Nanocrystalline metals go ductile under shear deformation

Materials Science and Engineering A 546 (2012) 248–257

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Nanocrystalline metals go ductile under shear deformation Markus Ames, Manuel Grewer ∗ , Christian Braun, Rainer Birringer Universität des Saarlandes, FR7.2 Experimentalphysik, 66041 Saarbrücken, Germany

a r t i c l e

i n f o

Article history: Received 10 January 2012 Received in revised form 15 March 2012 Accepted 16 March 2012 Available online 28 March 2012 Keywords: Shear deformation Activation volume Strain-hardening rate Nanocrystalline metals Plasticity

a b s t r a c t Nanocrystalline Pd90 Au10 (grain size <10 nm) has been deformed under dominant shear and superimposed compression to large plastic strains (ε> 20 %). Taking stress strain curves at different strain rates (10−4 < ε˙ < 100 ) allowed us to extract the shear activation volume and strain rate sensitivity as a function of plastic strain, which are of the order of 4b3 and 0.035, respectively. Particularly, the plastic strain-dependent shear activation volume evolves through a maximum value, which relates to yielding. Analyzing the strain hardening rate  as a function of applied stress enables to determine strain rate dependent macroyield stresses. Scaling of the applied stress with the macroyield stress yields in addition also the strain rate dependent onset stresses of microyield. Surprisingly, strain hardening associated with interfacial deformation modes dominates the stage of microplasticity, whereas, in the macroplastic regime the strain hardening rate saturates at strains >10% to approach a more or less generic value of /G = 0.015. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Intense research efforts have been made over the last decade to study and understand the novel deformation physics of nanocrystalline (NC) metals, which belong to the class of polycrystalline materials [1–3]. Ideally, they are made up of a space filling arrangement of randomly oriented equiaxed crystallites or grains. The jump of crystal lattice orientation between neighboring grains causes atomic mismatch (disorder) which is localized in the core region of grain boundaries (GBs) [4]. As a result, they carry an average specific excess volume (per unit area) of the order of 0.02 nm [5,6] that is correlated with an average specific excess energy of the order of 1 J m−2 [7]. Since the GB area per unit volume of polycrystal scales with the reciprocal average size of the grains, 1/L, it is expected, particularly when the grain size is reduced to the nanometer range <10 nm, that the primary deformation mechanism in conventional polycrystals, namely intragranular slip of lattice dislocation, becomes gradually or even fully replaced by GB processes. A variety of modes of plastic deformation related to GBs has been identified so far: GB slip [8] and sliding [9,10], grain rotation [11,12] and GB migration [13,14], shear transformation zone mediated plasticity [15,16], likewise diffusional creep [17,18] as well as twinning and faulting resulting from partial dislocations nucleated in/at GBs [19–22]. One of the intriguing aspects here is that these interfacial modes of deformation must partly operate

∗ Corresponding author. Tel.: +49 681 302 5190; fax: +49 681 302 5222. E-mail address: [email protected] (M. Grewer). URL: http://www.nano.uni-saarland.de (R. Birringer). 0921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2012.03.061

together in the sense that accommodation and deformation processes must coexist in order to avoid brittle fracture. Naturally, interface deformation processes should be in any case accompanied by dislocation activity such as nucleation of partial dislocations into grains helping in accommodating shearing of adjacent crystallites [23,24]. Even in cases when the flux of lattice or partial dislocations through the nanometer-sized grains ceases to contribute to inelastic strain in a dominant manner, disclination and GB dislocation activity should still operate as carriers of interfacial deformation modes [25]. Guided by the 1/L scaling behavior, one would aim at studying the emergence of GB mediated deformation in the limit of very small grain sizes L < 10 nm. However, such studies are scarce since the overwhelming number of experiments have been conducted on samples with grain sizes larger than 20 nm [26,27] implying that the volume fraction of GBs is on the order of 10% or less. On the other hand, to fully exploit the engineering potential of NC metals, for example generating materials having a combination of superior strength and good ductility as well as improved fatigue and wear resistance, it seems mandatory to study and develop a thorough understanding of the deformation physics underlying the interfacially mediated deformation mechanisms. Recently, torsion testing under high pressure [28] has been applied to NC Pd (L ≈ 14 nm) and allowed to explore the mechanical response and microstructure evolution in a wide range of strain and strain rates. In fact, the presence of shear banding, GB sliding, grain rotation, GB migration and twinning have been identified by electron microscopy and X-ray diffraction studies of the deformed specimens. There is however still a lack of understanding to which extent and in which stage of deformation these possible carriers of irreversible deformation

