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Indentation-induced localized deformation and elastic strain partitioning in composites at submicron length scale
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Acta Materialia 58 (2010) 6784–6789 www.elsevier.com/locate/actamat
Indentation-induced localized deformation and elastic strain partitioning in composites at submicron length scale R.I. Barabash a,b,⇑, H. Bei a, Y.F. Gao b,c, G.E. Ice a a
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA b Materials Science and Engineering Department, University of Tennessee, Knoxville, TN 37996, USA c Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received 3 August 2010; received in revised form 6 September 2010; accepted 7 September 2010 Available online 1 October 2010
Abstract Three-dimensional spatially resolved strains were mapped in a model NiAl/Mo composite after nanoindentation. The depth-dependent strain distributed in the two phases and partitioned across the composite interfaces is directly measured at submicron length scale using X-ray microdiffraction and compared with a detailed micromechanical stress analysis. It is shown that indentation-induced deformation in the composite material is distinct from deformation expected in a single-phase material. This difference arises in part from residual thermal strains in both phases of the composite in the as-grown state. Interplay between residual thermal strains and external mechanical strain results in a complex distribution of dilatational strain in the Mo fibers and NiAl matrix and is distinct in different locations within the indented area. Reversal of the strain sign (e.g., alternating tensile/compressive/tensile strain distribution) is observed in the NiAl matrix. Bending of the Mo fibers during indentation creates relatively large 1.5° misorientations between the different fibers and NiAl matrix. Compressive strain along the h0 0 1i direction reached 0.017 in the Mo fibers and 0.007 in the NiAl matrix. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Composites; Deformation; Micromechanics; X-ray synchrotron radiation; Micro-/nanoindentation
1. Introduction The behavior of composite materials under external load depends on deformation micromechanisms taking place in both the fibers and the matrix and, in particular, at the matrix/fiber interface [1–9]. The matrix/fiber interface plays a particularly crucial role in the mechanical properties of composite materials and in strain partitioning between the two phases. Despite the key role of interfaces in composite behavior, the strength of interfaces and strain partitioning mechanisms through an interface are largely unknown, particularly at the micron scale. Indeed, the complexity and inhomogeneous nature of deformation ⇑ Corresponding author at: Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. Tel.: +1 865 2417230; fax: +1 865 5747659. E-mail address: [email protected] (R.I. Barabash).
microstructure precludes micron-resolution studies over macroscopic length scales; however, nano- and microindentation of semi-infinite samples can be used to confine deformation fields to volumes in nano-/micrometer scale. Localization of the plastic deformation field under an indent is of profound importance [10,11]. Unlike randomly selected “representative volumes” from macroscopic deformed samples, all aspects of plastic deformation, including stochastic events (e.g., dislocation patterning, dislocation cells [12], different kinds of boundaries [13]) are confined within a volume that can be both probed by three-dimensional (3D) X-ray microscopy and simulated computationally. Therefore, measurements/calculations on the confined volume provide for a direct quantitative connection between 3D X-ray microscopy measurements and computer simulations and modeling. The unique capability of 3D polychromatic microdiffraction now makes it possible to study elastic strain at
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.09.004
R.I. Barabash et al. / Acta Materialia 58 (2010) 6784–6789
low forces and microstructural evolution at high force for direct comparison with simulations and modeling. Of course, the strong deformation gradients inherent in indentations present difficult experimental and theoretical challenges, but also provide stringent tests of controversial strain gradient effects and issues with respect to fundamental length scales in materials. Exploring these plastic/elastic phenomena and interfacial effects in the model NiAl–Mo composite under indentation is the focus of this study. 2. Experimental details 2.1. Specimen preparation The directionally solidified NiAl–Mo eutectic alloy in this study has a relatively simple microstructure, consisting of long uniformly spaced Mo fibers with approximately square cross-sections embedded in a NiAl matrix [14–17]. In this study, NiAl–Mo composite with the fiber sizes (edge lengths) of 550 nm is selected. Details of the directional solidification and the microstructures control have been described earlier [14], and their strain status in as-grown and uniaxially pre-strained condition has been measured previously with 3D X-ray microdiffraction [18,19]. Disk-shaped specimens, 2 mm thick, were cut from the directionally solidified NiAl–Mo eutectic perpendicular to the growth (fiber) direction and mounted in epoxy resin. After grinding with SiC paper through 1200 grit, the specimen was polished with Al2O3 slurries through 0.3 lm and was then polished with a colloidal silica suspension in a vibratory polishing machine. 2.2. Indentation Spherical indentation on the polished surface was conducted with a Nanoindenter XP equipped with sapphire tip with radius 100 lm to prescribed load P = 650 mN. A 15 3 array of the indents was put on the sample surface with spacing of 150 lm. Load–displacement curves were recorded for each indentation. A top view of the indented area with multiple indents and a close view of the indent studied with X-ray microdiffraction, as well as the load– displacement curve for this particular indent are shown in Fig. 1. The spherical indentation impression in Fig. 1b clearly shows the Mo fibers (contrast in bright), which are embedded in the NiAl matrix. 2.3. 3D polychromatic microdiffraction and differential aperture X-ray microscopy In order to test the effect of indentation impact on the twophase composite materials submicron-resolution 3D polychromatic and monochromatic X-ray microscopy was used to measure the elastic strains and dislocation densities in both alloy constituents in undeformed and post-nanoindentation states. Measurements were performed at ID-34-E at the Advanced Photon Source. Depth-dependent elastic strains
