## Abstract

Yield point phenomena (YPP) are widely attributed to discrete dislocation locking by solute atmospheres. An alternate YPP mechanism was recently suggested by simulations of Ta single crystals without any influence of solutes or discrete dislocations. The general meso-scale (GM) simulations consist of crystal plasticity (CP) plus accounting for internal stresses of geometrically necessary dislocation content. GM predicted the YPP while CP did not, suggesting a novel internal stress mechanism. The predicted YPP varied with crystal orientation and boundary conditions, contrary to expectations for a solute mechanism. The internal stress mechanism was probed by experimentally deforming oligocrystal Ta samples and comparing the results with independent GM simulations. Strain distributions of the experiments were observed with high-resolution digital image correlation. A YPP stress–strain response occurred in the 0–2% strain range in agreement with GM predictions. Shear bands appeared concurrent with the YPP stress–strain perturbation in agreement with GM predictions. At higher strains, the shear bands grew at progressively slower rates in agreement with GM predictions. It was concluded that the internal stress mechanism can account for the existence of YPP in a wide variety of materials including ones where interstitial-dislocation interactions and dislocation transient avalanches are improbable. The internal stress mechanism is a CP analog of various micro-scale mechanisms of discrete dislocations such as pile-up or bow-out. It may operate concurrently with strain aging, or either mechanism may operate alone. A suggestion was made for a future experiment to answer this question.

## Introduction

An alternate mechanism for yield point phenomena (YPP2) was recently suggested by Zhou et al. [5] from simulated stress–strain behavior of Ta single crystals with no accounting for interstitial solutes. The simulations [5] produced a yield stress followed by drop of stress and then recovery of a “normal” strain hardening pattern after the transient strain interval based naturally on the development of internal stresses of geometrically necessary dislocation (GND) content. Discrete dislocation effects, special rules for their multiplication/annihilation, or interactions with interstitial solutes were not considered in the simulations. The current work probes whether experiments are consistent with such a mechanism in place of strain aging, i.e., the locking and breakaway from interstitial atmospheres [6] or dislocation avalanche [7,8].

Internal stress development has been associated with a range of related macro phenomena in recent years including the Bauschinger effect [9,10], the Hall–Petch relationship [11,12], and anelasticity [5,9,10,1316]. No such mechanism has apparently been suggested for YPP prior to Ref. [5], although internal stresses and strain gradients have been invoked to refine the fits of solute-based mechanisms [1719]. Internal stresses arise inevitably from heterogeneous plastic deformation and can be considered analogs of specific dislocation configurations such as bow-out [9,2022] and pile-ups [9,11,12].

The new, non-solute YPP mechanism was predicted by a general meso-scale (GM) model [5] that is an outgrowth of a crystal plasticity (CP) formulation with the unusual capability to account for the internal stress of GND content without fit parameters or length scales3 [11]. As part of predictive GM simulations of Ta single crystals and bicrystals, a YPP was observed in Fig. 1(a). The YPP was absent from CP simulations that are identical except for lacking the internal stress aspect. Figure 1(b) presents corresponding experimental results from the literature [24] with a similar YPP aspect for a [111] Ta single crystal and polycrystal. Remarkably, the [100] single-crystal experimental results show little evidence of YPP, which seems inconsistent with a strain aging/interstitial mechanism, where substantially equal interstitial content would be expected for all dislocations. This variation of response is, however, qualitatively consistent with GM predictions for single crystals and bicrystals, which vary with orientation, specimen shape, and other boundary conditions (see, for example, Fig. 12 of [5]).

Fig. 1
Fig. 1
Close modal

A review of YPP [25] concluded that they are not well understood. YPP occurs in a wide range of materials not likely to have active interstitial solutes (typically C, O, H, and N in body centered cubic (BCC) steels) or even dislocations—materials such as ionic crystals, glasses, and rocks [25,26]. Such diversity seems to exclude strain aging as an exclusive mechanism. There is little doubt that YPP can be caused by strain aging, although the best evidence remains indirect after 70 years: BCC steels deformed and then low-temperature heat is treated to allow diffusion of interstitial atoms, see YPP return [19]. This same treatment also eliminates anelasticity, presumably for the same reason: locking of dislocation motion [16,22].