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contribute to overall strain. Likewise, their impact on the evolution of the strain hardening rate needs still to be settled. Our goal is to extract thermal activation parameters, e.g. strain rate sensitivity, shear activation volume and pressure activation volume. They are quite generally defined and thus applicable to all sorts of inelastic deformation processes and are of great interest for probing the mechanism(s) of activated processes [29]. This requires plastic deformation of NC metals at different strain rates and as function of temperature and pressure. Such studies can be achieved by utilizing a recently developed method of mechanical testing that allows one to investigate the shear deformation of small-sized NC specimens up to large strains and large strain rates [30]. The option of miniaturizing the geometry of test specimens in the range <1 mm even opens up a corridor for systematically investigating NC metal samples synthesized by inert gas condensation (IGC) [31]. Such samples represent polycrystals with a random texture, equiaxed grains, a lognormal grain size distribution function as well as a dominant share of random high-angle GBs and may therefore be considered as a model system for random polycrystals [32]. Moreover, IGC-prepared specimen enable to explore the grain size regime L < 10 nm. The paper is organized in the following way. At the outset we discuss specimen preparation and characterization and give a summary of the newly developed mechanical testing scheme. A central goal of this work is to demonstrate that shear deformation of NC metals enables to deduce thermal activation parameters that cover the regime of microplasticity as well as the range of macroplastic yield. The evolution of activation parameters as a function of strain and their absolute values will be discussed in the light of prominent deformation mechanisms operating in the limit of very small grain sizes. Finally we compare the different criteria to identify the onset of macroyielding (yield stress) and discuss the strain hardening behavior of NC metals in the light of the findings from conventional fcc metals.

2. Specimen preparation, characterization and mechanical testing Nanocrystalline Pd90 Au10 particles were prepared by IGC and consolidated at a pressure of 1.8 GPa to obtain disc-shaped samples with a diameter of 8 mm and a thickness of ≈0.5 mm [31]. The mean grain size of the as-prepared samples was determined from X-ray diffraction (Bragg-peak broadening) based on the method of Klug and Alexander [33]; actually, area-weighted average column lengths of the coherent scattering domains which we denote by L are deduced (for more details see [34,35]). We preferred to synthesize this alloy composition because thermally activated curvature-driven grain growth can so be suppressed up to 150 ◦ C through solute drag [36]. Pure NC Pd exhibits rapid grain growth even at room temperature [37] and thus would not allow to discriminate between stress-induced and temperature-induced changes of the microstructure. The relative density of as-prepared specimen has been determined to be ≈0.95 using the method of Archimedes. Taking into account a 2–3% overall density reduction due to the excess volume stored in the GBs [5,6] at L ≈ 10 nm, some residual porosity still remains. Pores may act as nucleation centers for cracks eventually leading to brittle fracture, a drawback that can be circumvented by shear compression testing to be discussed later. In a recent study of the scaling behavior of ultrasound propagation in NC Pd, we were able to extract the grain-size dependence of the overall shear modulus G and bulk modulus B. The ratio G/B is displayed in Fig. 1 and reveals a drop of G/B as the grain size decreases. The reduction of G/B results mainly from shear softening of GBs as discussed in detail in [38]. Since the intrinsic plasticity of metals correlates with the ratio G/B [39,40] and decreasing G/B

249

Fig. 1. Ratio of shear modulus G to bulk modulus B of NC Pd plotted as a function of grain size L that evolved during room temperature grain growth.

indicates increasing ductility, we expect absence of inherent brittleness of NC Pd90 Au10 . In fact, a 30% shear softening of GBs has been deduced suggesting that the onset of microplasticity should dominantly involve interfacial slip [8]. A recently introduced novel specimen geometry, the so-called shear compression specimen (SCS) [41], allows one to investigate dominant shear deformation of materials at large strains and over a large range of strain rates. We managed to miniaturize the SCS so that small-sized specimens – ideally obtained from a master sample with a well defined microstructure and chemical composition – are available for mechanical testing [30]. In Fig. 2c we display a miniaturized SCS of NC Pd90 Au10 at different deformation stages with increasing load. Obviously, when applying a uniaxial load perpendicular to the top surface of the SCS, the gauge section experiences a shear deformation. Although the mode of deformation appears as simple shear, the state of stress and strain in the gauge section is necessarily three-dimensional. The effect of the compressive stress also acting in the gauge section seems to effectively suppress the influence of processing faults as sources of material failure. In all cases, numerical finite element method (FEM) simulations of the material response under load are required to convert experimentally measured load–displacement curves into von Mises equivalent stress  equ versus equivalent strain εequ curves; the details of mechanical testing using the miniaturized SCS are discussed in [30]. In the following paragraph, we discuss how we extract strain-dependent activation parameters. 3. Thermal activation parameters To learn more about the microscopic mechanism(s) controlling plastic flow in NC metals, we first concentrate on determining the activation volume because it can vary by orders of magnitude for different rate-limiting deformation processes and therefore is highly sensitive to the active deformation mechanism. The shear activation volume va is a central and readily measurable thermal activation parameter and is defined as [29,42]



va = −

∂G(a , ˆ )   ∂a T,P

(1)

where G is the Gibbs free energy of activation that must be supplied by thermal fluctuations at constant applied stress  a , pressure P, and temperature T to reach a saddle point configuration of the shear barrier and so makes an activated process to take

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Fig. 2. (a) Engineering drawing of the shear compression specimen (SCS); the gauge section is colored, (b) SCS mounted on sample holder and loaded by P, (c) series of consecutive plastic deformation steps.