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were obtained using spatially resolved differential aperture X-ray microscopy (DAXM) [20–22]. The geometry of the polychromatic microdiffraction (PXM)/DAXM measurements is shown in Fig. 2. First, two-dimensional (2D) mapping of the indented area was performed with a polychromatic X-ray microbeam to determine the region affected by indentation. The size of the beam was 0.5 lm in diameter. Typically, strong plastic deformation results in the formation of geometrically necessary dislocations, causing strong lattice rotations in the indented area, accompanied by “streaking” and smearing of the Laue spots [22]. For 3D spatially resolved measurements, a platinum wire was used as a knife-edge to block part of the diffracted radiation (Fig. 2). The wire was translated parallel to the sample surface, and Laue patterns were collected at 400 wire positions. Changes in the patterns for small wire displacements are due to rays passing near the wire edge, hence the term “differential aperture”. Triangulation of the partially shadowed Laue pattern with wire position relative to the sample surface, using ray-tracing algorithms, determines the origin of the diffracted intensity along the incident beam path. Details of DAXM data collection can be found elsewhere [20]. Depth-resolved measurements were also performed with monochromatic radiation, which enables the depth-dependent lattice parameter and/or dilatational strain changes to be obtained. The diffraction geometry of the measurements employed in this study (Fig. 2) resolves the overlapping information from the “fiber forest” beneath the surface of the composite. At each depth, the beam probes either the matrix or a different fiber and provides unique information about the plastic/elastic status as a function of depth and distance from the indentation center. This technique is complementary to the transmission geometry technique employed in Refs. [23,24], and to the energy-variable diffraction, which provides one-dimensional cross-section (normal to the sample surface) for strain distribution across interfaces between different materials on a sub-micrometer scale [25]. 3. Results and discussion The elastic strain measurements in the NiAl/Mo eutectic composite demonstrate a new approach to the understanding of the load accommodation mechanisms in composite materials. In addition, this particular system is of interest because of the large (1%) compressive strains of the embedded fibers [17,18]. The melting point of the NiAl– Mo is 1600 °C. During solidification, thermal expansion mismatch between the Mo-alloy fibers and the NiAl matrix cause large thermal strain between two phases. Details of thermal mismatch strain and thermal response of individual phases have been reported elsewhere [9,18]. The thermal mismatch strain is 1.0%. Here, the focus is on the critical information about interfacial properties that govern the overall composite behavior. The sample surface was first mapped with depth-integrated PXM (2D microdiffraction) to determine the regions
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Fig. 1. (a) The general view of the indented area; (b) SEM image shows the indent studied with PXM and the Mo fibers (bright contrast) embedded in NiAl matrix; (c) the corresponding load–displacement curve.
Fig. 2. Sketch of the polychromatic X-ray microdiffraction of the indented NiAl/Mo sample.
of maximal deformation in the indented area and to identify locations for depth-resolved measurements. A typical indexed Laue pattern is shown in Fig. 3a. The enlarged regions around (0 0 h)-type reflection at two different locations near the indent are shown in Fig. 3b and c. In the first location, which is outside the indented area (Fig. 3b) both the NiAl and Mo (0 0 h) Laue spots are sharp. Although the beam intercepts several Mo fibers (typically up to 10
fibers), they all diffract X-rays into the same position on the CCD, indicating very high alignment of the fibers along the h0 0 1i NiAl and Mo growth direction. In the indented zone, the character of the Laue spots changes. The NiAl Laue spot is streaked, indicating the presence of lattice rotations in the indented zone. The largest observed lattice rotations in the NiAl matrix were <3°. The most dramatic changes in the indented area are observed for the Mo fibers: instead of a single Laue spot arising from different probed fibers, multiple Mo Laue spots are observed when the Xray beam probes a volume with the indent (Fig. 3c). It is known from microscopic studies [2] that, under compression of composite materials, different Mo fibers bend, and their local orientation changes significantly. Such fiber misorientation will lead to X-ray scattering into different directions for different locations on various fibers. The depth-resolved measurements of the dilatational strain along the fiber axis in both NiAl martix and Mo fibers were further performed along the vertical line passing through the center of the indented area where maximal deformation was observed. A total of 15 locations were probed along this line, with a step size of 2 lm in the central region of the indent with maximal deformation, and with a step size of 4 lm farther from the indent center. To determine the fiber axis strain, the (0 0 6) d-spacing
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Fig. 3. (a) Typical indexed Laue pattern from NiAl–Mo composite alloy; (b) enlarged region around (0 0 h) Laue spot for location 1 far from the indented area; and (c) location 2 in the center of the indented area.
