Orientation-dependent indentation response of magnesium single crystals: Modeling and experiments

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ScienceDirect Acta Materialia 81 (2014) 358–376 www.elsevier.com/locate/actamat

Orientation-dependent indentation response of magnesium single crystals: Modeling and experiments ⇑

Balaji Selvarajou,a,1 Joong-Ho Shin,b,1 Tae Kwon Ha,c In-suk Choi,b, Shailendra P. Joshia,* and Heung Nam Hand a

Department of Mechanical Engineering, National University of Singapore, Singapore b Korea Institute of Science and Technology (KIST), Republic of Korea c Department of Advanced Metal and Materials Engineering, Gangneung-Wonju National University, Republic of Korea d Department of Materials Science and Engineering and RIAM, Seoul National University, Republic of Korea Received 28 April 2014; revised 6 July 2014; accepted 19 August 2014

Abstract—We investigate the orientation-dependent characteristics of magnesium single crystals under localized contact using nanoindentation experiments and crystal plasticity finite element (CPFE) simulations. Nanoindentation experiments on (0 0 0 1) and ð1 1 2 0Þ planes exhibited distinct load–depth responses. Atomic force microscopy revealed material pile-up with sixfold symmetry in the former case and a sink-in phenomenon with twofold symmetry in the latter case. Our corresponding detailed CPFE simulations uncover the evolution of deformation activity in the indented volume, thereby providing insight into the interacting effects that cause the pile-up and sink-in phenomena. The simulations indicate the occurrence 1 2g extension twins in both cases, although their spatial locations are different. These observations strongly corroborate with our transmission of f1 0  electron microscopic analysis of the indented samples. Finally, our simulations also indicate that, depending upon the crystal orientation, elastic recovery upon unloading may play important role in final surface morphology around the indented region. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanoindentation; Elasto-plastic contact; Magnesium; Crystal plasticity; Pile-up

1. Introduction The hexagonal closed-packed (hcp) crystal structure of magnesium is host to about 50 slip and twinning systems, nearly 30 of which are commonly reported in experiments [1]. However, the stresses required to activate these systems vary over a wide range. For example, the critical resolved shear stress (CRSS) required to activate the plastically softest ð0 0 0 1Þ h1 1 2 0i basal slip is nearly two orders of magnitude lower than the plastically hard non-basal slip modes [2,3]. The low-symmetry hcp crystal structure of Mg often limits the number of independent slip systems available to accommodate plastic deformation, thereby promoting twinning. Under nominally uniform loading, the dominance of particular slip or twin systems within a single crystal of Mg is typically determined by the orientation and sense of the loading directions with respect to its c-axis. For instance, c-axis extension results in the prevalence of the plastically soft f1 0  1 2gh1 0  1 1i extension twinning; on the other hand, c-axis contraction results in the dominance

⇑ Corresponding

authors.; e-mail addresses: [email protected]; [email protected] 1 Equal contribution.

2 2gh1 1  2 3i pyramidal hc þ ai slip. of the much harder f1 1  However, given the extremely low basal slip CRSS, a small misalignment between the loading axis and the crystallographic c-axis can cause profuse basal slip activity, leading to interacting effects even under simple loading conditions. The scenario can become significantly more complicated if the loading condition is complex. An important situation is that of an Mg crystal subjected to localized contact, e.g. indentation. The complexity of the stress state around an indenter, which depends on the geometric characteristics of the indenter and the crystal orientation, may give rise to rich interactions between the various deformation modes. These interactions may influence the local material flow and the macroscopic load–deformation characteristics. While there are several detailed experimental reports and computational efforts that probe the micro–macro nexus in Mg single crystals under simpler loading conditions [2,3], there is much less fundamental understanding regarding the nature of plastic deformation activity under such localized loading conditions. Nanoindentation is a popular protocol for experimentally evaluating the mechanical properties of various classes of materials ranging from bulk materials to thin films [4–16]. The triaxial stress state induced during indentation varies spatially and evolves with increasing depth of indentation.

http://dx.doi.org/10.1016/j.actamat.2014.08.042 1359-6462/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