place. The shear resistance is a material property and is defined as  = − ∂G/∂, where  denotes the shear strain. In particular, ˆ is the athermal (rate-independent) threshold stress characterizing the maximum level of shear resistance at T. We note that va is an apparent activation volume that is to first order given by the true volume of the flow-defect (a defective region comprising a cluster of atoms carrying localized inelastic deformation) multiplied by the critical strain of activation that is needed to reach the saddle point configuration. For applied stresses close to ˆ the net strain rate ˙ is given as ˙ = ˙ 0 exp(−G(a , ˆ )/kT ) [29]. The preexponential factor ˙ 0 is proportional to a cluster frequency , that is related to the amount of atoms involved in making up a flow-defect. It is so typically much smaller than the Debye frequency of the atoms, and is proportional to the volume fraction of fertile material that can take part in crossing the barriers as well as to the associated shear strain. Thus, ˙ 0 represents the rate dependence of the operating deformation mechanism and is usually considered not to depend on stress. We point out here that care must be taken when analyzing NC materials under the latter supposition. In NC materials GBs form a contiguous and shear-softened network that serves as preferential zone to assist the onset of plastic deformation and may eventually carry shear in a dominant manner when the grain size becomes so small that dislocation nucleation and migration ceases to

propagate strain. Clearly, we then expect ˙ 0 to be proportional to the volume fraction of GBs which is equivalent to 2ı/L where ı is the GB thickness (ı ≈ 1 nm) [34]. Since there is ample evidence that shear stress-driven grain boundary migration is a carrier of deformation in NC metals [28,14], it consequently follows that L becomes dependent on the applied stress, L = L( a ) and therefore ˙ 0 changes its character into a stress-dependent quantity ˙ 0 = ˙ 0 (a ). Since the functional form of L( a ) is unknown to the best of our knowledge, we will for practical reasons assume in the following ˙ 0 ≈ constant but are aware of the task to extract L( a ) from experiment. With the continuum theory of plasticity we find √ the kinetic shear rate ˙ related to the tensile strain rate ε˙ p by ˙ = 3ε˙ p [29]. Solving for G and substituting ˙ by ε˙ p , we derive from the definition of va (Eq. (1)) the following expression



va = kT

∂ ln ε˙ p ∂a

   

(2) T,P,εp ,L

now given in terms of experimentally accessible quantities. Clearly, the applied stress is a function of the applied strain and strain rate a = a (εa , ε˙ a ) and therefore va has to be determined at given strain. Since va is characteristic of plastic deformation, we calculate va at given plastic strain εp which is deduced from the stress strain curve in  a − εa coordinates based on the relation

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Fig. 3. Applied stress–strain curves of miniaturized SCS of NC Pd90 Au10 that have been deformed at different strain rates ε˙a as indicated in the insert.

εp = εa −  a /C, where C is the effective elastic modulus obtained from the data points located in the regime of linear elasticity. Thus, we implicitly assume linear elastic unloading and neglect, for the sake of feasibility, possible nonlinear and anelastic contributions to unloading. Shear deformation experiments (Fig. 3) were carried out at room temperature and ambient pressure. Stress–strain curves were obtained from a set of four NC Pd90 Au10 miniaturized SCS (L = 6.5 nm) cut from a disc-shaped master sample. The pronounced strain rate dependence of the  a versus εa curves directly reveals the presence of thermally activated deformation. Specimens tested at room temperature and a prescribed strain rate ε˙ a = 3 s−1 failed at a stress of  ≈ 1.2 GPa (Fig. 4). Likewise, specimens tested at 77 K by applying a strain rate ε˙ a = 3 × 10−4 s−1 showed only little inelastic deformation to fail at  ≈ 1.7 GPa (Fig. 4). This evidence confirms the assertion that thermally activated processes dominate the deformation behavior of NC metals. In particular, the maximum stress before failure at ε˙ a = 3 s−1 is even smaller than the maximum stress at ε˙ a = 0.3 s−1 , suggesting that strain localization may be responsible for the failure, a ductile to brittle transition, at a strain rate of ε˙ a ≈ 1 s−1 . The slope of a semi-log plot of the four applied strain

Fig. 4. Applied stress–strain curves of two miniaturized SCS of NC Pd90 Au10 . The solid line shows a deformation at room temperature with a strain rate of 3 s−1 , whereas the dashed line corresponds to a deformation at 77 K with a strain rate of ε˙ a = 3 × 10−4 s−1 .