was measured for both the NiAl matrix and Mo fibers. The energy of the beam was scanned with a step of 3 eV over an energy range of 13.310–18.601 keV for the (0 0 6) Mo reflection, and within a range of 19.660–19.918 keV for the (0 0 6) NiAl reflection. Energy scans were synchronized with the movement of the special differential aperture (Pt wire). At each energy value, typically 400 partially shadowed diffraction patterns were obtained at different Pt wire positions relative to the sample surface. Triangulation of differential aperture position and energy allowed the energy of the scattered X-rays to be related to the depth of the scattering location. As a result of such measurements, the depth-dependent energy distributions were obtained for each location. The details of the triangulation process can be found in Ref. [20]. Depth-dependent dilatational strains were calculated from the above depth-dependent energy distributions (Fig. 4.) The dilatational lattice strain in both the Mo fibers and the NiAl matrix changes character with location (Fig. 4). Far from the indented area (location 1) where the probed volume is sufficiently far from the indent so that the indentation-induced stress fields are negligible, the strain distribution reveals residual strain caused by the mismatch in the coefficients of thermal expansion between Mo and NiAl which is consistent with previous measurement [18]. As discussed previously [17,18], the incompatible thermal expansion coefficients of the fiber and matrix introduce thermal strain, which causes the Mo fibers to be under compression along the h0 0 1i fiber direction (z axes), and NiAl matrix to
Fig. 4. (a) Distribution of lattice strain along h0 0 1i direction in Mo fibers and (b) NiAl matrix probed at three different locations: (1) blue filled circles; (2) pink triangles; (3) green triangles. Note that the three different location are indicated in Fig. 1b and also Fig. 5a. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
be under tension. Near the surface, these residual thermal strains relax and result in a large near-surface strain gradient (blue circles in Fig. 4). Such a near-surface relaxation is similar to a pull-out test, and the resulting interface slip strength can be determined as 180 MPa from the slip
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zone size (5 lm). Completely stress-free Mo lattice parameters are measured at the tops of the exposed Mo pillars when the matrix is etched away [18]. This value is also shown in Fig. 4a. In the second probed location (pink triangles in Fig. 4), the incident X-ray beam passes through highly deformed fibers (near surface zone). A mechanically introduced compressive residual strain along the h0 0 1i direction of Mo fibers is superimposed on the original thermal compressive strain. The resulting compressive strain in the Mo fibers along their axis reaches values up to 0.013 at a depth of 15 lm. At greater depths, the beam passes out of the indentation-affected area, and lattice strain levels off at the values typical for thermal strain. The most unexpected strain distribution is observed at location 3 (green inverse triangles in Fig. 4). The beam first probes regions close to the edge of the indent where the mechanically introduced compressive strain is superimposed on the thermal strain, suppressed near-surface strain relaxation. At the depth of 9–13 lm, mechanical strain even reduces thermal strain. At a depth of 15–32 lm, the beam passes through the region of maximal compression, and compressive strain reaches 0.017 in Mo fibers. The NiAl matrix without deformation is under tensile strain along the growth direction. After indentation, the sign of strain distribution in the matrix reverses and reaches 0.007 (compressive) strain in the NiAl matrix in the regions with maximal compression. The strain distribution in Fig. 4 can be understood by combining both thermal and indentation stress analyses. During spherical indentation, materials at a depth of about half the contact radius yield first, and the plastic zone size increases as the applied load increases. The resulting elastic stress fields resemble that caused by an expanding cavity located near the contact center. During unloading, the plastically deformed material tends to preserve its shape, while the surrounding elastic materials tend to spring back and thus exert compressive stress to the plastic zone. A detailed picture of residual stress distribution can be gained from finite element modeling. Because the contact size is significantly larger than the Mo fiber spacing, an indenter radius equal to 100 lm is simulated, and a homogeneous substrate is adopted, with Young’s modulus 200 Pa and Poisson’s ratio 0.3 (as calculated by the rule of mixture from properties of fiber and matrix materials [14]). In addition, the initial thermal residual stresses were not included in the finite element simulations; rather, only indentationinduced stress fields are of interest. On fitting to the measured surface residual impress after unloading, a yield stress of 1.15 GPa was found. The detailed finite element simulation finds compressive residual stress at the center and tensile stress near the contact edge, as shown in Fig. 5. Along line 2, material near the free surface experiences a large compressive stress peak load, and develops a slight tensile stress after unloading. As discussed previously [19], this is equivalent to the Mo–NiAl composite under a significant compressive pre-strain, which, according to previous analysis, leads to increase the h0 0 1i strain in Mo and
Fig. 5. (a) Contour plots of z-axis stress (normalized by the yield stress) after indentation. Three X-ray beam lines corresponding to locations shown in Fig. 4 are also included. (b) The normalized z-axis stress distribution along the X-ray beam lines. Indentation-residual stresses qualitatively agree with the change in measured lattice strains in Fig. 4.
