C. Zambaldi and D. Raabe

Acta Materialia 58 (2010) 3516–3530
 

Overview

This paper presents a combined experimental and computational study of the orientation-dependent plastic behaviour of the intermetallic phase γ-TiAl (L1₀ structure), which is a constituent of two-phase γ/α₂-microstructures used in high-temperature structural applications. Because near-stoichiometric single crystals of γ-TiAl cannot be grown in bulk form, the authors develop an indirect strategy: nanoindentation is performed on individual γ-phase grains within two-phase microstructures, and the resulting surface pile-up topographies are characterised by atomic force microscopy (AFM) and electron backscatter diffraction (EBSD). These measurements are then analysed using a three-dimensional crystal plasticity finite element model (CP-FEM) that explicitly incorporates all known deformation systems of γ-TiAl. The central scientific output is the pile-up inverse pole figure — a systematic graphical representation of orientation-dependent indent topographies covering the full unit orientation triangle of the tetragonal crystal structure. This map constitutes a new tool for assessing the relative activity of competing deformation mechanisms, and allows independent confirmation of the predominance of ordinary dislocation glide in stoichiometric γ-TiAl at room temperature.

Material and Crystallographic Background

Gamma-TiAl crystallises in the L1₀ structure, an fcc-derived tetragonal lattice with alternating (002) planes occupied exclusively by Ti or Al atoms and a c/a ratio of approximately 1.02. This near-unity tetragonality is crucial: it introduces a distinction between slip directions that preserves the ordered structure and those that do not. Three classes of deformation system are relevant at room temperature:

• Ordinary dislocations (b^O = ½〈110〉, 4 systems): shear lying entirely within the (001) plane, no c-component, no destruction of the intermetallic order.

• Superdislocations (b^S = 〈101〉, 8 systems): shear with a c-component; the full 〈101〉 Burgers vector avoids an order fault, but the Peierls stress for these dislocations is substantially higher.

• Mechanical twins (b^T = ⅙〈11̄2〉, 4 true twinning systems, one per {111} plane): operate unidirectionally; they do not generate an order fault at the twin interface.

Superdislocations with Burgers vector ½〈112〉 were disregarded because they form sessile configurations at room temperature and are not expected to contribute to plastic flow below elevated temperatures. The relative activities of ordinary dislocations, superdislocations, and twins depend strongly on Al content and interstitial impurity level: for near-stoichiometric and Ti-rich compositions, ordinary dislocations and twinning prevail, whereas Al-rich compositions favour superdislocation glide.

Experimental Methods

Cylindrical specimens of Ti–45.9Al–8Nb (at.%) were produced by optical floating zone growth at IFW Dresden at a withdrawal rate of 10 mm h⁻¹. The as-grown microstructure was nearly fully lamellar with a colony size of several hundred micrometres, plus small equiaxed γ-TiAl grains along selected grain boundaries. These equiaxed γ-grains, typically ~20 µm in diameter, were used as indentation targets.

Surface preparation involved standard metallographic grinding to 1000 grit, polishing with 3 µm diamond suspension, and final electrolytic polishing at −30 °C/35 V in 6% perchloric acid in ethanol to produce a strain-free, highly reflective surface.

Indentations were performed with a Hysitron TriboScope 900 using both Berkovich (three-sided pyramidal) and sphero-conical (axisymmetric) indenters at maximum loads of 3–10 mN. Crystallographic orientations were determined by EBSD using a recently developed fit-rank indexing procedure that correctly identifies the order variants of γ-TiAl, which are otherwise difficult to distinguish by conventional pattern matching. Residual indent topographies were characterised by AFM (Veeco Dimension 3100) with vertical resolution below 1 nm.

A key experimental observation is that the Berkovich indenter — despite its widespread use in nanoindentation — is not ideal for single-crystal pile-up analysis. The geometric anisotropy of the three-sided pyramid interacts with the crystallographic plastic anisotropy of the material, systematically suppressing or promoting pile-up hillocks depending on the relative orientation of indenter edges and active slip directions. The sphero-conical, axisymmetric indenter eliminates this coupling, so that all measured topographic asymmetry can be attributed unambiguously to the crystallographic anisotropy of the material.