251

Fig. 5. Shear activation volume va and strain-rate sensitivity m plotted versus plastic strain εp . The maximum value of va appears at εp ≈ 4 % and is indicated by an arrow.

rates (specified in the insert of Fig. 3) versus  a yields, as defined in Eq. (2), va at a fixed value of plastic strain εp . Such plots are shown in the Appendix Fig. A.11 for a selection of different plastic strains. The data set displayed in Fig. 3 also enables to determine the strain rate sensitivity m which is defined as [43]



m=

∂ ln a  ∂ ln ε˙ a T,P,ε

(3)

p

For reasons of comparison we evaluate m at fixed εp . We note, that in the pertinent literature m is usually determined at fixed εa instead at εp . The so obtained m values significantly differ from the ones deduced from Eq. 3 as shown in the Appendix (Fig. B.12). The results extracted for va and m are displayed in Fig. 5. Fundamentally, the activation volume for crystalline materials is bounded by ≈103 b3 (≈2 × 101 nm3 ) when forest dislocation cutting dominates plasticity and on the lower end by ≈0.02 − 0.1b3 (≈2 × 10−3 − 4 × 10−4 nm3 ) which is indicative of point defect migration characteristic of creep processes [44]; we assumed a value of 0.275 nm for b. In the pertinent literature a general trend prevails indicating a decreasing activation volume as grain size decreases. Specifically, the activation volume approaches values in the range 1–10b3 (0.02–0.2 nm3 ) for grain sizes ranging from 10 to 20 nm [45,46]. The values we find for va seem to agree with those found in the literature. The relatively small value of the activation volume found in NC metals has been attributed to the nucleus size of dislocations emitted from a GB [19] or the local volume that is involved in de-pinning dislocations that are stuck to segregated impurities or GB ledges [47]. Hence, Coble creep type diffusionalflow requiring point defect migration can be ruled out on the basis of the obtained va values. In fact, the high stress levels present at room temperature suggest that shear mechanisms should almost instantly overtake processes of diffusional matter transport. This view is also supported by the magnitude of the deduced strain-rate sensitivity, m, being much smaller than the expected value for Coble creep (m = 1.0) [48]. Plastic deformation controlled by GB sliding (m = 0.5) [49] seems, based on the m value, also not to contribute in a dominant manner to overall deformation. This assessment also agrees with the finding that GB sliding becomes the dominant mode of GB response at a crossover temperature beyond 0.5Tm [14]. Other likely scenarios will be discussed in the following paragraphs where we will concentrate on shear transformation zone (STZ) mediated deformation as observed in bulk metallic glasses (BMG) and

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stress-driven GB migration (SDGBM) that has been detected to accompany plastic deformation of NC metals. The deformation of BMG [50,51] occurs locally and involves strain propagation by activation of shear transformation zones (STZ) acting as flow-defect in amorphous metals [52,53]. BMG are globally disordered systems with loss of translational symmetry and characterized by heterogeneities on a mesoscale. On the other hand, the disorder in NC metals is confined to the GBs. The atomic mismatch located in the core region of GBs manifests as shear softening rendering GBs sensitive to assist the onset of inelastic shear. As a result, it suggests itself to compare activation volumes of BMG and NC metals. For a whole variety of different glass compositions activation volumes have been found in the range between 0.1 and 0.3 nm3 . We computed these bounds according to va ≈ 3kT /mH, having used experimentally determined values for the strain rate sensitivities and indentation hardnesses H given in [54]. Obviously, there is some difference in the range of activation volumes between BMG and NC metals. However, taking into account that relaxing BMG near the glass transition temperature entails a reduction of activation volume by a factor of two [55], the similarity between the activation volumes of BMG and NC metals becomes quite obvious. This view is supported by an observed tension/compression strength asymmetry in NC metals [56], a phenomenon that is generic to BMG and relates to STZ-mediated shear deformation. We note that the STZ-concept has been developed to understand deformation of amorphous materials since they cannot sustain structural defects such as dislocations or vacancies, the generic carriers of strain in conventional polycrystalline materials. Therefore, unlike conventional polycrystals, it seems plausible that NC metals in the limit of small grain sizes, due to the 1/L scaling of shear-softened interfacial matter and the concomitant gain in statistical homogeneity, are susceptible to STZ-mediated shear [52,53]. In which stress regime of plastic flow shear is propagated by STZ-based flow confined to the GBS and which share of strain is thereby carried is an issue that needs to be explored. On the other hand, if we, for the sake of argument, associate with the decreasing strain rate sensitivity displayed in Fig. 5 an increase of grain size due to SDGBM [14], the related activation volume would be rather correlated with the nucleation and propagation of disconnections [57,58]. They are suggested to be possible elementary carriers of shear deformation evolving in the material volume swept out by the moving GB. If this conjecture was right, the true volume s of the flow-defect (disconnection) can be approximated by s ≈ va /ˇ where ˇ is the so-called coupling factor given as the ratio of v⊥ /v where v is the velocity component of a moving GB induced by the resolved shear stress acting parallel to the GB plane. The essence of coupling lies in the fact that the latter stress also causes the GB to migrate with v⊥ perpendicular to v . Therefore, the coupling factor seems to be an appropriate measure of the critical shear strain needed to make a GB migrate [59]. Regarding geometry, the volume of a deformed zone that has been sheared by a disconnection (Fig. 6) has a size of r2 h where r2 is a measure of the area of a grain facet bounded by the step character of the disconnection with step height h. Note that h manifests the elementary step of migration of the considered grain facet parallel to v⊥ and hence contributes to incremental growth of the associated grain along v⊥ . Referring to the literature values for ˇ [60], we find ˇ ≈ 0.3 for high angle GBs. Based on the maximum value for va ≈ 4.5b3 , we compute va /ˇ ≈ s ≈ 0.31 nm3 for the true volume of the flow defect, and its areal size amounts to r2 = s /h ≈ 1.13 nm2 , where h has been approximated by the interplanar spacing of 1 1 1 lattice planes; for comparison, the magnitude of s corresponds to about 30 atomic volumes. Assuming for the sake of argument that disconnections are nucleated across grain facets, we expect r to be roughly bounded by the incircle radius R of a grain facet. Approximating a grain by a Kelvin body, the relations L ≈ 1.7e and R ≈ 1.2e apply,