decrease the h0 0 1i strain in NiAl. Along line 2, materials around the plastic zone will experience compressive stresses at peak load and after unloading. This stress history will lead to enhanced compressive strains in both phases, as shown in the measurements along line 2 in the regime of 10–20 lm. Along line 2, materials below the plastic zone will experience compressive stress at peak load and a noticeable tensile stress after unloading. Thus both phases will see increases in their lattice strains. In contrast, traversing along line 3 gives a tensile–compressive–tensile residual stress. Consequently, the diffraction results for Mo along line 3 at Fig. 5 exhibit a reduced compression zone near 10 lm, an enhanced compression zone at 25 lm, and a slightly reduced compression zone near 40 lm (which is difficult to detect because the residual tensile stress caused by indentation is very small). Accordingly, the diffraction results for NiAl along line 3 in Fig. 5 exhibit an enhanced compression zone near 25 lm. The predicted transition occurs near 20 lm, which is slightly smaller than the experimental observation. Despite the qualitative agreement in the transition of additional tension/compression stresses, it is noted that the predicted indentation-induced residual stress is on the order of 0.4 GPa, which leads to a change in residual strain of 0.002. This predicted change is 1/3 of the
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observed strain change. This discrepancy is largely due to the complicated interaction of thermal and indentation stress fields, which is not incorporated in the finite element simulations and is assumed to be additive in the discussion in the preceding paragraph. The tensile thermal residual stress in NiAl is one order of magnitude smaller than the compressive thermal residual stress in Mo. Thus, during indentation, Mo fibers tend to yield first, and the contact load will be supported by a larger plastic zone than that predicted in a homogeneous solid. It is anticipated that an explicit modeling of composite microstructure, as well as the consideration of the initial thermal residual stresses, will yield better agreement. Finally, it is also noted that the slopes of the strain change along lines 2 and 3 are similar, and they both agree with the stress variation in the indentation stress fields.
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3D DAXM measurements also call for a more detailed finite element simulation with explicit description of the 3D microstructure. Acknowledgements Research sponsored by the US Department of Energy, Office of Basic Energy Science, Materials Sciences and Engineering Division (RIB, HB and GEI) and the Center for Defect Physics, an Energy Frontier Research Center (YFG). X-ray microbeam measurements were performed at ID-34-E at the Advanced Photon Source. The use of the APS was supported by US Department of Energy, Office of Basic Energy Science, Scientific Users Facilities Division. References
4. Conclusion This study of a model composite response to nanoindentation demonstrates a powerful new approach to exploring stress accommodation mechanisms in composite materials. The study follows the gradual transition from undeformed regions of the samples to the highly deformed regions near the center of the indent. Analysis of different stress zones under the indent is used to characterize strain partitioning in the composite which is extremely difficult to measure with other techniques. Using depth-resolved X-ray microdiffraction, it is shown that the deformation behavior of the NiAl–Mo composite is distinct from the single-phase materials and dependent on the strain partitioning between both phases of the composite. A dramatic change in the Mo fiber orientation distribution is observed after indentation. Before indentation, Mo fibers are highly aligned with h0 0 1i of the NiAl matrix. Indentation causes the Mo fibers to bend under the external load. Lattice misorientations between the Mo fibers are several times higher then misorientation observed in the NiAl matrix. The superposition of mechanically applied external stresses with the original thermal mismatch induced stresses cause the non-monotonic depthdependent residual strain distribution both in the NiAl matrix and Mo fibers. Moreover, in the matrix, a reversal of the strain sign is observed with depth. Tensile–compressive–tensile residual strain distributions along a line crossing through the most deformed regions are observed in the NiAl matrix. A simple micromechanical model which assumes addition of thermal residual strains and indentation-induced strain fields leads to a fair prediction of the change in measured lattice strains in both fiber and matrix phases. The
[1] Chawla N, Chawla KK. Metal matrix composites. Berlin: Springer; 2006. p. 401. [2] Metcalfe Arthur G. In: Broutman Lawrence J, Krock Richard H, editors. Interfaces in metal matrix composites. New York: Academic Press; 1974. [3] Howe James M. Interfaces in materials. New York: John Wiley; 1997. [4] Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Acta Metall Mater 1994;42:475. [5] Olsen RJ, Judd G, Ansell GS. Met Trans 1971;2:1353–7. [6] Ohashi T, Barabash RI, Pang JWL, Ice GE, Barabash OM. Int J Plast 2009;25:920–41. [7] Medvedeva NI, Gornostyrev YN, Kontsevoi OY, Freeman AJ. Acta Mater 2004;52:675–82. [8] Fox MR, Ghosh AK. Mater Sci Eng A 1999;259:261–8. [9] Bei H, George EP, Brown DW, Pharr GM, Choo H, Porter WD, et al. J Appl Phys 2005;97:123503. [10] McElhaney KW, Vlassak JJ, Nix WD. J Mater Res 1998;13:1300. [11] Nix WD, Gao HJ. J Mech Phys Solids 1998;46:411. [12] Mughrabi H. Acta Metall 1983;31:1367. [13] Hansen N. Metall Mater Trans A 2001;32:2917–35. [14] Bei H, George EP. Acta Mater 2005;53:69. [15] Bei H, Shim S, Pharr GM, George EP. Acta Mater 2008;56:4762. [16] Misra A, Wu ZL, Kush MT, Gibala R. Philos Mag 1998;78:533. [17] Misra A, Gibala R. Metall Mater Trans A 1997;28A:795. [18] Bei H, Barabash RI, Ice GE, Liu W, Tischler J, George EP, et al. Appl Phys Lett 2008;93:071904. [19] Barabash RI, Bei H, Gao YF, Ice GE, George EP. J Mater Res 2010;25:2. [20] Larson BC, Yang W, Ice GE, Budai JD, Tischler JZ. Nature 2002;415:887. [21] Barabash RI, Gao YF, Sun Y, Lee SY, Choo H, Liaw P, et al. Philos Mag Lett 2008;88:553. [22] Ice GE, Barabash RI. In: Nabarro FRN, Hirth JP, editors. Dislocations in solids, vol. 13. Amsterdam; 2007. p. 500–601. [23] Maass R, Van Petegem S, Van Swygenhoven H, Derlet PM, Volkert CA, Grolimund D. Phys Rev Lett 2007;99:145505. [24] Zimmermann J, Van Petegem S, Bei H, Grolimund D, George EP, Van Swygenhoven H. Scripta Mater 2010;62:746–9. [25] Zolotoyabko E. Thin Solid Films 2008;516:8036–41.
