internal stress distributions induced by the opposite signs of their lattice misfits.
4.2 SEM observations Figs. 6 and 7 show the SEM secondary electron images of some CMSX-4 and alloy-A micropillars after compression, respectively. Multiple slip traces parallel to more than two {111} planes are observed to have developed rather homogeneously during a relatively early stage of the plastic deformation (about 0.25% compressive strain (Fig. 6(b)). Most of the slip traces extend continuously from one edge to another, indicating that, not only the γ channels, but also the γ′ precipitates were sheared during a very early stage of the plastic deformation. Similar trends are also observed for the micropillars of alloy-A (Fig. 7). No apparent differences are observed between the characteristics of the slip traces of the two alloys.
4.3. Internal stress calculation 4.3.1 Bulk model Typical results of the internal stress calculations for the bulk models are presented in Fig. 8, which shows the two-dimensional contour maps of the resolved shear stress of the (111)[011] slip system, generated by the lattice misfits of -0.002 (CMSX-4) and +0.004 (alloy-A). The resolved shear stress generated by a lattice misfit is hereafter 9
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designated the internal shear stress
int . The (111)[011] slip system is one of the eight
operative slip systems with the highest Schmid factor of 0.408 under uniaxial compression in the [001] direction. Three contour plots with mutually orthogonal cross sections passing through the center of the cubic unit block of model 1 are presented for each model. Figs. 8(a)–8(c) and 8(g)–8(i) show the contour maps of the values of
int
calculated using the bulk model for the negative and positive lattice misfits, respectively. In the contour maps, the positive maximum (+600 MPa), zero, and negative maximum (600 MPa) values of
int are indicated in green (8-bit RGB code (0, 255, 0)), white (255,
255, 255), and red (255, 0, 0), respectively. The sign of the resolved shear stress is assigned such that the application of a positive shear stress on the slip system yields a compressive strain in the [001] direction. Consequently, the external applied stress in the [001] direction also yields a positive resolved shear stress on the (111)[011] slip system (hereafter designated the external shear stress ext ). The effective (or total) shear stress
eff acting on the slip system is simply given by eff int ext
(4).
Figs. 9(a,b) schematically illustrate the distributions of
int on the (111) slip plane for
the bulk models with the negative and positive lattice misfits, respectively. In the figures, red, white and green colors qualitatively indicate negative, zero and positive values of
int , respectively and blue circles indicate the positions exhibiting the highest int values. For a negative lattice misfit (Figs. 8(a) and 9(a)), only the vertical γ channels perpendicular to the [010] axis (hereafter referred to as the Vy channel) exhibit relatively 10
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high positive values (green regions). The (111)[011] slip is thus expected to be initially activated and propagate inside the Vy channels, because
eff is able to reach the critical
value for slip activation under a relatively low external compressive load in the [001] direction, according to eq. (4). In contrast, for a positive lattice misfit (Figs. 8(c) and 9(b)), the regions favorable for the (111)[011] slip activation (green regions) under a [001] compressive load are initially limited in the horizontal γ channels (H channels) perpendicular to the [001] axis. The distributions of the shear stress can be more quantitatively appreciated from Fig. 10, which shows histograms of
int for the γ and γ′ phases as obtained from the entire
grid points inside the model micropillars. In addition to the total distributions shown as bar charts, the
int distributions inside the various γ channels (Vx (vertical channel
perpendicular to the [100] axis), Vy, H, and their intersections (designated mixed channels)) are also indicated by the line plots in Figs. 10(a) and 10(e). The
int data from
the mixed channels are excluded from the plots for the Vx, Vy, and H channels. As can be clearly seen from Figs. 10(a) and 10(e), the three geometrically different major γ channels, namely, the Vx, Vy, and H channels with relatively large volumes, have different
int
distribution profiles. In the case of the negative lattice misfit, the average values of
int
in the major Vx, Vy and H channels are zero, positive, and negative, respectively, whereas they are zero, negative, and positive for the positive lattice misfit, respectively. It is important to note that the average value of
int for the (111)[011] slip inside the γ′
precipitate of the bulk model is zero with a relatively small standard deviation irrespective of the sign of the lattice misfit. This is quite reasonable considering that the internal stress 11
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field inside the γ′ precipitates is virtually isotropic for an ideal bulk model. These results for the present bulk models are consistent with those in a recent report [30].