B. Selvarajou et al. / Acta Materialia 81 (2014) 358–376

Such a systematic, but highly heterogeneous, stress state offers a platform for the extraction of quantitative information regarding the deformation mechanisms as a function of varying levels of triaxiality. Furthermore, it is a valuable method, which may broadly resemble the complex stress states within individual grains of polycrystalline Mg subjected to intricate deformation processes. In particular, indentation of single crystals can provide insight into the orientation-dependent interacting effects arising from the dominant deformation mechanisms in their highly textured polycrystalline analogs. Although the micromechanics that prevails during indentation is reasonably well understood for face-centered cubic [17–21] and body-centered cubic crystals [22,23], similar detailed analysis for hcp single crystals in general [24,25], and Mg in particular [24,26–30], is nascent. Notably, none of the hcp-CPFE indentation simulations to date account for the role of deformation twinning [24,25,30]. A recent investigation on Mg single crystals included nanoindentation experiments and a theoretical analysis based on anisotropic elastic contact theory to rationalize the observed slip modes [28]. In this work, we present a detailed micromechanical analysis of the orientation-dependent Mg single crystal indentation response. To that end, we performed systematic three-dimensional finite-element-based hcp crystal plasticity (CPFE) simulations and nanoindentation experiments on Mg single crystals with different orientations. It is well known that twinning is one of the most important deformation modes in Mg and needs to be taken into account [29,31–33]. Therefore, we adopted an hcp-CPFE framework that includes constitutive descriptions of both slip and twinning (extension as well as contraction twinning) modes [3] to elucidate the manner in which plastic deformation evolves in the neighborhood of an indenter as a function of the crystal orientation. Recent nanoindentation experiments indicate a possible dichotomy with regards to the occurrence of extension twinning: the cono-spherical indentation experiments of Shin et al. [29] exhibited extension twinning when indented on both the basal and the f1 1  2 0g prismatic planes. On the other hand, the spherical nanoindentation experiments of Catoor et al. [28] showed extension twinning only in the case of indentation on the f1 0  1 0g plane, but no twinning in the case of indentation on the basal plane. By corroborating our simulations with experiments, we provide insight into these observations. We also assess the implications of the slip and twinning evolution on the load–depth curves, and the pile-up and sink-in behaviors. In the following, we first discuss the experimental method adopted for indentation experiments on two distinct crystallographic planes of Mg single crystals: the ð0 0 0 1Þ basal plane and the ð1 1  2 0Þ second-order prismatic plane. These experiments give primary information that includes the indentation force vs. depth ðP  dÞ response and the indentation-induced surface deformation characteristics (pile-up/sink-in) as a function of the crystal orientation. Further, we also discuss the observations from transmission electron microscopy (TEM) analysis performed on these indented crystals that highlight the deformation characteristics induced in the vicinity of the indenter. We corroborate these and other recent experimental observations with the predictions from our hcp-CPFE simulations.

359

2. Experimental procedure Single-crystalline Mg was fabricated by a modified Bridgman method, whereby 99.9% pure polycrystalline Mg was heated in a boron-nitride-sprayed graphite crucible in an argon atmosphere above its melting point. The fabricated single-crystal Mg was then sliced with a low-speed cutting wheel along two planes with different crystallographic orientations: ð0 0 0 1Þ basal plane and ð1 1  2 0Þ second-order prismatic plane. Using electron backscatter diffraction (EBSD; HKL Nordlys Channel 5), we confirmed that the misorientation in the samples deviates less than 5 from the desired orientations (Fig. 1a and b). For the nanoindentation test, the samples were prepared by a standard metallographic grinding and polishing procedure, finishing with a 0.1 lm diamond suspension followed by electropolishing with a 5% perchloric acid to remove the mechanically damaged layer. The surface roughness was measured at less than 90 nm. Finally, the samples were annealed at 350  C under an argon atmosphere to relieve the possible residual stress. Additional EBSD analysis showed no signs of recrystallization. Using a Hysitron TriboLabÒ 750 Ubi nanoindenter equipped with a cono-spherical diamond tip of radius equal to 663 nm, series of load-controlled nanoindentation tests were performed on the basal and prismatic planes. The single crystals were loaded at a nominally constant indentation rate of 30 lN=s. Multiple indentation tests were performed decreasing the peak load from 6000 lN to 50 lN. For the indentation on ð0 0 0 1Þ plane, the measured indentation depths were 178 nm and 1300 nm for the peak loads of 6000 lN and 50 lN, respectively. Likewise, for the indentation on the ð1 1  2 0Þ plane, the indentation depths were 178 nm (at 50 lN peak load) and 1500 nm at 6000 lN peak load. Some of the indents were sectioned using focused ion beam milling (FIB; Nova Nanolab 200, FEI) and these regions were observed using TEM (JEM3000F). The surface morphologies of the indentation sites were analyzed using AFM (XE-70, Park Systems Co.) in the non-contact mode. A commercial cantilever (PPPNCHR, NANOSENSORSe) having tip radius of curvature of less than 10 nm was used for this measurement. The measured area and scan resolution were 10  10 mm2and 512  512 squared pixels, respectively. 3. Numerical simulation procedure We performed three-dimensional finite element (FE) simulations using ABAQUS/STANDARDÒ. The next subsection describes the key numerical aspects of the FE model and SubSection 3.2 briefly discusses the underlying single CP constitutive model that includes slip and twinning.