Acta Materialia 58 (2010) 6784–6789 www.elsevier.com/locate/actamat
Indentation-induced localized deformation and elastic strain partitioning in composites at submicron length scale R.I. Barabash a,b,⇑, H. Bei a, Y.F. Gao b,c, G.E. Ice a a
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA b Materials Science and Engineering Department, University of Tennessee, Knoxville, TN 37996, USA c Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received 3 August 2010; received in revised form 6 September 2010; accepted 7 September 2010 Available online 1 October 2010
Abstract Three-dimensional spatially resolved strains were mapped in a model NiAl/Mo composite after nanoindentation. The depth-dependent strain distributed in the two phases and partitioned across the composite interfaces is directly measured at submicron length scale using X-ray microdiffraction and compared with a detailed micromechanical stress analysis. It is shown that indentation-induced deformation in the composite material is distinct from deformation expected in a single-phase material. This difference arises in part from residual thermal strains in both phases of the composite in the as-grown state. Interplay between residual thermal strains and external mechanical strain results in a complex distribution of dilatational strain in the Mo fibers and NiAl matrix and is distinct in different locations within the indented area. Reversal of the strain sign (e.g., alternating tensile/compressive/tensile strain distribution) is observed in the NiAl matrix. Bending of the Mo fibers during indentation creates relatively large 1.5° misorientations between the different fibers and NiAl matrix. Compressive strain along the h0 0 1i direction reached 0.017 in the Mo fibers and 0.007 in the NiAl matrix. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Composites; Deformation; Micromechanics; X-ray synchrotron radiation; Micro-/nanoindentation
1. Introduction The behavior of composite materials under external load depends on deformation micromechanisms taking place in both the fibers and the matrix and, in particular, at the matrix/fiber interface [1–9]. The matrix/fiber interface plays a particularly crucial role in the mechanical properties of composite materials and in strain partitioning between the two phases. Despite the key role of interfaces in composite behavior, the strength of interfaces and strain partitioning mechanisms through an interface are largely unknown, particularly at the micron scale. Indeed, the complexity and inhomogeneous nature of deformation ⇑ Corresponding author at: Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. Tel.: +1 865 2417230; fax: +1 865 5747659. E-mail address: [email protected] (R.I. Barabash).