4.3.2 Effects of micropillar size and in-plane configuration of γ′ precipitates on internal shear stress The determined distributions of
int for model 1 (n = 1) with surface relaxation
are significantly different from those for the bulk models. The trends of the contour maps and distributions of
int for model 1 for negative (Figs. 8(b), 9(c), 10(c), and 10(d)) and
positive (Figs. 8(d), 9(d), 10(g), and 10(h)) lattice misfits are clearly opposite. Specifically, the values of
int tend to be negative (red region in Figs. 8(b) and 9(c)) and
positive (green region in Figs. 8(d) and 9(d)) for negative and positive lattice misfits, respectively. Similar trends were observed for all the micropillar models n (n = 1, 2, 3, 4, and 8). The average values of
int for the various γ channels and γ’ precipitates for the
two in-plane configurations of the γ′ precipitates (configurations A and B) are plotted as a function of the number of unit blocks (n) in the x and y directions in Fig. 11, in which the horizontal axis is plotted on a logarithmic scale. The error bars in Fig. 11 indicate the standard deviations. In the case of the negative lattice misfit (Figs. 11(a) and 11(b)), only the Vy channels has a positive average value of
int ; the other γ channels and the γ′
precipitates have negative or nearly zero average values of
int for all the models (n = 1,
2, 3, 4, and 8 approximately represented micropillars with side length L of 0.5, 1, 1.5, 2, and 4.0 μm, respectively) irrespective of the in-plane configuration. This suggests that the activation and propagation of [011] dislocations on (111) are suppressed in most of the γ 12
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channels (except for the Vx channels) as well as the γ′ precipitates by the internal stress
int . The situation is completely opposite for the positive misfit. In this case, most of the γ channels and γ′ precipitates, with the exception of the Vy channels and the mixed channels between the Vx and Vy channels, have positive
int values. This indicates that,
for a positive lattice misfit, slip activation and propagation in most of the γ channels are strongly assisted by the internal stress
int . The varying int distributions of the model
1 micropillars are thus considered to significantly affect their yield behavior. With regard to the size dependence, the values of
int for all the γ channels and
γ′ precipitates vary with the micropillar size exhibiting different size dependences depending on the channel. Among the major γ channels, i.e., the Vx, Vy, and H channels, the size dependences of
int are greater for the latter two. The int values for the Vy, and
H channels gradually decrease and increase with decreasing n for the negative and positive lattice misfits, respectively. In addition, the size dependence of
int for the H
channels appears stronger for the in-plane configuration A, while that in the Vy channels is slightly stronger for the in-plane configuration B. These results indicate that the surface relaxation effects are greater in the H channels when the γ channels are located on the side surfaces of the smaller micropillars. The
int values for the most of the mixed γ
channels, with the exception of the mixed channel between the Vx and Vy, channels, exhibit greater and opposite size dependences compared to those for the major γ channels. However, the internal stresses
int for the mixed γ channels are expected to give minor
contributions to the macroscopic yielding because of relatively small volume fractions 13
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and higher
int of the mixed γ channels compared to those of the major γ channels with
the lowest average
int . The internal stress int for the γ′ precipitates does not exhibit
apparent size dependence in the size range of n = 1 ~ 8.