The foregoing background motivates the current research to verify or disprove the internal stress mechanism of YPP. A direct observation is unlikely in view of solely indirect evidence supporting the widely studied and long-established solute/strain aging mechanism. An indirect grand experiment4 to address the proposed internal stress mechanism for YPP is pursued here in two stages: (1) a predictive simulation of suitable physical tensile test carried out independently of YPP or solutes and (2) conduct and characterization of a matching physical test. This predict-and-probe style of research was the central motivation for the construction of the GM model. The benefits and drawbacks of predict-and-probe are discussed in Ref. [5].

Tantalum is an obvious candidate for the experiment because the new mechanism was predicted for pure Ta single crystals without complicating solutes, precipitates, or second phases. Ta is also suitable because its strain rate sensitivity at room temperature (m = 0.08) exceeds the lower limit for stable crystal viscoplastic simulations, ∼0.05 [27]. Finally, the group has experience with preparing and characterizing columnar Ta oligocrystals suitable for testing [2830].

In the literature, YPP has been observed in Ta single crystals and polycrystals in tension [24,3133] although its occurrence in single crystals can depend on orientation, as shown in Fig. 1(b). Similarly, Byron [31] found YPP in compression of single crystals, while Lim et al. [34] found YPP in compression of polycrystals, but not single crystals. YPP also occurs over a wide range of interstitial content (C, O, or N) in Ta, from hundreds of ppm [35] to 25–50 ppm [24]. All of these indications seem inconsistent with the standard strain aging mechanism, thus suggesting that another mechanism might be responsible in some cases.

The stress–strain aspect of YPP has been associated from the very first [36,37] with localized deformation in bands referred to as Luders bands, deformation bands, microbands, or shear bands, among other descriptors. This correlation has been observed specifically in Ta polycrystals, with inhomogeneous deformation appearing at ∼1% strain and diminishing after ∼2% strain [38].

The principal drawback of Ta for the present purposes is that, being BCC and therefore having no close-packed planes, the slip behavior is poorly understood [28,39]. Observed slip planes vary with conditions and wavy slip (indeterminate or multiple slip planes) is common.5 A pencil glide mechanism [40] anticipates that all slip planes are equally favored; a refinement selects the slip plane that maximizes the (local) resolved shear stress [28,39]. Slip planes of type {110} are always found [31], although there are indications of active {112} [38,41] and {123} [28] slip planes that are present and perhaps even dominant under some conditions, particularly in compression. Nonetheless, Schmid’s law appears to capture most of the observed behavior at room temperature [28,39], while at lower temperatures, non-Schmid effects are more significant [4244]. Improving on Schmid’s law, predictions will likely require stress determination taking into account microstructure, grain orientations, and shapes [28,38]. The grand experiment attempted here does that by applying GM/CP simulation to a simple single-phase microstructure, i.e., an oligocrystal.

## Experimental Procedures

Four oligocrystal specimens (specimens 1–4 or S1–S4 for short) having a small number of very coarse columnar grains were prepared and characterized for research now published [2830]. The multi-crystal nature of oligocrystals confers some of the physical generality of a polycrystal while the columnar grains allow precise measurement of grain shape and orientation. Details of the preparation and characterization techniques appear in the earlier publications: a summary is provided here sufficient only to interpret comparison with GM and CP simulations. The original specimen numbers have been retained for easy reference.

Dog-bone tensile specimens, Fig. 2, were machined from rolled polycrystal sheet of 99.9% pure Ta, then heat treated at 2000 °C and ∼10−4 Pa (∼10−6 Torr) vacuum for 10 h to obtain the grain structures in the specimen neck area as shown in Fig. 2. They were tensile tested using a custom-built in situ load frame [29] at strain rates of 10−4 s−1 (S2) or 10−3 s−1 (S4), with high-resolution digital image correlation (HR-DIC)6 recorded at approximately 4% and 2% strain intervals, respectively. No extensometer was used and the machine stiffness was uncertain, so the existence of YPP was originally unknown and unnoticed.