Fig. 6. Sketch of a disconnection [57] with step height h that propagates across a grain facet with edge length e; for more details see text.

where e is the edge length of a grain facet. As a result, r < R is effective at a grain size of L ≈ 6.5 nm in agreement with our conjecture. In this context, Bobylev et al. [25] recently suggested that stress-driven GB migration provides an active accommodation mechanism for cooperative GB sliding. They also point out that stress-driven GB migration is energetically preferred compared to accommodation through dislocation emission from triple junctions. Yet at this point, we primarily need more careful experiments in order to identify at which stress levels or stages of the stress strain curve GB migration is triggered and at which rate grains coarsen during irreversible deformation. If grains would double or even triple their size to enable plastic flow, we consequently expect that the NC material would self-organize evolve into a regime where dislocation flow would progressively take over the propagation of strain. Refraining from the absolute values of va but regarding now the evolution of va as a function of plastic strain, we identify a shallow maximum of va at ≈4% plastic strain. As expected, va and m behave in a roughly correlated manner, here m approaches its minimal value when va reaches its maximum value. Intuitively, we may argue that this maximum is related to the crossover from micro- to macroplasticity. However, care has to be taken here since we do not know how possible shear stress-driven GB migration and hence a non constant ˙ 0 would influence the evolution of va . Not surprisingly that there is an ongoing debate in the literature [61–63] related to the onset of macroplasticity in NC metals, with the notion that the yield stress taken at εp = 0.2% is not appropriate to specify the beginning of macroyield in the nanoscale regime. In what follows, we discuss an approach, based on a recently introduced phenomenological analysis by Saada et al. [43], that helps clarifying the onset of micro- and macroplasticity in NC metals. 4. Micro-, macroplasticity and strain hardening When a polycrystal is deformed at constant applied strain rate ε˙ a , the machine equation defines a relation between rates of stress and strain in the following way: ˙ a = C(ε˙ a − ε˙ p ), where ε˙ p is the plastic strain rate, ˙ a the average applied stress rate and C is an effective elastic modulus corresponding to the mechanical testing mode. The tangent modulus or strain hardening rate  is a measure of the local slope of a given stress–strain curve and is defined as  : = d a /dεa . Combining the equations for ˙ a and , we find  = C(1 − (ε˙ p /ε˙ a )) implying that  is actually a measure of the

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evolution of ε˙ p . It characterizes the response of the material to the imposed ε˙ a that generates the applied stress  a . As a consequence, ε˙ p /ε˙ a equals zero in the elastic regime and it so follows that  = C. The onset of microplasticity comes along with the emergence of finite values for ε˙ p /ε˙ a inducing a deviation of  from C. In fact,  decreases dramatically in the regime of microplasticity to approach values of about 0.1C near the onset of macroplasticity. The limiting case of steady state deformation of a material is characterized by ε˙ p /ε˙ a = 1. In terms of stress, macroplastic deformation of a material emerges when ε˙ p ≈ 0.9ε˙ a and the associated applied stress is commonly denoted as the yield stress  y . On the other hand, the onset of microplasticity that coincides with the onset of deviation from linear elastic behavior is characterized by the stress  m , the so-called microyield stress. To extract yield stress from experimentally obtained stress–strain curves, it is in order to look into more detail how ε˙ p /ε˙ a depends on stress. As discussed in [43], ε˙ p /ε˙ a can be quite generally expressed as a function of ( a −  m )/ s where  s =  u −  m is a saturation stress describing the difference between the microyield stress  m and the ultimate stress  u . The functional form of ε˙ p /ε˙ a = f ((a − m )/s ) depends on the material system. By substituting ε˙ p /ε˙ a = (1 − /C) into the LHS of the above equation we obtain an differential equation that can be solved by integration to give the wanted expression for  a =  a (εp ). For a broad range of metallic systems, stress–strain curves can be sufficiently well represented by a modified Voce-law [64] that has the following form:  a (εp ) =  m + s εp + ( s* −  m ) [1 − exp (− εp (C − s )/( s* −  m ))], where s is the slope in the macroplastic regime and  s* is given by the intersection of the line with slope s , taken at the maximum stress  u *, with the ordinate located at εp = 0. To extract , we take the derivatives of the Voce-functions fitted to the experimentally deduced stress–strain curves. Knowing , it is straightforward to generate plots of ε˙ p /ε˙ a = (1 − /C) versus ( a −  m )/ s that then allow one to identify the micro- as well as macroplastic regime; note that the normalized stress ( a −  m )/ s covers only the micro- and macroplastic regime and is hence related to εp .