Crystal Plasticity Model

An elasto-viscoplastic crystal plasticity framework, originally formulated by Kalidindi et al. (1992) and implemented in the commercial FEM code MSC.Marc via the user subroutine hypela2, was adapted for γ-TiAl. The kinematic description follows the multiplicative decomposition of the deformation gradient F = F*F^p, where F* contains elastic stretch and lattice rotation and F^p represents the plastic distortion accumulated by crystallographic shear.

The plastic velocity gradient is:

L p = F ˙ p ( F p ) − 1 = ∑ α γ ˙ α   d 0 α ⊗ n 0 α Lp=F˙p(Fp)−1=α∑γ˙αd0α⊗n0α

where d₀^α and n₀^α are the slip direction and slip plane normal of system α in the reference configuration. Shear rates follow a rate-dependent power law:

γ ˙ α = γ ˙ 0 ∣ τ α τ c α ∣ 1 / m s i g n ( τ α ) γ˙α=γ˙0τcατα1/msign(τα)

with strain rate sensitivity parameter m = 20 (approximating rate-independent behaviour) and reference shear rate γ˙0=0.001γ˙0=0.001 s⁻¹. Slip resistance evolution is governed by:

τ ˙ c α = ∑ β h α β ∣ γ ˙ β ∣ , h α β = q α β h ( β ) τ˙cα=β∑hαβ∣γ˙β∣,hαβ=qαβh(β)

where the self-hardening modulus h(β)h(β) follows a Voce-type saturation law. Crucially, system-specific hardening parameters were assigned to each of the three dislocation/twin classes, reflecting the physically different strengths of ordinary dislocations, superdislocations, and twins:

The ratio τ₀^S/τ₀^O ≈ 3 encodes the substantially higher Peierls barrier for superdislocation glide in near-stoichiometric γ-TiAl. Twinning systems were implemented as unidirectional slip systems: shear is permitted only when the resolved shear stress on the twinning plane is positive; negative resolved shear stress sets the twinning rate to zero for that increment. The twinning reorientation of the twinned volume was not incorporated, but the high saturation resistance makes reaching the physical twinning shear limit of ~0.75 in any significant volume virtually impossible.

Elastic constants were taken from measurements on mono-crystalline Ti–56Al: c₁₁ = 182.8 GPa, c₃₃ = 176.9 GPa, c₁₂ = 75.2 GPa, c₄₄ = 103.5 GPa, c₆₆ = 76.5 GPa. The FE mesh comprised 4320 hexahedral eight-node elements (validated against a 15,060-element mesh), with the sphero-conical indenter modelled as a rigid body. The coefficient of friction was set to 0.3 after systematic variation confirmed that pile-up height decreased by approximately 30% as friction increased from 0 to 0.3 and remained nearly constant above that value. The indenter tip radius was calibrated iteratively by minimising the difference between simulated and measured residual topographies.

Orientation Convention and Pile-Up Inverse Pole Figure

A central methodological contribution of this work is the definition of an unambiguous orientation convention that allows direct comparison of pile-up topographies measured or simulated at different crystallographic orientations.

For an axisymmetric indenter, specifying only the indentation axis [uvw] leaves a rotational degree of freedom about that axis. To remove this degeneracy, the authors introduce a spherical-coordinate thought experiment: starting from a reference orientation in which the crystal c-axis is parallel to the indentation direction Z, and the a-axis is parallel to X, the crystal is rotated to bring any desired indentation direction [uvw] to Z via the shortest angular trajectory. The two spherical angles (ζ, η) convert to Bunge Euler angles as (φ₁, Φ, φ₂) = (270° + ζ, η, 90° − ζ). Any experimentally measured topography can be rotated into this canonical frame by correcting for the in-plane misorientation determined from EBSD.