Fig. 2
Fig. 2
Close modal

Engineering strain data for S2 and S4 were recovered from machine displacement measurements by identifying an effective specimen deforming length that reproduces the last DIC strain measurement for each specimen. To adjust for machine compliance, the total machine stiffness Em was obtained by matching the initial measured stress–displacement slope with the simulated elastic slope (Young’s modulus for the specimen). The accuracy of the result was verified by comparing the other DIC strains (not used in the calibration procedure) with computed ones. The average of (DIC-measured strain–displacement computed strain) was 0.2% strain for S2 and 0.3% strain for S4. Such error magnitudes make only a slight difference in the overall stress–strain representation in view of the low strain hardening rate for these tests.

The results of the reconstructions for S2 and S4 are shown in Fig. 3 along with original DIC stress–strain points for comparison.7 A YPP is seen for both specimens, in the strain range 0–1% for S2 and 0–2% for S4, with stress magnitudes of ∼10 MPa for S2 and ∼25 MPa for S4. These results are consistent with the experiments and simulation of Ta single crystals that show varying YPP magnitude with orientation. The detailed subsequent analyses in this work focus on specimen 4, with the more prominent YPP, for space limitations.

Fig. 3
Fig. 3
Close modal

## Modeling

The GM model [5] was developed from an earlier version [1012] that incorporated internal stresses of GND content in a well-established CP model [45,46]. It operates as a user-defined material model (UMAT) subroutine within abaqus standard (6.14). GM differs from CP only in adding consideration of the internal stress of GND content. The method is determinant and explicit: no adjustable parameters or length scales are added to the scale-independent and GND-independent CP model.

The meso-scale approach reflected in GM is special: the goal is to predict macro mechanical behavior of metals knowing only micro-scale properties, such as for a single crystal. It accounts for lower and higher order strain gradient effects without invoking or using macro-scale information. It differs from methods that seek to reproduce known macro phenomena by fitting adjustable parameters. The advantages and disadvantages of each approach have been discussed elsewhere [5].

The viscoplastic shear rate in the α slip system of a metal crystal is as follows:
$γ˙α=γ˙0(|τeffαgα|)1/msign(τeffα)$
(1)
where $τeffα$ as defined below is the effective shear stress on the α slip system, $γ˙0$ is a reference shear rate, m is the strain rate sensitivity, and gα is the slip resistance.
$τeffα$ represents the interaction of two or three components:
$τeffα=τα+τintα(standard)$
(2)
or
$τeffα=τα+τintα+τobsα(optionalGBtreatment)$
(3)
where $τintα$ is the internal shear stress corresponding to singular defect stress fields and $τobsα$ is an optional obstacle stress for grain boundary (GB) elements, both known independently. In the current work, unless otherwise stated, $τobsα=0$, i.e., Eq. (2) is used.
The internal stress is evaluated from the Nye tensor αij in the neighboring elements having volumes Vremote [4750]8:
$σhmint(R→)=CijklϵlnhCpqmnGkp,q(R→)αijVremote$
(4)
where Cijkl is the fourth-order elastic constant tensor, Gkp is Green’s function, and a comma “,” indicates the spatial derivative. For the isotropic elasticity assumed here, the term CpqmnGkp,q can be expressed as follows:
$CpqmnGkp,q(R→)=−18π(1−υ)[(1−2υ)δnk(Rm)+δkm(Rn)−δmn(Rk)R3+3RmRnRkR5]$
(5)
where δij is the Kronecker delta.
The slip resistance $gα$ evolves with the dislocation density as follows [51]:
$gα=Aμb∑β=1NShαβρβ$
(6)
where μ is the shear modulus, b is the magnitude of the Burgers vector, $ρβ$ is the dislocation density9 for the slip system β, NS is the number of slip systems, and A is a material constant with a representative value of 0.4 adopted uniformly. A common initial forest dislocation density of $ρo$ on each slip system corresponds to an initial slip resistance go that is typically determined by reproducing a measured yield stress for a single crystal with a corresponding GM simulation. The hardening matrix is calculated from the slip system geometry [51].
The well-known Kocks equation [52] is applied to model the dislocation density evolution in each slip system:
$ρ˙α=1b(∑βNSρβka−kbρα)|γ˙α|$
(7)
where ka and kb are material constants related to dislocation generation and annihilation, respectively. The only constants that need to be identified from a single-crystal stress–strain curve for either GM or CP models are go from the yield stress and strain hardening parameters ka and kb from the shape of the stress–strain curve in the plastic range. Equation (7) with constant ka and kb, as required here, cannot predict or even reproduce YPP.