microstructure precludes micron-resolution studies over macroscopic length scales; however, nano- and microindentation of semi-infinite samples can be used to confine deformation fields to volumes in nano-/micrometer scale. Localization of the plastic deformation field under an indent is of profound importance [10,11]. Unlike randomly selected “representative volumes” from macroscopic deformed samples, all aspects of plastic deformation, including stochastic events (e.g., dislocation patterning, dislocation cells [12], different kinds of boundaries [13]) are confined within a volume that can be both probed by three-dimensional (3D) X-ray microscopy and simulated computationally. Therefore, measurements/calculations on the confined volume provide for a direct quantitative connection between 3D X-ray microscopy measurements and computer simulations and modeling. The unique capability of 3D polychromatic microdiffraction now makes it possible to study elastic strain at
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.09.004
R.I. Barabash et al. / Acta Materialia 58 (2010) 6784–6789
low forces and microstructural evolution at high force for direct comparison with simulations and modeling. Of course, the strong deformation gradients inherent in indentations present difficult experimental and theoretical challenges, but also provide stringent tests of controversial strain gradient effects and issues with respect to fundamental length scales in materials. Exploring these plastic/elastic phenomena and interfacial effects in the model NiAl–Mo composite under indentation is the focus of this study. 2. Experimental details 2.1. Specimen preparation The directionally solidified NiAl–Mo eutectic alloy in this study has a relatively simple microstructure, consisting of long uniformly spaced Mo fibers with approximately square cross-sections embedded in a NiAl matrix [14–17]. In this study, NiAl–Mo composite with the fiber sizes (edge lengths) of 550 nm is selected. Details of the directional solidification and the microstructures control have been described earlier [14], and their strain status in as-grown and uniaxially pre-strained condition has been measured previously with 3D X-ray microdiffraction [18,19]. Disk-shaped specimens, 2 mm thick, were cut from the directionally solidified NiAl–Mo eutectic perpendicular to the growth (fiber) direction and mounted in epoxy resin. After grinding with SiC paper through 1200 grit, the specimen was polished with Al2O3 slurries through 0.3 lm and was then polished with a colloidal silica suspension in a vibratory polishing machine. 2.2. Indentation Spherical indentation on the polished surface was conducted with a Nanoindenter XP equipped with sapphire tip with radius 100 lm to prescribed load P = 650 mN. A 15 3 array of the indents was put on the sample surface with spacing of 150 lm. Load–displacement curves were recorded for each indentation. A top view of the indented area with multiple indents and a close view of the indent studied with X-ray microdiffraction, as well as the load– displacement curve for this particular indent are shown in Fig. 1. The spherical indentation impression in Fig. 1b clearly shows the Mo fibers (contrast in bright), which are embedded in the NiAl matrix. 2.3. 3D polychromatic microdiffraction and differential aperture X-ray microscopy In order to test the effect of indentation impact on the twophase composite materials submicron-resolution 3D polychromatic and monochromatic X-ray microscopy was used to measure the elastic strains and dislocation densities in both alloy constituents in undeformed and post-nanoindentation states. Measurements were performed at ID-34-E at the Advanced Photon Source. Depth-dependent elastic strains
6785
were obtained using spatially resolved differential aperture X-ray microscopy (DAXM) [20–22]. The geometry of the polychromatic microdiffraction (PXM)/DAXM measurements is shown in Fig. 2. First, two-dimensional (2D) mapping of the indented area was performed with a polychromatic X-ray microbeam to determine the region affected by indentation. The size of the beam was 0.5 lm in diameter. Typically, strong plastic deformation results in the formation of geometrically necessary dislocations, causing strong lattice rotations in the indented area, accompanied by “streaking” and smearing of the Laue spots [22]. For 3D spatially resolved measurements, a platinum wire was used as a knife-edge to block part of the diffracted radiation (Fig. 2). The wire was translated parallel to the sample surface, and Laue patterns were collected at 400 wire positions. Changes in the patterns for small wire displacements are due to rays passing near the wire edge, hence the term “differential aperture”. Triangulation of the partially shadowed Laue pattern with wire position relative to the sample surface, using ray-tracing algorithms, determines the origin of the diffracted intensity along the incident beam path. Details of DAXM data collection can be found elsewhere [20]. Depth-resolved measurements were also performed with monochromatic radiation, which enables the depth-dependent lattice parameter and/or dilatational strain changes to be obtained. The diffraction geometry of the measurements employed in this study (Fig. 2) resolves the overlapping information from the “fiber forest” beneath the surface of the composite. At each depth, the beam probes either the matrix or a different fiber and provides unique information about the plastic/elastic status as a function of depth and distance from the indentation center. This technique is complementary to the transmission geometry technique employed in Refs. [23,24], and to the energy-variable diffraction, which provides one-dimensional cross-section (normal to the sample surface) for strain distribution across interfaces between different materials on a sub-micrometer scale [25]. 3. Results and discussion The elastic strain measurements in the NiAl/Mo eutectic composite demonstrate a new approach to the understanding of the load accommodation mechanisms in composite materials. In addition, this particular system is of interest because of the large (1%) compressive strains of the embedded fibers [17,18]. The melting point of the NiAl– Mo is 1600 °C. During solidification, thermal expansion mismatch between the Mo-alloy fibers and the NiAl matrix cause large thermal strain between two phases. Details of thermal mismatch strain and thermal response of individual phases have been reported elsewhere [9,18]. The thermal mismatch strain is 1.0%. Here, the focus is on the critical information about interfacial properties that govern the overall composite behavior. The sample surface was first mapped with depth-integrated PXM (2D microdiffraction) to determine the regions
6786
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Fig. 1. (a) The general view of the indented area; (b) SEM image shows the indent studied with PXM and the Mo fibers (bright contrast) embedded in NiAl matrix; (c) the corresponding load–displacement curve.