Fig. 1. EBSD orientation maps for (a) (0 0 0 1) basal plane, (b) f1 1 2 0g second-order prismatic plane.

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3.1. Finite element mesh Fig. 2 shows an enlarged view of the three-dimensional FE rendering of the substrate and indenter. The substrate radius and height were set to rs ¼ 6 lm and hs ¼ 8:5 lm, respectively (Fig. 2a). A cono-spherical indenter (Fig. 2b) was modeled as a near-rigid, elastically isotropic solid (Young’s modulus = 1200 GPa, Poisson’s ratio = 0.07) with a tip radius of 663 nm and an included angle of 30 , which agrees with the indenter geometry used in the experiment. Magnesium possesses a sixfold symmetry about the h0 0 0 1i axis and a twofold symmetry about the h1 1  2 0i axis. We exploited this symmetry to ease the computational expense by modeling only one half of the cylindrical substrate. Before adopting it as a standard model for full-scale analysis, we ensured the validity of the reduced model by comparing it with the full model for smaller indentation depth (Appendix A). The substrate was discretized using nearly 27,000 eight-node hexahedral FEs with reduced integration scheme (C3D8R). In constructing the FE mesh, we referred to the guidelines elaborated in the work of Giannakopoulos et al. [34] on indentation simulations. Accordingly, the FEs beneath the indenter were designed to possess an elongated morphology with an aspect ratio (longest-to-shortest dimension) of  5 while maintaining a sufficiently fine mesh in the region of contact to minimize the errors in the calculated displacements and tractions. Away

from the indenter the mesh density was gradually coarsened to keep the simulations computationally tractable. These dimensions were chosen so that the stresses arising from the maximum indentation depth did not significantly interfere with the boundary effects, thereby mimicking a bulk response. All the simulations were performed under quasi-static, isothermal displacement-controlled conditions. This is different from the experiments, which were in a load-controlled mode. A displacement-controlled setup was chosen in the simulations as it provides a stable analysis environment. In all the simulated cases, we prescribed a velocity boundary condition (v ¼ 30 nm s1) to the indenter and restricted the maximum time increment (Dt) to 0.01 s to ensure convergence. Further, we assumed frictionless contact between the indenter and the substrate as frictional effects are not expected to affect the qualitative aspects of the problem [35,36]. For the indenter–substrate contact interaction, we adopted the penalty approach, described in the ABAQUSÒ Theory Manual [37]. In addition to the material nonlinearity, we also invoked geometric nonlinearity to account for finite stretches and rotations. In accordance with the experiments, we considered two canonTable 1. Elastic constants (MPa) of Mg single crystal at 300 K. C 12

C 13

C 33

C 44

59,400

25,610

21,400

61,600

16,400

(b)

663nm

(a)

C 11

300

(c) Fig. 2. Finite element model showing (a) an overview of the 3-D FE mesh, (b) mesh refinement in the vicinity of the contact area and (c) the geometry of the cono-spherical indenter.

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361

Table 2. Key slip and twin constitutive equations and calibrated parameters using the experimental loading curves of the basal and prismatic plane indentations (Fig. 3; strength and hardening parameters are in MPa). Constitutive equations

Parameters

Slip in parent (a systems) and slip in twinned region (~a systems)  m  n c_ n ¼ c_ 0 gsn  sgnðsn Þðn ¼ a or ~aÞ (T-1) c_ 0 – reference slip rate m – rate sensitivity parameter

c_ 0 ¼ 1  103 s1 m ¼ 50 s

 Rt  gn ¼ sn0 þ t0 g_ nslsl þ g_ ntwsl dt

(T-2)