In the current work, no single-crystal specimens from the same lot of Ta were available for testing. However, the stress–strain deformation of each oligocrystal was dominated by a large grain in the neck region, thus the measured stress–strain behavior closely matches that of a single crystal. The values of go, ka, and kb were therefore fit to the stress–strain response of S4, which had more recorded data of better quality than S2.

The finite element mesh representing S4 is shown in Fig. 4. It includes 98,566 C3D8R elements (linear brick elements with reduced integration) with finer elements at the GBs. The parameters obtained from S4 are used for GM and CP simulations as follows: go = 73.3(GM)/54.1(CP) MPa, ka = 8.5(GM)/25.5(CP), and kb = 350(GM)/30(CP) mm. The strain rate sensitivity parameter, m = 0.08, was obtained from published compression tests for single-crystal Ta [5,34]. The other material parameters have been presented elsewhere [5]. Simulations were repeated assuming slip planes of {112} alone, {110} alone, or {112} and {110}, with only minor differences in results. Slip of {112}<111> was used for the results shown here because the shear bands were delineated better, allowing for more precise determination of plane and direction of the bands.10

Fig. 4
Fig. 4
Close modal

## Results

General meso-scale- and CP-predicted stress–strain curves are compared with the experiments in Fig. 5. GM predicts YPP in both specimens, CP in neither. This difference between GM and CP simulations matches results for the Bauschinger effect and anelasticity. This tends to confirm the presence of a common internal stress mechanism for all three macro phenomena. However, it should be noted that, consistent with the literature [19], the details of band location and spacing observed here, but not the existence of bands or not, depend on mesh characteristics. In particular, simulated band width and spacing cannot be resolved with fewer than two elements, a limitation that is seen clearly with coarser meshes.

Fig. 5
Fig. 5
Close modal

Axial strain (ɛxx) maps for S4 were compared from HR-DIC, GM, and CP (Fig. 6) to probe the detailed meso-scale predictions and experiment. The strains shown are incremental. Marked heterogeneity of strain across GBs is apparent in DIC and GM, but not in CP. The other striking difference is the development of deformation bands in the region circled. This appears experimentally in the YPP strain increment, 0–2%, as predicted by GM, but not by CP. In the 2–4% strain increment, both DIC and GM simulation show the origination of new bands but the diminished growth of existing bands. All subsequent deformation increments show the progressively reduced development of the bands in DIC and GM corresponding to the strain hardening range.

Fig. 6
Fig. 6
Close modal

The correspondence between the stress–strain response and the appearance of banding in GM simulation is shown in Fig. 7. These comparisons allow assessment at smaller strain intervals. In each image, the ɛxx strain maps represent the strain increments from the previous step. Increments of 0.71%, 1.1%, and 1.1% were chosen as shown to capture changes of sign in the stress–strain slope. During the first increment there is no banding. This corresponds with a region of monotonically rising stress, i.e., macro strain hardening. During the second increment, banding initiates and develops markedly. This corresponds with a region of dropping stress, i.e., macro strain softening. During the third period, the existing bands progress but at a diminished rate. This corresponds with a region of rising stress, i.e., macro strain hardening. Until necking occurs, simulations at subsequent strains show diminished band growth and nearly identical concave-down stress–strain behavior in GM and CP simulations, similar to the experiments. These observations are similar to those from a wide range of experiments in the literature, including stagnation effects.

Fig. 7
Fig. 7
Close modal

Figure 8 compares the orientation of the measured and predicted deformation band traces in the XY plane. Directions N and B denote traces of the band normal direction and band direction obtained from visual inspection of the high strain areas from the DIC map, Fig. 8(a), or the GM map, Fig. 8(b). Directions n and b are the projections onto the XY plane of the dominant slip system plane normal and slip direction determined from simple Schmid analysis in Fig. 8(a) and by identification of the slip system with maximum slip activity for GM in Fig. 8(b). These slip systems are the same for any of the three analytical methods: GM, CP, and Schmid. The components for the 3D vectors for n and b are shown numerically while the trace directions are shown by the arrows, for comparison to the trace-only N and B directions.