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Before we apply this criterion to NC materials, we like to demonstrate that the notion of yield stress in conventional polycrystals, given as the applied stress corresponding to εp = 0.2%, agrees with the criterion that identifies the yield stress as the stress value associated with ε˙ p ≈ 0.9ε˙ a . We selected stainless steel (ASTM 304) as the representative of a conventional engineering material and display a plot of ε˙ p /ε˙ a versus normalized stress in Fig. 7. Clearly, ε˙ p is linearly increasing with applied stress to eventually change into a plateau value ε˙ p /ε˙ a = 0.99 revealing fully developed macroscopic yielding. The stress value where the increase of ε˙ p starts deviating from linearity is associated with the onset of macroplasticity; the regime of strictly linearly increasing ε˙ p hence manifests microplasticity. Based on these arguments, we determine ε˙ p /ε˙ a = 0.95 to mark the onset of macroplasticity, thus implying that the corresponding  a value on the abscissa defines a lower bound (–) for the yield stress y− = 561 MPa. For comparison with the engineering εp = 0.2% criterion, we also show in Fig. 7 the evolution of ε˙ p /ε˙ a as function of εp . Note that the two abscissa scales are not simply related to each other, since εp is linked to  a and needs to be extracted for given  a values from the stress–strain curve (Fig. C.13). Evidently, εp = 0.2% corresponds to ε˙ p /ε˙ a = 0.75 and consequently to a yield stress  y (εp = 0.2 %) = 446 MPa. Obviously, the engineering criterion, since relevant to security, yields a conservative estimate for  y clearly located in the microplastic regime. When applying the ε˙ p /ε˙ a criterion to NC Pd90 Au10 , we gained the data displayed in Fig. 8. Again, we find as a general trend that ε˙ p is gradually increasing with increasing  a to virtually approach the values of the respective imposed ε˙ a . In contrast to stainless steel, the extent of the microplastic regime has been significantly increased. For convenience, we identify the stresses  a at which ε˙ p /ε˙ a = 0.95 with the yield stresses y− . Their values are increasing with rising



strain rate obeying the relation y− 



ε˙ a =3×10−4

≤ y− ≤ y− 

ε˙ a =3×10−1

.

Referring to an medium strain rate ε˙ a = 3 × 10−3 corresponding to  − y −3 = 1.19 GPa, we deduce from Fig. 3 an applied strain ε˙ a =3×10

εa = 5.2% that is linked to a plastic strain εp = 3.2%. The plastic strain value εp ≈ 4 % associated with the maximum of the shear activation volume is somewhat larger than the value obtained from the ε˙ p /ε˙ a = 0.95 criterion but still reflects the right order of magnitude and so should have upper bound character. Clearly, the engineering criterion yields a considerably lowered yield stress  ( y (εp = 0.2 %) −3 = 916 MPa) and would thoroughly fail to ε˙ a =3×10

identify the strain level at the onset of macroyielding.

Fig. 7. Evolution of the strain rate ratio ε˙ p /ε˙ a = (1 − /C) as a function of normalized stress ( a −  m )/ s for stainless steel (black curve). The dashed red line is a linear fit to the straight section of the curve. In addition, the dotted blue curve displays the variation of the strain rate ratio ε˙ p /ε˙ a as a function of plastic strain εp . Note that the blue graph only shows εp up to 2.2% whereas the black curve covers the whole range of plastic deformation up to 20% strain and, therefore, the two abscissa are not simply related. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Fig. 8. Variation of the strain rate ratio ε˙ p /ε˙ a = (1 − /C) of NC Pd90 Au10 (L = 6.5 nm) as a function of normalized stress ( a −  m )/ s and strain rates as given in the insert. For convenience, we identify the stress  a at ε˙ p /ε˙ a = 0.95 with the yield stress  y ; for more details see text.

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(a)

(b)

(c)

(d)

Fig. 9. (a) Normalized strain hardening rate /G plotted versus normalized applied stress  a /G, G = 42.7 GPa is the shear modulus. Straight line behavior of /G is observed in the regime of microplasticity. Colored lines indicate the extrapolation of the linear behavior to cross the abscissa at  = 0. Macroplasticity (horizontally arranged data points) is discussed in figure (d). (b) Modified version of figure (a) with each curve rescaled by the corresponding macroyield stress y (ε˙ a ). The intersections of the colored lines with the dashed line located at /G = C/G mark the respective microyield stresses m (ε˙ a ). (c) Variation of the macroyield–microyield ratio y (ε˙ a )/m (ε˙ a ) as a function of the applied strain rate ε˙ a . (d) Evolution of the strain hardening rate  as a function of applied stress  a at increasing strain values (7%, 10%, 13%, 16%, 20%) in stage P. The gray lines are guides to the eyes, connecting data points measured at constant strain rate but gradually increasing applied strain εa . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