5. Discussion In the present study, we experimentally determined that the yield stress (0.2% flow stress) in smaller micropillars (L < ~2 μm) exhibited different trends depending on the sign of the lattice misfit. Moreover, our calculation results clearly indicated that the effects of surface relaxation on the internal stress distributions were significant for micropillars. Here, we discuss the possible influences of the internal stress distributions on macroscopic yielding of micropillar specimens. Figs. 9(c) and 9(d) schematically illustrate the distributions of
int on the (111)
slip plane for model 1 (configuration A) with the negative and positive lattice misfits, respectively. In the case of the negative lattice misfit (Fig. 9(c)), the positions with the highest
int (marked with a blue circle) was found to be located inside the Vy channel
near the edge of the γ’ precipitate. Thus, it is reasonable to assume that dislocations inside the Vy channel near the position exhibiting the highest
int are initially activated when
eff exceeds the critical value for the (111)[011] slip activation. In addition, considering the fact that γ’ precipitates were sheared during a very early stage of the plastic deformation of the micropillars of CMSX-4 (Fig. 6), dislocations are required to glide through the entire slip plane including the H channels with the lowest average well as the γ’ precipitates with a negative average 14
int as
int at the 0.2 % flow stress level. To
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achieve this, the effective stresses eff for dislocation activation in both the H channels and the γ’ precipitates needs to exceed certain critical values under external compressive loading, which indicates that a dominant controlling factor for macroscopic yielding of micropillars with the negative lattice misfit is considered to be the internal stress either the H channels or the γ’ precipitates. Although the critical values of
int for
eff required
for dislocation activation in the γ channels and the γ’ precipitates are unknown, it is reasonable to assume that these two
eff values does not differ much as a first
approximation, considering the fact that dislocation activation in both phases at early stages of plastic deformation is generally observed in bulk-size single crystals of Ni-based superalloys tested in tension at room temperature [31-32]. Under this assumption, the H channels with the lowest average
int are considered to play a key role for macroscopic
yielding. As seen in Figs. 11(a) and 11(b), the
int values for the H channels gradually
decrease with decreasing n for the negative lattice misfit. According to eq. (4), a decrease in internal stress
int should increase the minimum external stress ext to make eff
exceed the critical values for macroscopic yielding with the decrease in the specimen size. This is consistent qualitatively with the experimentally observed size dependence of the yield stress. Thus, the assumption for
eff is considered to be reasonable. If the similar
argument is applied for the cases of the positive lattice misfit, the
int value of the Vy
channels is expected to be the controlling factor for macroscopic yielding (Fig. 9(d)). Because the
int values for the Vy channels slightly increase with decreasing n for the
positive lattice misfit (Fig. 11(c) and 11(d)), the minimum external stress 15
ext is
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considered to decrease with the specimen size, which is also consistent fairly well with the experimentally observed tendency of the size dependent yield stress. Thus, the variations of the
int distributions in the major γ channels with the lowest average int
is considered to predominantly affect the size dependence of the yield stress of the Nibased superalloy micropillars. However, the above discussions may only be valid for the earliest stage of the plastic deformation. Indeed, a previous study on the compression of René N5 singlecrystalline micropillars oriented for single-slip activation revealed that the power law factor of the size-dependent flow stress increased with increasing plastic strain [22]. A similar observation has also been made for CMSX-4 micropillars [10]. With the progression of plastic deformation, the distribution of the internal stress
int is expected
to vary gradually because the lattice misfit can be partly relaxed by the interaction between activated dislocations and interfaces between the γ matrix and γ′ precipitates. Because the distribution of
int affects various dislocation-related factors such as the
activation of dislocation sources in the γ matrix, the dislocation velocity, the occurrence of cross-slip and shearing of the γ′ precipitates, the quantitative estimation of the effects of
int on the observed flow behavior is very complex. Further study involving, for
example, simulation of the dislocation dynamics and modeling based on dislocation kinetics is required to obtain a full understanding of the observed variations of the sizedependent flow stress with respect to the plastic strain.
6. Conclusions
16
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We conducted systematic micropillar compression tests on [001]-oriented Ni-based superalloys, namely, CMSX-4 (with negative misfit) and alloy-A (with positive misfit), and used the micro-elasticity theory to calculate the internal stresses with respect to the specimen size. Following is a summary of the main findings: 1.