Fig. 8
Fig. 8
Close modal

Digital image correlation and GM show band traces oriented about 40 deg (±1 deg) from the X axis, which corresponds closely to the ∼38 deg (±2 deg) for the dominant slip system (1 −2 −1)[−1 −1 1] from any of the Schmid analysis or GM or CP simulations. At 2% strain, these results tend to confirm that observed bands follow crystal planes for {112} <111> slip bands, although at larger strains both predictions and measurements reveal curved bands that cannot correspond solely to a single slip system.

## Discussion

The foregoing comparisons show that the GM model is able to predict the main aspects of YPP without considering interstitials or discrete dislocations that are required by the strain aging hypothesis. The agreement tends to confirm the broad outline of the proposed internal stress mechanism. In particular, strains at which bands appear and begin to diminish are predicted reasonably well, along with their orientations.11 The internal stress mechanism matches the instability of homogeneous strain and the slow recovery of stability that is observed experimentally. No stress slope reversal or banding is noted when identical simulations are carried out without considering the internal stress of GND content.

However, the predicted intensity and spacing of the bands is considerably different from the measured ones—the predicted ones are widely spaced and more intense. Such a difference is to be expected in view of the known mesh dependence of FE descriptions of bands [19,24]. One only needs to consider that the minimum band width represented by an FE mesh must be at least 2–3 element distances in order to capture the intense shear strain within the band and the stagnant shear strain outside the band. Figure 6 shows close correspondence to this expectation. Finer and finer meshes would presumably show band refinement toward the HR-DIC observed limit, but the computation time would be forbidding. (The S4 simulation in Fig. 4 required approximately 17,000 s (5 h) on a Xeon E5-2687W computer with eight cores.) However, simulations with coarser meshes are practical and were carried out. They showed more intense, more widely separate bands.

While accurate band spacing and intensity for typical microstructures are beyond prediction with the current computer power and standard FE methods, the dependence of the GM prediction on other, less-mesh-dependent aspects can be probed. Figure 9 evaluates the role of strain rate sensitivity on the form of the YPP effect for oligocrystals. Based on the previous single-crystal simulations and theoretical considerations [5,16], anything that reduces heterogeneity should diminish internal stress and its role in mechanical behavior. This result is demonstrated by the GM simulations of Fig. 9 comparing values of one half and double the physical value (0.08).

Fig. 9
Fig. 9
Close modal

The role of strain hardening on GM simulations was also explored. Similar to higher strain rate sensitivity, higher strain hardening rate encourages more uniform deformation which would in turn be expected to reduce the YPP effect according to the internal stress YPP mechanism. Figure 10 confirms this relationship. Figure 10(a) shows the stress-strain responses constructed with altered values of ka and kb to exhibit higher strain hardening (denoted “high n” in the figure) and lower strain hardening (denoted “low n” in the figure).

Fig. 10
Fig. 10
Close modal

In view of the foregoing, the internal stress mechanism predicts YPP in materials for which no solute–discrete dislocation mechanism is likely. In terms of Ta specifically, there are indications that some reported YPP behavior is inconsistent with a solute mechanism, but it cannot be ruled out. Both mechanisms may operate concurrently, or either one alone. Separating these possibilities may be possible by applying a low-temperature bake insufficient to cause dislocation recovery but sufficient to diffuse solute atoms distances of the order of dislocation spacing. Return of YPP would indicate the presence of strain aging, no return would rule it out.

## Conclusions

Oligocrystal simulations and experiments were undertaken to test a new mechanism of YPP recently proposed on the basis of internal stress of GND content. The new mechanism does not rely on interstitial solutes or discrete dislocations. The analysis is based on comparing parallel but independent predictive simulations and well-characterized physical measurements for a deforming Ta oligocrystal. The simulations use no information related to YPP or to macro measurements of its effects.

The following conclusions were reached:

1. The new internal stress YPP mechanism is confirmed in terms of the form of GM-predicted stress–strain response and the associated deformation banding, both of which match HR-DIC measurements of the strain and stress range and band evolution and orientation.

