A crystallographic constitutive model for Ni3Al (L12) intermetallics

Materials Science and Engineering A 400–401 (2005) 256–259

A crystallographic constitutive model for Ni3Al (L12) intermetallics Y.S. Choi a , D.M. Dimiduk b,∗ , M.D. Uchic b , T.A. Parthasarathy a b

a UES, Inc. 4401 Dayton-Xenia Rd., Dayton, OH 45432-1894, USA Air Force Research Laboratory, AFRL/MLLM, 2230 Tenth Street, Wright-Patterson AFB, OH 45433-7817, USA

Received 13 September 2004; received in revised form 26 January 2005; accepted 28 March 2005

Abstract A constitutive model was developed in order to capture the unique thermo-mechanical flow behavior of L12 -structured Ni3 (Al, X) alloys. This model utilized a framework for flow-stress partitioning, which was previously proposed by Ezz and Hirsch, and incorporated a model for exhaustion hardening proposed by Caillard. The simulation results well represent the major aspects of the thermo-mechanical flow behavior of Ni3 (Al, X) alloys, such as a flow-stress anomaly, its strain dependence and a work-hardening rate anomaly. Selected limitations are discussed along with our current efforts toward extending the present model. Published by Elsevier B.V. Keywords: Ni3 Al; Constitutive model; Yield anomaly; Cross-slip locking; Exhaustion hardening

1. Introduction Ni3 (Al, X) intermetallic alloys exhibit unusual thermomechanical flow behavior, even in single-crystal form. Some key features include an anomalous temperature-dependence of the flow-stress and its variation in the micro-strain regime, an anomalous change of work-hardening rate (WHR) with temperature, a tension–compression asymmetry, and a partially-to-fully reversible flow behavior under CottrellStokes (CS) type [1] two-step deformation sequences at different temperatures. Mostly, these unique features originate from the cross-slip locking of screw-character dislocations, a process which is usually referred to as Kear-Wilsdorf (KW) locking [2]. Several mechanistic models have successfully described key phenomena and mechanisms associated with Ni3 (Al, X) alloys [3,4]. In a flow model proposed by Ezz and Hirsch, the flow-stress was partitioned into two (or three) contributions [5,6], which represent a temperature-dependent, thermallyreversible contribution and both temperature and strain-rate ∗

Corresponding author. Tel.: +1 937 255 9839; fax: +1 937 255 3007. E-mail addresses: [email protected] (Y.S. Choi), [email protected] (D.M. Dimiduk), [email protected] (M.D. Uchic), [email protected] (T.A. Parthasarathy). 0921-5093/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.msea.2005.03.040

dependent forest-like work-hardening contribution(s), respectively. Caillard proposed an exhaustion-hardening model for the plastic flow at small strains [7]. In his model, workhardening is a consequence of the physical processes that ensue from the formation of KW locks that have various cross-slip distances, w (a multiple of b), in the cube plane (w = 2b to mb, where mb is the maximum cross-slip distance allowed) and the athermal defeat of these locked dislocations upon an increment of the applied stress. The present study aims at building a crystallographic constitutive model that captures the major flow features of Ni3 (Al, X) alloys, and which is linked to experimentally observed dislocation behavior. In this effort, we have critically examined the merits and limitations of existing mechanistic models, extracted selected formulations and concepts, and have re-formulated these ideas into a combined constitutive model that is based upon a collective understanding of plastic deformation in this class of alloys.

2. Structuring the constitutive descriptions The constitutive description for the present model has been formulated from known mechanisms that are responsible for flow. We have separated the known flow mechanisms into

Y.S. Choi et al. / Materials Science and Engineering A 400–401 (2005) 256–259

two broad categories—exhaustion of mobile dislocations via cross-slip locking, and the evolution of ‘forest’ dislocations. Following the conceptual framework of Ezz and Hirsch [5,6], the flow-stress is defined by three contributions: ˙ T ). τ = τp (T ) + τf (γ) + τt (γ,

(1)