Fig. 2. Sketch of the polychromatic X-ray microdiffraction of the indented NiAl/Mo sample.
of maximal deformation in the indented area and to identify locations for depth-resolved measurements. A typical indexed Laue pattern is shown in Fig. 3a. The enlarged regions around (0 0 h)-type reflection at two different locations near the indent are shown in Fig. 3b and c. In the first location, which is outside the indented area (Fig. 3b) both the NiAl and Mo (0 0 h) Laue spots are sharp. Although the beam intercepts several Mo fibers (typically up to 10
fibers), they all diffract X-rays into the same position on the CCD, indicating very high alignment of the fibers along the h0 0 1i NiAl and Mo growth direction. In the indented zone, the character of the Laue spots changes. The NiAl Laue spot is streaked, indicating the presence of lattice rotations in the indented zone. The largest observed lattice rotations in the NiAl matrix were <3°. The most dramatic changes in the indented area are observed for the Mo fibers: instead of a single Laue spot arising from different probed fibers, multiple Mo Laue spots are observed when the Xray beam probes a volume with the indent (Fig. 3c). It is known from microscopic studies [2] that, under compression of composite materials, different Mo fibers bend, and their local orientation changes significantly. Such fiber misorientation will lead to X-ray scattering into different directions for different locations on various fibers. The depth-resolved measurements of the dilatational strain along the fiber axis in both NiAl martix and Mo fibers were further performed along the vertical line passing through the center of the indented area where maximal deformation was observed. A total of 15 locations were probed along this line, with a step size of 2 lm in the central region of the indent with maximal deformation, and with a step size of 4 lm farther from the indent center. To determine the fiber axis strain, the (0 0 6) d-spacing
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Fig. 3. (a) Typical indexed Laue pattern from NiAl–Mo composite alloy; (b) enlarged region around (0 0 h) Laue spot for location 1 far from the indented area; and (c) location 2 in the center of the indented area.
was measured for both the NiAl matrix and Mo fibers. The energy of the beam was scanned with a step of 3 eV over an energy range of 13.310–18.601 keV for the (0 0 6) Mo reflection, and within a range of 19.660–19.918 keV for the (0 0 6) NiAl reflection. Energy scans were synchronized with the movement of the special differential aperture (Pt wire). At each energy value, typically 400 partially shadowed diffraction patterns were obtained at different Pt wire positions relative to the sample surface. Triangulation of differential aperture position and energy allowed the energy of the scattered X-rays to be related to the depth of the scattering location. As a result of such measurements, the depth-dependent energy distributions were obtained for each location. The details of the triangulation process can be found in Ref. [20]. Depth-dependent dilatational strains were calculated from the above depth-dependent energy distributions (Fig. 4.) The dilatational lattice strain in both the Mo fibers and the NiAl matrix changes character with location (Fig. 4). Far from the indented area (location 1) where the probed volume is sufficiently far from the indent so that the indentation-induced stress fields are negligible, the strain distribution reveals residual strain caused by the mismatch in the coefficients of thermal expansion between Mo and NiAl which is consistent with previous measurement [18]. As discussed previously [17,18], the incompatible thermal expansion coefficients of the fiber and matrix introduce thermal strain, which causes the Mo fibers to be under compression along the h0 0 1i fiber direction (z axes), and NiAl matrix to
Fig. 4. (a) Distribution of lattice strain along h0 0 1i direction in Mo fibers and (b) NiAl matrix probed at three different locations: (1) blue filled circles; (2) pink triangles; (3) green triangles. Note that the three different location are indicated in Fig. 1b and also Fig. 5a. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
be under tension. Near the surface, these residual thermal strains relax and result in a large near-surface strain gradient (blue circles in Fig. 4). Such a near-surface relaxation is similar to a pull-out test, and the resulting interface slip strength can be determined as 180 MPa from the slip
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zone size (5 lm). Completely stress-free Mo lattice parameters are measured at the tops of the exposed Mo pillars when the matrix is etched away [18]. This value is also shown in Fig. 4a. In the second probed location (pink triangles in Fig. 4), the incident X-ray beam passes through highly deformed fibers (near surface zone). A mechanically introduced compressive residual strain along the h0 0 1i direction of Mo fibers is superimposed on the original thermal compressive strain. The resulting compressive strain in the Mo fibers along their axis reaches values up to 0.013 at a depth of 15 lm. At greater depths, the beam passes out of the indentation-affected area, and lattice strain levels off at the values typical for thermal strain. The most unexpected strain distribution is observed at location 3 (green inverse triangles in Fig. 4). The beam first probes regions close to the edge of the indent where the mechanically introduced compressive strain is superimposed on the thermal strain, suppressed near-surface strain relaxation. At the depth of 9–13 lm, mechanical strain even reduces thermal strain. At a depth of 15–32 lm, the beam passes through the region of maximal compression, and compressive strain reaches 0.017 in Mo fibers. The NiAl matrix without deformation is under tensile strain along the growth direction. After indentation, the sign of strain distribution in the matrix reverses and reaches 0.007 (compressive) strain in the NiAl matrix in the regions with maximal compression. The strain distribution in Fig. 4 can be understood by combining both thermal and indentation stress analyses. During spherical indentation, materials at a depth of about half the contact radius yield first, and the plastic zone size increases as the applied load increases. The resulting elastic stress fields resemble that caused by an expanding cavity located near the contact center. During unloading, the plastically deformed material tends to preserve its shape, while the surrounding elastic materials tend to spring back and thus exert compressive stress to the plastic zone. A detailed picture of residual stress distribution can be gained from finite element modeling. Because the contact size is significantly larger than the Mo fiber spacing, an indenter radius equal to 100 lm is simulated, and a homogeneous substrate is adopted, with Young’s modulus 200 Pa and Poisson’s ratio 0.3 (as calculated by the rule of mixture from properties of fiber and matrix materials [14]). In addition, the initial thermal residual stresses were not included in the finite element simulations; rather, only indentationinduced stress fields are of interest. On fitting to the measured surface residual impress after unloading, a yield stress of 1.15 GPa was found. The detailed finite element simulation finds compressive residual stress at the center and tensile stress near the contact edge, as shown in Fig. 5. Along line 2, material near the free surface experiences a large compressive stress peak load, and develops a slight tensile stress after unloading. As discussed previously [19], this is equivalent to the Mo–NiAl composite under a significant compressive pre-strain, which, according to previous analysis, leads to increase the h0 0 1i strain in Mo and
Fig. 5. (a) Contour plots of z-axis stress (normalized by the yield stress) after indentation. Three X-ray beam lines corresponding to locations shown in Fig. 4 are also included. (b) The normalized z-axis stress distribution along the X-ray beam lines. Indentation-residual stresses qualitatively agree with the change in measured lattice strains in Fig. 4.