2. Standard CP simulations, identical to GM simulations except not considering the role of internal stress of GNDs, show no indications of YPP, i.e., neither characteristics of stress–strain behavior nor deformation banding.

3. The internal stress mechanism for YPP appears closely related to internal stress mechanisms for the Hall–Petch effect, the Bauschinger effect, and anelasticity. All are predicted by CP simulations accounting for GND stresses; none are predicted without such accounting.

4. The predict-and-probe method followed here provides advantages for identifying unexpected connections or disconnections between micro mechanisms and macro behavior.

5. The new internal stress mechanism may operate in parallel with, or instead of, the widely accepted solute atmosphere mechanism. A simple future test is suggested to rule out the solute mechanism in a material for which YPP is observed to provide more direct evidence.

## Footnotes

2

In this paper, “YPP” is defined to include any drop of stress at the onset of plastic deformation that is temporary such that normal concave-down strain hardening is recovered after some strain interval. Included are phenomena with clear, sharp upper and lower yield stresses and a measurable Luders strain before “normal” strain hardening is recovered. More ambiguous temporary periods of diminished strain hardening that occur initially or following path changes or reversals are sometimes referred to as stagnation [13] and are also included. Excluded, however, is continuous serrated flow (Portevin–LeChatelier effect [4]) that is related in classical theory with dynamic strain aging.

3

In the solid mechanics realm, this capability can also be classified as accounting for both lower-order and higher order strain gradient effects [23] without introducing any arbitrary length scales or constants, hence the shorthand descriptor used here is “predictive.”

4

The term “grand experiment” refers to the goal of probing predictions from a proposed theory by constructing and conducting a critical independent experiment. Successful historical examples of such predict-and-probe experiments include the Michelson–Morley experiment (Michelson–Morley experiment—Wikipedia) and the Millikan oil drop experiment (oil drop experiment—Wikipedia).

5

Slip behavior in BCC alloys varies with temperature; in this paper, we are concerned only with room temperature.

6

HR-DIC was performed by secondary electron imaging of copper powder (300 nm diameter) speckling. Details are presented elsewhere [28].

7

DIC stress–strain points for S4 were presented in Fig. 14 of Ref. [30] but the stresses reported there differ in part because the previous ones correspond to the instant that the DIC image was captured, after stress relaxation had occurred. In the current presentation, the loads for DIC points are taken just prior to stopping the machine for a DIC image.

8

The internal stress equations for arbitrary GND content were provided, along with a subroutine for computing them, by Hussein Zbib and his co-worker Mehdi Hamid. The authors are honored to provide this paper for this volume honoring Dr. Zbib’s life and contributions.

9

The dislocation density on a slip system β in Eq. (6), $ρβ$, can optionally represent both statistically stored dislocation (SSD) and GND contributions, or SSD only. In the current work, both contributions are included.

10

Our current work suggests that {110}and {112} slip planes have differing critical resolved shear stress (CRSS) and hardening behavior. Taking this into account allows single-crystal simulations to match experiments better, but such a constitutive model was not verified and available by the deadline for this paper.

11

Although not specifically analyzed here, it should be noted that the deformation bands predicted by GM, particularly at larger strains, not shown, are curved. (See, for example, the configuration of the central band predicted by GM in the 2–4% or 4–6% interval.) This sort of indication of “wavy slip” from cross slip on 112 planes or multiple slip on alternate slip planes (such as 110 and 123) is consistent with the observations of Carroll et al. [28] and with Kocks’s [40] concept of pencil glide.

## Acknowledgment

The authors are thankful for the support by U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Awards DE-SC00012483 and DE-SC0012587.

The authors are grateful to Hussein and are honored by the collaborative interactions with Hussein Zbib in the last years of his life, and for the continued contributions of his former student Mehdi Hamid. The first author is very grateful for office support of Dayong Li and Shanghai Jiaotong University to carrry out this work.

The authors appreciate the help of Brad Boyce, Joseph Michael, and Bonnie Mckenzie for help with oligocrystal experiments.

This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

## Conflict of Interest

There are no conflicts of interest.

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