In Eq. (1), τ p is the flow-stress contribution from cross-slip locking of the screw-character dislocations, τ f the flow-stress contribution from formation of ‘FCC-like’ forest obstacles and τ t is the flow-stress contribution from obstacles that can be overcome with aid from thermal activation. The first term (τ p ) is a function of temperature alone and is primarily responsible for the yield-stress anomaly. This term is instantaneously set by the current temperature and thus carries a reversible part of the flow-stress. However, the formulation of τ p differs from that used in Ezz and Hirsch’s original model [5], as τ p in the present model was not intended to provide a critical configuration for the operation of Frank-Read (FR) sources (see Fig. 11 in [5] for details). In the present effort, we have treated τ p of Eq. (1) as being controlled by Caillard’s exhaustion-hardening description [7], which was expressed for single slip on an octahedral slip system, α, by  
 dτpα τpα − τp1 GαCS = θ10 exp − , (2) exp − ∗ dγ kT τp − τp1 where

 √ 4r 2α r , = +K b α 3τcs   √ Γ111 A 3 Γ100 α + τcb = − b A + 2 Γ111

GαCS α τCS



Ucα

2 3

(3)

(4)

and Ucα =

µb3 b µb3 α α Γ111 h+ s [τpe − κτse ] . 4π 4π Γ111 ΓCSF

(5)

In Eq. (2), GCS is the activation energy for cross slip over distances of w > b, and τ p1 and τp∗ are the stress levels re¯ + b, quired to unlock the KW locks having w = b and w = w ¯ is the average cross-slip distance in respectively, where w the cubic plane at a given temperature. Also, k is the Boltzmann’s constant and θ 10 is a pre-exponential factor. Note that Eq. (2) is only valid for w > b, the importance of which is treated later. In Eq. (3), τ CS is a stress projected on the cubic plane in the slip direction [4], Uc , the constriction energy on the superpartial dislocation [3], K, the energy factor (=[0.5C44 (C11 − C12 )]1/2 ), α , the geometric factor for line tension and r, the recombination energy [7]. In Eq. (4), A is the anisotropy factor (2C44 /(C11 − C12 )), τ cb , the resolved shear stress on a corresponding cubic slip system and Γ 111 , Γ 100 and Γ CSF , the antiphase-boundary energies for octahedral and cubic planes, and the complex-stacking-fault energy, respectively. In Eq. (5), µ is the shear modulus, h, a nondimensional constant, s and κ, the parameters that determine

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the relative magnitude of the tension–compression asymmetry and the orientations for which the tension–compression asymmetry vanishes in a stereographic triangle, respectively. The stresses τ pe and τ se , are the resolved shear stresses for the primary and the secondary octahedral slip systems, respectively [8]. In Caillard’s exhaustion-hardening description Eq. (2), the first and second factors of the RHS are related to the probability of locking for all KW configurations having b < w ≤ mb. The third factor on the RHS in Eq. (2) corresponds to the fraction of the KW configurations that remain locked under the stress τ p , in which these KW configurations correspond to those having w > wp . Note that this factor decays exponentially with increasing τ p . Thus, this expression represents the formation and gradual defeat of KW locks with increasing applied stress, which results in a change in the WHR. In the original model of Ezz and Hirsch [5,6], τ f and τ t of Eq. (1) were combined and treated as a work-hardening stress that controls the strain-rate sensitivity through specific forest obstacles, and they showed that this term obeys a Cottrell-Stokes law. In their model, the mechanism for the formation of forest obstacles was described by the multiplication of FR sources and their interactions with cross-slip locking and bypass-unlocking events. In the present model, τ f was introduced in order to account for the influence of forest obstacles, however, no effort was made to define such obstacles in detail. Rather, all hardening mechanisms (such as dipole debris, antiphase-boundary-tube formation, cubicplane glide debris, etc.), which are expected to contribute a hardening rate comparable to forest-dislocation defeat mechanisms, are phenomenologically gathered into this term. In this sense, τ f , becomes an ‘ordinary’ FCC-like hardening contribution and the carrier of the irreversible part of the flowstress during CS-type deformation sequences. The evolution of τ f is expressed by combining a modified one-parametertype dislocation-density-evolution description (dρα /dγ = kf Λαf /b) and a Taylor-type relation (τfα = ηµb(ρα )1/2 ) as: dτfα η 2 µ2 b = kf Λαf . dγ 2τfα