decrease the h0 0 1i strain in NiAl. Along line 2, materials around the plastic zone will experience compressive stresses at peak load and after unloading. This stress history will lead to enhanced compressive strains in both phases, as shown in the measurements along line 2 in the regime of 10–20 lm. Along line 2, materials below the plastic zone will experience compressive stress at peak load and a noticeable tensile stress after unloading. Thus both phases will see increases in their lattice strains. In contrast, traversing along line 3 gives a tensile–compressive–tensile residual stress. Consequently, the diffraction results for Mo along line 3 at Fig. 5 exhibit a reduced compression zone near 10 lm, an enhanced compression zone at 25 lm, and a slightly reduced compression zone near 40 lm (which is difficult to detect because the residual tensile stress caused by indentation is very small). Accordingly, the diffraction results for NiAl along line 3 in Fig. 5 exhibit an enhanced compression zone near 25 lm. The predicted transition occurs near 20 lm, which is slightly smaller than the experimental observation. Despite the qualitative agreement in the transition of additional tension/compression stresses, it is noted that the predicted indentation-induced residual stress is on the order of 0.4 GPa, which leads to a change in residual strain of 0.002. This predicted change is 1/3 of the
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observed strain change. This discrepancy is largely due to the complicated interaction of thermal and indentation stress fields, which is not incorporated in the finite element simulations and is assumed to be additive in the discussion in the preceding paragraph. The tensile thermal residual stress in NiAl is one order of magnitude smaller than the compressive thermal residual stress in Mo. Thus, during indentation, Mo fibers tend to yield first, and the contact load will be supported by a larger plastic zone than that predicted in a homogeneous solid. It is anticipated that an explicit modeling of composite microstructure, as well as the consideration of the initial thermal residual stresses, will yield better agreement. Finally, it is also noted that the slopes of the strain change along lines 2 and 3 are similar, and they both agree with the stress variation in the indentation stress fields.
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3D DAXM measurements also call for a more detailed finite element simulation with explicit description of the 3D microstructure. Acknowledgements Research sponsored by the US Department of Energy, Office of Basic Energy Science, Materials Sciences and Engineering Division (RIB, HB and GEI) and the Center for Defect Physics, an Energy Frontier Research Center (YFG). X-ray microbeam measurements were performed at ID-34-E at the Advanced Photon Source. The use of the APS was supported by US Department of Energy, Office of Basic Energy Science, Scientific Users Facilities Division. References
4. Conclusion This study of a model composite response to nanoindentation demonstrates a powerful new approach to exploring stress accommodation mechanisms in composite materials. The study follows the gradual transition from undeformed regions of the samples to the highly deformed regions near the center of the indent. Analysis of different stress zones under the indent is used to characterize strain partitioning in the composite which is extremely difficult to measure with other techniques. Using depth-resolved X-ray microdiffraction, it is shown that the deformation behavior of the NiAl–Mo composite is distinct from the single-phase materials and dependent on the strain partitioning between both phases of the composite. A dramatic change in the Mo fiber orientation distribution is observed after indentation. Before indentation, Mo fibers are highly aligned with h0 0 1i of the NiAl matrix. Indentation causes the Mo fibers to bend under the external load. Lattice misorientations between the Mo fibers are several times higher then misorientation observed in the NiAl matrix. The superposition of mechanically applied external stresses with the original thermal mismatch induced stresses cause the non-monotonic depthdependent residual strain distribution both in the NiAl matrix and Mo fibers. Moreover, in the matrix, a reversal of the strain sign is observed with depth. Tensile–compressive–tensile residual strain distributions along a line crossing through the most deformed regions are observed in the NiAl matrix. A simple micromechanical model which assumes addition of thermal residual strains and indentation-induced strain fields leads to a fair prediction of the change in measured lattice strains in both fiber and matrix phases. The
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