An important symmetry result follows: the pile-up pattern for an indentation axis related to another by an improper symmetry operation (involving inversion) exhibits mirror symmetry rather than identity. Therefore, the correct orientation triangle for pile-up representation is defined using only proper symmetry operations and is twice as large as the fundamental zone used for uniaxial properties. This result applies to any crystal structure.

Using this convention, CP-FEM simulations were performed for 51 orientations distributed throughout the unit triangle ​–– of the tetragonal γ-TiAl structure, following a near-equidistant discretisation with an angular resolution of ~9° after Helming et al. The resulting set of topographic maps, placed at the corresponding positions in an inverse pole figure projection, constitutes the pile-up IPF. This representation encodes the three-dimensional plastic anisotropy of the material as a two-dimensional atlas of orientation-dependent surface deformation patterns.

Key Results

Orientation Dependence of Pile-Up Patterns

The simulated pile-up IPF reveals systematic trends across orientation space:

• ​ indentation axis: fourfold pile-up pattern with four dominant hillocks in the 〈110〉 directions, similar to the fcc case.

• ** indentation axis**: similar fourfold topology to ​, but with broken fourfold symmetry; only twofold symmetry remains, reflecting the tetragonality.

• ** indentation axis**: two major hillocks on opposite sides of the impression.

• ** indentation axis**: two dominant hillocks in adjacent positions; most sensitive orientation for distinguishing ordinary vs. superdislocation activity.

• ** indentation axis**: a single dominant protuberance — entirely unlike the threefold-symmetric pattern expected for an fcc crystal.

These departures from fcc behaviour are a direct consequence of the strong plastic anisotropy introduced by the L1₀ ordering. The fact that and axes — equivalent in the fcc structure — produce strikingly different topographies in γ-TiAl illustrates how the tetragonal symmetry and the different dislocation slip system strengths jointly control the material flow pattern around the indent.

Identification of Dominant Deformation Mechanism

The indentation axis proved particularly diagnostic. For stoichiometric γ-TiAl (τ₀^O/τ₀^S < 1), pronounced pile-up forms in the [2̄11̄] and [1̄21̄] directions (right-hand side of the indent as displayed in the convention). CP-FEM analysis of subsurface plastic shear shows that two ordinary dislocation glide systems — systems no. 9 (½1̄10) and no. 12 (½11̄0) — are responsible for these two hillocks. The spatial distribution of plastic shear, visible as distinct lobes below the contact zone, directly maps onto the observed hillocks at the surface.

When the slip system strengths are adjusted to τ₀^O/τ₀^S ≈ 1 (generic fcc-equivalent) or τ₀^O/τ₀^S > 1 (Al-rich γ-TiAl), the dominant hillocks shift to the opposite side of the impression ([2̄11] and [12̄1] directions). This means that the position of the pile-up hillocks relative to the crystallographic reference frame acts as a sensitive binary indicator of whether ordinary or superdislocation glide is the easier mechanism. The simulations place the critical ratio between the initial slip resistances in the range τ₀^S/τ₀^O ≈ 2–3 for near-stoichiometric γ-TiAl.

Experimental AFM topographies from indentations close to and axes showed good agreement with the simulated pile-up patterns. The presence of glide steps on {111} planes, observed in AFM images, is consistent with slip on the closest-packed planes but does not by itself distinguish dislocation glide from twinning.

Assessment of Twinning Activity

The CP-FEM analysis demonstrates that twinning is not the primary deformation mechanism in single-phase near-stoichiometric γ-TiAl under the multi-axial stress state generated by indentation, in contrast to an earlier report by Mahapatra et al. (1995) who identified twinning as dominant in compression experiments. The present work provides a reconciling explanation: in the Mahapatra experiments, the crystal was oriented such that the Schmid factors for ordinary dislocation glide on all active systems were minimal. Given the high Peierls barrier for superdislocations, twinning was thus the only mechanism capable of accommodating the applied ​ compression. For general, multiaxial deformation — as in indentation — ordinary dislocation glide is the dominant shape change mechanism. This is consistent with TEM observations of two-phase alloys with low interstitial content.