P s g_ nslsl ¼ Ng¼1 hng ðcÞ_cg  hðcÞ; ðn ¼ gÞ ðno sum on nÞ hng ¼ qhðcÞ; ðn – gÞ ( h0 ;   ðbasal slipÞ   c  hðcÞ ¼ ðnon-basal slipÞ h0 sech2 sahs a ; s

sn0 – CRSS of the nth slip system

Basal

g_ nslsl – slip–slip interaction hardening

Prismatic hai

125

g_ ntwsl – twin–slip interaction hardening

Pyramidal hai Pyramidal hc þ ai

125 170

(T-3) (T-4)

q¼1

q-parameter for latent hardening

h ¼ 100 (basal slip) (T-5)

h0 – initial hardening modulus Prismatic hai Pyramidal hai Pyramidal hc þ ai

Twin in parent (b systems) (TT – extension twin, CT – contraction twin] 8  b mt < f_ btt sb ; ðTTÞ b stt m (T-6) f_ btt - reference TT v.f. rate f_ ¼ : f_ b sbb t ; ðCTÞ ct s f_ bct – reference CT v.f. rate ct c_ b ¼ f_ b ctw

s_ btwtw

g_ btwsl

(T-7)

Rt  t0

s_ btwtw þ s_ bsltw



dt

  8  b  > < hbtt sech2  b htt cttb c_ b ; ss tt s0 tt ¼ P b > N ct n : c_ b ; H ct n¼1 f

¼

8 < :

h

ss

1500 1500 6000

250 250 225



sa0 – CRSS for ath slip system sas – saturation stress for ath slip system

sb ¼ sb0 þ

15

(T-8)

ðTTÞ (T-9) ðCTÞ

 b   h sl ctt  b _ ; ðTTÞ hbtt sl sech2 sb tt s b c s tt

 tt

0:5H ct sl ðcct Þ0:5 c_ ct ;

ðCTÞ

(T-10)

f_ btt ¼ 1  102 f_ btt ¼ 1  104 mt ¼ 50  0:129; ctw ¼ 0:138;

c_ b – plastic shearing rate f_ b – twin v.f. evolution rate ctw – twinning shear sb - TT or CT CRSS at t0 ¼ 0

s

tt

¼ 10

s_ btwtw – twin–twin interaction hardening

s0

ct

¼ 120

s_ bsltw – slip–twin interaction hardening

s_ bsltw ¼ 0

hbtt – initial hardening for b TT system

htt ¼ 25

sb0 tt sbs tt

– CRSS of b TT system

ss

– saturation stress for b TT system

H ct ¼ 9000

H ct ,b – CT system hardening parameters

b ¼ 0:05 hbtt

cct – shear strain on all CT systems

H bct

3.2. Crystal plasticity constitutive model The hcp-CPFE model adopted here was developed and implemented by Zhang and Joshi [3]. This model takes into account the four key slip modes commonly observed in Mg: (1) three basal hai slip systems (ð0 0 0 1Þh1 0  1 2i); (2) three prismatic hai slip systems (f1 0  1 0gh1 0  1 2i); (3) six pyramidal hai slip systems (f1 0  1 1gh1 0  1 2i); and (4) six pyramidal hc þ ai slip systems (f1 1  2 2gh1 1  2 3i). In addition to slip, two twin modes are also included: (1) six extension twin systems (f1 0 1 2gh1 0 1 1i); and (2) six contraction twin systems (f1 0 1 1gh1 0 1 2i). The total deformation gradient F is decomposed as F ¼ Fe Fp , where Fe and Fp are the elastic and plastic parts

s1 ðTTÞ ðCTÞ

¼ 20

tt

ctt - shear strain on all TT systems

ical cases: (1) indentation on the basal plane (i.e. along the h0 0 0 1i direction); and (2) indentation on the second-order prismatic plane (i.e. the h1 1  2 0i direction). In the next section, we briefly describe the single crystal plasticity approach developed for hcp-Mg.

s1

sl

¼ htt

sl

¼ 15

of the deformation gradient, respectively.2 The spatial velocity gradient comprises an elastic (Le ) and a plastic (Lp ) part _ 1 ¼ Le þ Fe Lp ðFe Þ1 L ¼ FF ð1Þ where F_ is the material time derivative of F. The plastic velocity gradient is constitutively prescribed, which is assumed to be additively decomposed into three parts: slip in the parent region, twin in the parent region and slip in the twinned region:  XN tw XN s XN tw Lp ¼ 1  fb c_ a ðsa  ma Þ þ c_ b ðsb  mb Þ b¼1 a¼1 b¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} þ

X N

slip in parent

tw

XN

s

fb c_ ~a ðs~a  m~a Þ ~ b¼1 a¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

twin in parent

ð2Þ

slip in twinned region

2

Throughout the text, uppercase bold Latin and bold Greek alphabets refer to second-order tensors, while lowercase bold Latin alphabets indicate first-order tensors. Italicized, plain Greek and Latin variables denote scalar components.