A less sophisticated approach that allows one to extract a measure for the yield stress is shown in Fig. 9a, where the normalized strain hardening rate is plotted against normalized applied stress. Since for all strain rates the data points above /G ≈ 0.1 fall on straight lines, we extrapolate these straight lines to cross the abscissa and so define an applied stress that is identified with the upper bound (+) of the yield stress y+ . Again referring to a

medium strain rate ε˙ a = 3 × 10−3 s−1 , we extract y+ = 1.20 GPa corresponding to an applied strain εa = 5.5% which compares favorably with the respective value deduced from the ε˙ p /ε˙ a = 0.95 criterion. It is tempting to interpret the set of curves in Fig. 9a as a single parameter set, then a normalization of the abscissa with the scaling stress y (ε˙ a ) should condense the whole set into a single master curve. As shown in Fig. 9b this is not the case. Nevertheless, scaling  a by y (ε˙ a ) enables to straightforwardly determine the explicitly strain-rate dependent microyield stresses m (ε˙ a ) which are given by the intersections of the strain-rate dependent straight lines (Fig. 9b) with the horizontal line located at /G = C/G, where C denotes the ensemble average of C. The reason for the failure of scaling is related to the evidence that the ratio y (ε˙ a )/m (ε˙ a ) is not constant. In fact, we can infer from Fig. 9c that this ratio strongly decreases with increasing strain rate. Extrapolating the linear behavior on the log-strain-rate scale to even higher strain rates, we find that y (ε˙ a )/m (ε˙ a ) ≈ 1 so indicating that deformation at ε˙ a ≈ 101 s−1 would completely override microplasticity. However, as seen from Fig. 5, such high strain rates induce failure of the material, implying that thermal activation also plays a crucial role in the macroplastic regime.

Overall we were able to identify three distinct deformation stages (Fig. 10) related to the stress–strain curves of NC metals in the limit of small grain sizes. The elastic regime (E) that ends at  m , the onset stress of the microplastic regime (M) and M itself

Fig. 10. Three distinct deformation stages of NC Pd90 Au10 are observed: a linear elastic regime (E), a microplastic regime (M), and a stage (P) that is characteristic of macroplasticity. The microyield stresses  m and the macroyield stresses  y are strongly strain-rate dependent as indicated by the ordinate. In the macroplastic region P1 ,  is decreasing to assume a constant value /G ≈ 2 × 10−2 in regime P2 (εa ≥ 13 %). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

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crosses over to macroplasticity (P) at the yield stress  y . Unlike conventional polycrystals,  m and  y are sensitively strain-rate and temperature dependent. Whereas, when loading the material appreciably beyond  y , a transition into a deformation stage (εp  10 %) characterized by a relative strain-rate independence of the hardening rate is observed as evidenced by the /G≈ constant branches in Fig. 9a. The strain hardening behavior in the macroplastic regime is shown in more detail in Fig. 9d. Macroplasticity, depending on strain, eventually assumes a constant strain hardening rate. With rising strain rate, the regime of /G = const . is entered at decreasing applied strain values. Nevertheless, these strain values and the associated stresses are undoubtedly larger than the respective yield stresses. In other words, the onset and incipient macroplasticity is accompanied by a decrease in the strain hardening rate which saturates at εa ≈ 13 %. If we compare this behavior with the phenomenology of strain hardening in conventional fcc metals [29,65], we notice the following: In conventional fcc metals, macroscopic flow (stage III) manifests thermally activated strain hardening that can be understood in the realm of dislocation physics. On the contrary, the regime of macroscopic flow in NC metals (L < 10 nm) exhibits a rather strain-rate independent hardening rate. It compares with the insensitiveness to strain rate of conventional fcc metals in stage IV. The strain hardening rate of conventional fcc metals in stage IV approaches a generic value of /G ≈ 2 − 5 × 10−4 , whereas, the investigated NC metals reach values between /G ≈ 1 − 2 × 10−2 in regime P (Fig. 9d). In other words, the macroplastic regime in NC metals displays an apparent saturation of the plastic shear resistance. Interestingly, it is however two orders of magnitude larger than the shear resistance of conventional metals in stage IV. The observation of saturation behavior is related to a hardening recovery balance which has been investigated in great detail in conventional metals [66] and needs still to be explored for NC metals. To end the discussion, we conclude that strain hardening in NC metals seems to dominantly appear in the microplastic regime ( m <  a <  y ) that extends in the limit of small strain rates (ε˙ a ≈ 10−4 ) up to ≈5% applied strain, whereas, microplasticity in conventional fcc metals is usually difficult to resolve from stress–strain curves. To further elucidate the nature of the microscopic processes that give rise to unconventional strain hardening behavior in NC metals, we need more information about the evolution of microstructure during deformation. To this end, we may speculate that SDGBM does not contribute in a significant manner to microplasticity since the yield stress deduced from the maximum of the shear activation volume agrees within acceptable error bars with the values extracted based on the ε˙ p /ε˙ a = 0.95 criterion, thus implying that ˙ 0 remains practically unaffected by the stress acting in the microplastic regime. Exploring deformation mechanism, we like to point out that the SCS geometry is predestined to investigate plasticity in conjunction with in situ diffraction that probes the gauge section in transmission geometry. Moreover, by varying the inclination angle of the gauge section, that is accompanied by a variation of the hydrostatic pressure in the gauge section, we have the opportunity to determine the pressure activation volume which is defined as vP := ∂G/∂P at fixed  a , T and is given in terms of measurable quantities as vP = va (∂a /∂P) at fixed strain rate and temperature. We expect that for NC metals the strength differential (∂ a /∂P) should significantly deviate from ideal von Mises deformation, characterized by (∂ a /∂P) = 0, due to interfacial deformation modes carrying strain in a dominant manner. First results on these issues, particularly on the evolution of the friction coefficient in the Mohr–Coulomb yield criterion, will be communicated in a forthcoming paper.