The absence of pronounced twinning in single-phase indentation, combined with the high twin activity observed in microstructures with densely spaced γ/α₂ interfaces, provides indirect support for the hypothesis that twin nucleation in two-phase microstructures occurs predominantly at interface stress concentrations rather than within the interior of γ-grains.

Discussion and Scientific Significance

Pile-Up as a Probe of Plastic Anisotropy

The pile-up topography around an axisymmetric indent is a two-dimensional projection of the three-dimensional anisotropic plastic flow field of the crystal. The height, shape, and angular distribution of the hillocks encode the relative activities of all operating deformation systems in a form that is accessible by routine surface characterisation techniques (AFM). This is complementary to TEM, which provides definitive identification of dislocation types and interactions but is limited in statistical sampling of slip system activity. The pile-up IPF approach, by contrast, yields statistically averaged information over the deforming volume beneath the indent, and can be applied systematically across the full orientation space.

The method inherits the precision of AFM surface metrology: vertical resolutions below 1 nm, compared to indenter displacements of hundreds of nanometres. Crucially, the analysis does not rely on contact area estimation (which introduces uncertainty through pile-up or sink-in effects); it uses only the direct measurement of the remaining topography, the applied load, and the residual depth.

Hardness Anisotropy

Experimental indentation depths at a fixed maximum load of 10 mN vary across orientations, indicating orientation-dependent hardness. The data suggest lower hardness for orientations close to ​ and higher hardness near , with a predicted maximum between and from CP-FEM. However, the orientation dependence of hardness is substantially weaker than that of the pile-up topography. This is expected: during indentation, all stress components act simultaneously, averaging out the directional differences in flow stress that dominate uniaxial loading. Additionally, cross-hardening between non-coplanar slip systems contributes to the indentation load at a level comparable to self-hardening, making the cross-hardening matrix parameters a significant source of uncertainty in hardness predictions.

Practical Implications

The pile-up IPF has direct practical applications:

1. Crystallographic orientation identification: The characteristic topographic signature of each indentation axis can serve as a rapid indicator of crystallographic orientation, potentially supplementing EBSD or even serving as an independent check for order variant identification in γ-TiAl.

2. Constitutive parameter identification: The sensitivity of the pile-up pattern to the ratio τ^O/τ^S provides a means to constrain slip system strengths in materials for which bulk single-crystal experiments are not feasible — a situation directly relevant to γ-TiAl and other ordered intermetallics.

3. Indenter geometry selection: The study establishes that axisymmetric indenters must be used for single-crystal pile-up analysis. Berkovich and other non-axisymmetric geometries introduce a systematic coupling between indenter edge orientation and the observed pile-up that is difficult to deconvolve and makes the results strongly dependent on the in-plane rotational alignment.

Conclusions

1. For near-stoichiometric γ-TiAl with low interstitial concentrations, easy activation of ordinary dislocation glide at room temperature is an intrinsic property of the single-phase material, independent of interface effects from γ/α₂ boundaries.

2. Crystal plasticity finite element simulation, with system-specific hardening parameters for ordinary dislocations, superdislocations, and twins, accurately reproduces the orientation-dependent pile-up topographies observed by AFM across all indented grain orientations.

3. The pile-up inverse pole figure — constructed from 51 CP-FEM simulations distributed across the unit orientation triangle — provides a comprehensive atlas of expected topographic signatures for all possible indentation axes in γ-TiAl.

4. The indentation axis is a particularly sensitive discriminator of the relative strengths of ordinary and superdislocation glide; the simulations indicate that superdislocations in near-stoichiometric γ-TiAl resist shear stresses two to three times higher than those required to initiate ordinary dislocation glide.

5. A formal orientation convention was established to display pile-up topographies unambiguously. The convention, expressed via spherical coordinates (ζ, η) and convertible to Bunge Euler angles as (270° + ζ, η, 90° − ζ), is applicable to any crystal structure and any axisymmetric indenter geometry.

6. Axisymmetric indenters are preferable for single-crystal indentation studies. Non-axisymmetric pyramid indenters suppress or modify pile-up hillocks depending on the relative orientation of their edges to the active slip directions, introducing systematic artefacts.