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B. Selvarajou et al. / Acta Materialia 81 (2014) 358–376

where c_ a denotes the shear rate on the ath slip system, and f b and c_ b denote the twin volume fraction and shear rate for the bth twin system, respectively. N s and N tw denote the total number of slip and twin systems, respectively; ma is the slip/twin plane normal vector and sa is the corresponding slip/twin direction vector in the parent region; and m~a is the slip plane normal vector and s~a is the corresponding slip direction vector of the slip systems in the twinned region. Table 1 gives the elasticity coefficients for Mg single crystal and Table 2 consolidates the constitutive equations describing plasticity on individual slip and twin systems, together with the material parameters adopted in this work. The shear strain rate on an ith slip/twin system depends on the resolved shear stress si acting on it and its current system strength gi . The high value of rate-sensitivity parameters in the power-law expressions (m ¼ mt ¼ 50) ensure that the strain on a particular slip/twin system is nearly zero before yield. The physical basis of the constitutive equations used in the simulations can be found in Ref. [3]. The material parameters for these slip and twin modes are obtained by fitting the simulated force–depth curve with the experimental curve (Fig. 3), which is briefly discussed further in Section 4. Eq. (T-6) in Table 2 tracks the evolution of twin volume fraction (v.f.) on a particular twin system at a material point and the equivalent shear strain at that material point (equivalently, Gauss point (GP) in FE) is tracked via Eq. (T-7). For a particular twin mode, when the total twin v.f. on all its twin systems reaches a critical value f cr (set to 0.9) the FE volume represented by that GP is reoriented from its original orientation to the twinned one. Amongst the n twin systems in a particular twin mode, the system that possesses the largest twin v.f. is chosen as the orientation of the twinned lattice. The reader is referred to Ref. [3] for further details on algorithmic description of this scheme and its FE implementation.

experimental results wherever possible. The elastic and plastic anisotropy of Mg, underscored by its low-symmetry hcp crystal structure, results in complex coupling between the different deformation modes. While our experiments provide post-mortem (after unloading) details of the indentation-induced effects, the CPFE simulations allow us to trace the manner in which these deformation mechanisms evolve during the indentation process and to relate them to the macroscopic observations. An aspect of interest in the following discussions is the manner in which strain accumulation and hardening of different slip/twin systems affect the evolution of pile-up or sink-in. The primary information obtained from an indentation experiment is the force–depth curve (P  d), which gives insight into some of the basic mechanical characteristics of a material. Fig. 3 shows the experimental and simulated P  d curves, corresponding to the indentation on the basal and prismatic planes. Notably, the response for the basal indentation is harder than the prismatic indentation, which is due to the orientation-dependent activation of slip and twin systems. Fig. 4a and b qualitatively indicate the manner in which the crystal c-axis may plastically deform upon indentation on the ð0 0 0 1Þ and ð1 1  2 0Þ planes. Accordingly, the primary deformation systems adopted in the parameter fitting procedure for CP simulations were: pyramidal hc þ ai slip, pyramidal hai, prismatic hai and contraction twinning (basal indentation), and extension twinning (prismatic indentation). Basal slip was expected to be

I

II

III

IV

III

IV

4. Results and discussion In this section, we discuss in detail the observations from CPFE indentation simulations on the basal and secondorder prismatic planes of Mg single crystal. Results from the simulations are compared with the corresponding

(a) (0001) indentation

I

II

(b) (11¯ 20) indentation

Fig. 3. Experimental and simulated load–depth (P  d) curves. The loading portion of the experimental curves was used in calibrating the slip and twin material parameters given in Table 2.