(6)

Here η is the geometric parameter in a Taylor-type relation and kf is a dimensionless parameter. Also, Λαf is the inverse of a mean-free-path for the slip system α. We assumed a parabolic evolution of Λf , which is typical for FCC-like forest hardening;
 Λs − Λαf dΛαf , (7) = θΛ dγ Λs where θ Λ and Λs are the initial magnitude of (dΛf /dγ) and the saturation value of Λf , respectively. Returning to Eq. (1), τ t is strain-rate and temperature dependent. In the present model, we treat this term through an adjustable parameter in a constitutive rate formulation, since the strain-rate-controlling mechanisms for Ni3 (Al, X) alloys are not adequately identified. In Eq. (1), τ p initially domi-

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Y.S. Choi et al. / Materials Science and Engineering A 400–401 (2005) 256–259

nates in the micro-strain regime. However, the influence of τ f and τ t gradually increase with increasing strain. The combined hardening formulations (1), (2) and (6) were incorporated into the strain-rate formulation based on thermally activated plastic flow proposed by Kocks et al. [9];   
  α |τ | − gˆ aα p q Fo α . (8) γ˙ = γ˙ o exp − 1− kT gˆ tα Here, x was defined such that x ≡ x if x > 0, otherwise x ≡ 0, where x is the quantity inside the braces. In Eq. (8) gˆ a and gˆ t are the strengthening contributions from the athermal (long-range character) and the thermally activated (shortrange character) obstacles, respectively. In the present model, τ p and τ f in Eq. (1) were both assigned to gˆ a in Eq. (8) such that dτpα dτ α dˆgaα = + f . dγ dγ dγ

(9)

We have incorporated τ p into gˆ a , since the nature of the cross-slip process in Ni3 Al is so rapid that it can be modeled as rate-independent (for the rates of quasi-static tests). Of course, the frequency of cross slip is also strongly temperature dependent. However, screw-character dislocation locking appears to take place instantaneously at any given temperature (within the anomalous temperature regime) so that one may view its behavior as a rate and time-independent deformation process. The strengthening contribution, gˆ t , in Eq. (8) was considered as an adjustable parameter by directly tying it to τ t in Eq. (1). The constitutive model was structured for an L12 crystal structure, having twelve octahedral- and six cubic-slip systems. However, the constitutive descriptions described above were applied only to the octahedral slip systems. The cubic-slip systems were described as almost perfectly plastic, with only negligible work-hardening. The temperature dependence of the initial yield stress for the cubic-slip systems was obtained from experimental data (for Ni3 (Al, 0.2%B)) from Allan [10]. The present constitutive descriptions were implemented into the finite element software ABAQUS via a User MATerial subroutine (UMAT). A tangent modulus method [11] was used for the time integration.

3. Simulation results and discussion The model was used to simulate the uniaxial compres¯ sion of [123] oriented Ni3 (Al, 0.25%Hf) crystals over a tem-

Table 1 Input parameters for GCS and hardening formulations for octahedral slip systems GCS Γ 111 (J/m2 ) Γ 100 (J/m2 ) Γ CSF (J/m2 ) α h s κ

0.25 0.20 0.32 0.1 0.15 −0.3 0.4

{111} Slip systems τ o (MPa) gˆ t (MPa) Fo (J) γ˙ o (s−1 ) p q η kf Λs (m−1 ) θ Λ (m−1 )