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5. Summary and conclusion Unlike the common notion that nanocrystalline metals exhibit only little plastic strain when tested in tension, we could demonstrate that NC metals – here in the limit of very fine grain sizes L < 10 nm – manifest large uniform strain deformation when tested under dominant shear and superimposed compression loading. By varying the applied strain rate we deduced the evolution of the strain-rate sensitivity and activation volume along the plastic strain coordinate. The intricate coupling of microyield- and macroyield stress to strain rate renders the construction of scaling variables and hence condensation of strain-rate dependent stress–strain curves into a single master curve impossible. We identified a maximum in the activation volume at εa ≈ 6 % which indicates the crossover to macroscopic plasticity. A much more robust signature of the onset of macroplasticity is obtained when linearly extrapolating the normalized strain hardening rate, plotted as a function of normalized applied stress, to zero hardening rate. For NC Pd90 Au10 , the so deduced upper bound of the yield stress y+ ≈ 1.20 GPa cor-

relates with an applied strain εa ≈ 5.5 % at ε˙ a = 3 × 10−3 s−1 . In view of the overall deformation behavior, we can clearly separate three deformation stages: an elastic regime followed by an expanded microplastic regime, exhibiting pronounced strain hardening, and eventually a crossover to macroplastic deformation manifesting saturation of strain hardening beyond ≈10% strain. Regarding shear stiffness, NC metals have a “cellular” microstructure with relatively lower shear resistance across the core region of GBs and correspondingly higher shear resistance inside the grains. Since we could demonstrate that NC metals deform plastically in a compatible manner, it seems conceivable that the observed strain hardening is correlated to hardening associated with interfacial deformation modes. As dominant strain hardening appears in the microplastic regime, it may imply that particularly in the initial stage of microplasticity the overall deformation is carried by strain increments localized in/at GBs. Such strongly inhomogeneous deformation could be sustained by a single dominating or different coexisting interfacial deformation modes, which should be coupled to concomitant accommodation modes to guarantee deformation in a compatible manner. As a result, this may suggest that a hierarchy of onset stresses could be responsible for the observed strain hardening, rather than flow defect interaction, accumulation and self-organization. These onset stresses may vary locally on a scale set by the heterogeneities such as triple junctions or structural units which are the building blocks of GBs. The saturation of strain hardening in the macroplastic regime calls for recovery processes which still need to be identified.

Acknowledgments Financial support for this research was provided by the German Science Foundation (DFG-FOR714). The authors thank the INM Leibniz Institute for New Materials Saarbrücken for their support in mechanical testing and X-ray diffraction measurements. M.A. is deeply grateful to Prof. Böhlke and his co-workers (Karlsruhe Institute of Technology) for their help and support to implement the SCS deformation into the FEM simulations.

Appendix A. Determination of the activation volume Fig. A.11 shows plots of kB T ln(ε˙ p ) versus  a for three different plastic strains εp . The slope of the fits yields the activation volume in [m3 ] which is easily converted to the more common unit b3 using the burgers vector of Palladium b = 275 pm.

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the definition refers to fixed applied strain εa ⇒ ma or fixed plastic strain εp ⇒ mp . Appendix C. Stainless steel data The data of the stainless steel tensile test is presented in Fig. C.13. Also shown is a fit using the modified Voce-law which gives an excellent representation of the experimental data. References [1] [2] [3] [4] [5] [6] Fig. A.11. Plots of kB T ln(ε˙ p ) versus  a with the corresponding linear fits. [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

Fig. B.12. Comparison of the strain rate sensitivity curves taken at fixed εa and fixed εp .

[25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

[35] [36]

[37] [38] [39] [40] [41] [42] Fig. C.13. Stress–strain curve of a stainless steel (ASTM 304) tensile test (sample geometry: DIN 10002-1:2001(D)) at a strain rate of 3 × 10−4 s−1 . The grey line is a fit to the experimental data using the modified Voce-law.

[43] [44]

Appendix B. Evaluation of the strain rate sensitivity

[45] [46] [47]

In Fig. B.12 we display the difference  between the evolution of  the strain rate sensitivity mx = ∂ ln ˙ a  depending on whether

[48] [49] [50]

∂ ln εa T,P,ε x

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