Fig. 4. Schematic of expected deformation state for (a) basal plane indentation: Region I – c-axis under direct contraction; Region II – transition from c-axis contraction to c-axis extension; Region III – caxis under extension; and (b) prismatic (1 1 2 0) plane indentation: Region I – c-axis under direct extension; Region II – transition from caxis extension to c-axis contraction; Region III – c-axis under contraction. In both cases, region IV is elastic; the elasto-plastic boundary is shown by a solid green curve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

B. Selvarajou et al. / Acta Materialia 81 (2014) 358–376

profuse in both cases; therefore, we considered both cases for fitting its response. Our initial analysis indicated that the P  d curves with the parameters reported by Zhang and Joshi [3] for pure Mg single crystals resulted in significantly softer responses for both the orientations compared to the experimental counterparts. Therefore, we obtained the set of parameters (Table 2) via iterative trials, starting with the parameters in Ref. [3] as the basis. These were arrived at by fitting only the loading portion of the experimental P  d curves of both the basal and prismatic plane indentations, as shown by dotted curves in Fig. 3; against that backdrop, the unloading responses, which were not fitted, compare well with the experimental results. As can be noted from Table 2, the CRSS and hardening parameters in this work are somewhat higher than those of Zhang and Joshi [3]. There may be several reasons for the differences; we identify three possible causes that could influence the experimental results but are not explicitly modeled in our simulations. First, the purity of Mg single crystals in our experiments was somewhat lower than those used as a basis by Zhang and Joshi [3]; this can result in a harder initial material. Second, Mg is susceptible to oxidation when exposed to an ambient environment or under stress, causing formation of magnesium oxide layer on the surface, which is plastically harder than Mg. Third, the maximum indentation depth in our experiments was in the range of  1 lm, which could introduce size effects leading to a harder P  d response compared to conventional microindentation experiments [38–40]. From the viewpoint of simulations, the modified CRSS and hardening parameters adopted in the current simulations could be reflective of these possible unmodeled phenomena. However, obtaining quantitative information about these effects is beyond the scope of this work. In the following section, we discuss in detail the computational observations obtained from indentation of Mg single crystal along the h0 0 0 1i direction. In Section 4.2, we discuss the simulation results of indentation along the h1 1 2 0i direction. For both cases, we compare the

simulation results observations.

363

with

corresponding

experimental

4.1. Indentation on the basal plane In general, the stress state induced by an indenter results in a spatially varying triaxiality that evolves with increasing depth of indentation. In view of the anisotropic elasticity and plasticity of Mg single crystals, it is useful to discuss some of the qualitative features that can be expected upon indentation of a Mg single crystal. Referring to Fig. 4a, at a given indentation depth the region directly beneath the indenter will be under c-axis contraction. If the force is sufficiently large, this is expected to activate non-basal pyramidal hc þ ai slip and/or contraction twin modes. In comparison, regions that are off the indenter axis will experience a stress state that results in the c-axis being less compressed to the extent that the region around the indenter circumference will transition from c-axis contraction to caxis extension. If this is so, it implies that there is a region where the c-axis is neither in extension nor under contraction (inflection point). Further, in the region where the caxis experiences extension, if the resolved shear stress on the f1 0  1 2g twin systems reaches the CRSS, it will cause peripheral extension twinning to occur in the vicinity of the indenter. Fig. 5 shows a series AFM images taken during ½0 0 0 1 indentation experiment, which indicate the evolution of pile-up around the indenter. Fig. 6 shows the evolution of pile-up obtained from the corresponding CPFE simulation. In Fig. 5, at d 350 nm the material pile-up is more or less uniformly distributed around the indenter. Interestingly, a sixfold symmetric pile-up emerges with increasing depth. The peaks of this newly formed discrete pile-up coincide with h1 0  1 0i, which is clear in the AFM images for d  622 nm. The amount of pile-up at these peaks is not equal, which is likely due to experimental limitations, such as misalignment between the tip and the surface or manufacturing offset of the indenter tip geometry. Nonetheless,

Fig. 5. AFM images showing evolution of pile-up observed in the indentation experiment on the basal plane.