18 600 4.07 × 10−19 107 1/2 3/2 1/3 1 2 × 106 1 × 106

perature range from RT to 800 ◦ C, for a fixed strain rate of 5 × 10−5 s−1 . Preliminary parametric studies were performed in order to identify the physical nature of the major input parameters, and to evaluate reasonable ranges for the values of these parameters. Input parameters were categorized into either temperature-dependent or temperatureindependent parameters, and further into adjustable parameters and non-adjustable parameters (i.e., those that must be determined from the physical properties and chemistry of the material). Tables 1 and 2 summarize input parameters that were affiliated with the cross-slip activation enthalpy (GCS ) and hardening formulations for octahedral slip systems, and that were varied with temperature, respectively. Fig. 1 shows the simulated flow-stress at different offset strains from 10−5 to 0.003. In this figure, the simulation results show the expected offset-strain dependence of the flow-stress anomaly [12]. At extremely low offset strains (10−5 for the current case), the flow-stress is almost independent of temperature. For larger values of offset strain up to 0.003, the flow-stress increased anomalously with temperature. The simulated flow-stress at 0.002 offset strain matched well with the experimental data from Shi [13] (Fig. 1). The flow-stress peaks at around 700 ◦ C, which was also observed experimentally for the Ni3 (Al, 0.25%Hf) alloy [13]. Fig. 2 shows the variation of the simulated WHRs with temperature at 0.01 and 0.02 axial strains. Experimental WHR data,

Table 2 Input parameters that were varied with temperature θ 10 (×1012 Pa) τp∗ (MPa) r A τ o(cube) (MPa)

27 ◦ C

100 ◦ C

200 ◦ C

300 ◦ C

400 ◦ C

500 ◦ C

600 ◦ C

700 ◦ C

800 ◦ C

3.0 20.5 0.001 3.28 330

3.8 22.7 0.0011 3.30 330

4.8 29.2 0.0013 3.33 330

5.9 35.6 0.0014 3.36 330

6.9 42.1 0.0015 3.39 330

8.0 51.0 0.0016 3.42 330

9.0 56.0 0.0017 3.45 330

9.0 56.0 0.0018 3.49 297

9.0 56.0 0.0018 3.53 239

Y.S. Choi et al. / Materials Science and Engineering A 400–401 (2005) 256–259

Fig. 1. Simulation results for the variation of flow-stress with temperature for selected amounts of offset strain. Experimental data (closed circles) for Ni3 (Al, 0.25%Hf) [13] is also shown in this figure with a solid line interpolation.

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to occur by renewed octahedral slip, not cubic slip as observed in the simulation. In order to incorporate Caillard’s unlocking model, the exhaustion-hardening formulation of Eq. (2) needs to be modified to include the effect of the defeat of incomplete KW locks at a smaller scale; specifically, the athermal defeat of KW locks having w = b. We are currently examining how to incorporate this mechanism into the exhaustion-hardening formulation. In the exhaustion-hardening formulation, the contribution of thermally activated dislocation exhaustion to the flow-stress increases with increasing temperature. However, the contribution of the athermal defeat of the KW locks also increases with increasing stress. Thus, the balance between these two contributions is expected to result in a WHR peak at a temperature well below the flow-stress peak temperature [15], but the detailed expressions for this are not complete.

Acknowledgements The present work was supported by the U.S. Defense Advanced Research Projects Agency and the Air Force Office of Scientific Research. YSC and TAP acknowledge support from the Materials and Manufacturing Directorate under contract # F33615-96-C-5258 and F33615-01-5214. The authors acknowledge fruitful discussions with Drs. P.M. Hazzledine and S.I. Rao. The computations described in this study were performed using computer resources at the Ohio Supercomputer Center (grant # PAS0647) with the assistance of Professor G. Daehn of The Ohio State University. Fig. 2. Simulation results for the variation of work-hardening rate with temperature. Experimental data for Ni3 (Al, 0.25%Hf) [13] is also shown in this figure.

which were obtained at 0.02 shear strain [13], are also shown in Fig. 2 for comparison. The simulation reasonably represents the temperature dependence of WHR up to the peak temperature. However, the peak temperature for the WHR was about 600 ◦ C for the simulation, which is approximately 100 ◦ C higher than the experimentally-observed value [13]. Furthermore, the decrease of WHR after the peak temperature was not represented in the simulation results. Further examination of the simulation results shows that the temperature range for both the flow-stress and WHR peaks (600–700 ◦ C) also corresponds to the temperature range for the activation of cubic slip. Recently, Caillard has indicated that the formation of the first WHR peak is related to the athermal unlocking of incomplete KW locks which roughly have w = b [14]. Mechanistically this is understood

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