The yield stress of the [001]-oriented CMSX-4 and alloy-A micropillars are nearly independent on the specimen size, for side length L values of 2–8 μm. In contrast, smaller (L ≤ ~2 μm) CMSX-4 (negative lattice misfit) and alloy-A (positive lattice misfit) micropillars exhibit “smaller is stronger” and “smaller is (slightly) weaker” trends, respectively.
2.
The calculated internal stress fields of the Ni-based superalloy micropillars vary with the specimen size because of the surface relaxation effects on the lattice misfit strain. The overall distributions of the internal shear stress
int resolved for a slip system
favorable to uniaxial compression shift toward the opposing and assistive sides for slip activation in micropillars with negative and positive lattice misfits, respectively. The numerically determined trends of the
int distributions are consistent with
experimental results, which indicate an increase and decrease in the yield stress with decreasing micropillar size for negative and positive lattice misfits, respectively. These results clearly indicate that the variation of the
int distributions is a
determinant of the size-dependent yield behavior of Ni-based superalloy micropillars under compression.
Acknowledgements 17
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This work was supported by the Elements Strategy Initiative for Structural Materials (ESISM) and Grants-in-Aid for Scientific Research (Nos. 26289258, 26630346, 15H02300 and 15K14162), both funded by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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Figure captions
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Fig. 1. Schematic illustrations of the internal misfit stresses for (a) negative and (b) positive lattice misfits. Fig. 2. Backscattered electron images of (a) CMSX-4 and (b) Alloy-A. Fig. 3. Schematic illustrations of the models used for the internal stress calculations. (a,b) Two types of cubic unit blocks with different geometrical configurations of the γ′ phase in the γ matrix, and (c) structure of a model comprising n × n × 1 cubic unit blocks surrounded by a buffer layer. Fig. 4. Stress-strain curves of the [001]-oriented (a) CMSX-4 and (b) alloy-A micropillars. Fig. 5. Side length L dependences of the yield stress (0.2% flow stresses) of the [001]oriented (a) CMSX-4 and (b) Alloy-A micropillars. The dashed lines indicate the yield stresses determined for the bulk specimens tested in compression. Fig. 6. Deformation microstructures of the [001]-oriented CMSX-4 micropillars for (a) L = 0.7 μm and 1.2% plastic strain, (b) L = 0.8 μm and 0.25% plastic strain, (c) L = 1.5 μm and 1.1% plastic strain, and (d) L = 4.0 μm and 2.7% plastic strain. Fig. 7. Deformation microstructures of the [001]-oriented Alloy-A micropillars for (a) L = 0.5 μm and 1.5% plastic strain, (b) L = 0.8 μm and 0.8% plastic strain, (c) L = 2.0 μm and 1.5% plastic strain, and (d) L = 4.0 μm and 3.9% plastic strain. Fig. 8. Contour maps of the internal shear stress
int values resolved on the (111)[011]
slip system, induced by lattice misfits of (a,b) -0.002 and (c,d) +0.004. (a) and (c) were
22
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obtained using the bulk model, while (b) and (d) were obtained using model 1 with configuration A of the cubic unit blocks (see Fig. 3). Fig. 9. Schematic illustrations of the distributions of the internal shear stress
int resolved
on the (111)[011] slip system. Blue circles indicate the positions exhibiting the highest
int values. Red, white and green colors qualitatively indicate negative, zero and positive values of
int , respectively.
Fig. 10. Histograms of the γ- and γ′-phase internal shear stress
int values for (a–d)
negative (δ = -0.002) and (e–h) positive (δ = +0.004) lattice misfits. (a), (b), (e), and (f) were obtained using the bulk model, while (c), (d), (g), and (h) were obtained using model 1 with configuration A of the cubic unit blocks (see Fig. 3). Fig. 11. Variations of the average internal shear stress
int values in the various γ
channels and γ′ precipitates with respect to the number (n) of unit blocks in the x and y directions, for (a,b) negative (δ = -0.002) and (c,d) positive (δ = +0.004) lattice misfits. (a) and (c) were obtained using configuration A of the cubic unit blocks, while (b) and (d) were obtained using configuration B.
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