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350

600

Fig. 6. Contour plots showing the evolution of pile-up in the basal plane indentation simulation.

the distinct nature of these peaks is clearly discernible. In comparison, the image sequence in Fig. 6 shows an interesting trend. It indicates an initial discrete nature of the pileup, which exhibits a sixfold symmetry (this was not observed in the experiment). With increasing indentation depth, the pile-up transitions toward becoming isotropic instead of remaining discrete, which is consistent with the experiment at a similar stage of indentation. This isotropic pile-up around the indenter prevails over a certain indentation depth range, beyond which it transitions again to discrete peaks, again consistent with the experiment. From this basic observation, we may divide the indentationinduced plasticity into four stages: 4.1.1. Stage 1: initial plasticity and embryonic signature of hardening evolution Fig. 7a shows the relative positions in the simulation that show the first sign of yield on particular slip/twin systems. Table 3 quantifies these positions from the origin in terms of the current radius of contact (rc ) at a given indentation depth, i.e. for the ith point, xi ¼ gx rc ; y i ¼ gy rc , where gx and gy are non-dimensional scaling factors in the x- and y-directions, respectively. The table indicates that, over the range 0 nm < d 6 300 nm, except for the pyramidal hc þ ai slip, the activation of the remaining deformation systems occurs primarily in the vicinity of the surface. Fig. 7b–f shows the corresponding evolution of shear strain (ca ) on these systems at those locations. Incipient plasticity occurs at d  10 nm (point I (Fig. 7)) and is initiated by activation of the basal slip system, consistent with the recent experimental observation of Catoor et al. [28]. Our simulations indicate that, at such small indentation depths, the plastic deformation in the vicinity

of the indenter does not manifest as pile-up. The pyramidal hc þ ai slip is also activated early at d  15 nm (point II (Fig. 7c)). This is facilitated by the necessity to accommodate c-axis contraction underneath the indenter and is a preferred mode in comparison to the f1 0  1 1gh1 0  1 2i contraction twinning, given that the latter has a higher non-saturation-type hardening. Indeed, the contraction twin v.f. and the corresponding total accumulated slip over the entire loading stage are negligibly small compared to the slip on the other systems. It is interesting to note that the pyramidal hc þ ai slip is active not only beneath the indenter (point II (Fig. 7c)), where the c-axis is under direct compression, but also around the indenter (points I (Fig. 7c)), and III (Fig. 7(d)). This indicates the inability of activating slip with the Burgers vector along the a-direction (i.e. prismatic hai and pyramidal hai) despite their lower CRSS compared to the pyramidal hc þ ai slip. The prismatic hai (point IV (Fig. 7)) and pyramidal hai slip systems (point V (Fig. 7f)) activate much later, at d  100 nm. Although f1 0  1 2gh1 0  1 1i extension twinning is activated as early as d  20 nm (point III (Fig. 7d)), the shear strain on that system tends to saturate well below the twinning shear. With progressive indentation, twinning evolves at a different location (point V (Fig. 7f)), but also saturates. In other words, over this range of indentation depth the twinning activity is still in incipient stages and does not result in any twinned volume. Consequently, it also does not contribute appreciably to the overall plasticity or the pile-up. As we show in SubSections 4.1.2 and 4.1.3, extension twinning begins to evolve rapidly at larger indentation depths and strongly influences the pile-up behavior. Embryonic pile-up appears at d  150 nm, beyond which it continues to increase with indentation depth.

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Fig. 7. (a) Schematic indicating locations where first signs of yield are observed in CPFE simulations of basal plane indentation: basal slip (I), pyramidal hc þ ai slip (II), f1 0 1 2g extension twin (III), prismatic hai slip (IV), pyramidal hai slip (V). Panels (b–f) show the evolution of ca at these points during Stage 1 (Section 4.1.1). Table 3 summarizes the positions of these points as a function of the current radius of contact (rc ) of the indenter.

Fig. 6a shows the contour plot of the pile-up at d  200 nm. A notable feature in the figure is that the pile-up exhibits a sixfold symmetry with peaks corresponding to h1 1  2 0i directions. In the case of indentation of polycrystals, the

resulting surface morphology has been explained based on the overall strain hardening behavior [41,42]. In the case of indentation of single crystals, it is the strain hardening behavior and the strain accumulation of individual slip

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Table 3. Relative positions gx ; gy of the points exhibiting first occurrence of yield on particular slip/twin systems for basal indentation (cf. Fig. 7) and prismatic indentation (cf. Fig. 14). Indentation mode

Deformation system

Point

Indentation depthd (nm)

Contact radius rc (nm)

gx

gy

Basal

Basal slip Pyramidal hc þ ai slip Extension twin Prismatic hai slip Pyramidal hai slip Extension twin Basal slip

I II III IV V I II

10 15 20 100 100 5 10

110 140 160 350 350 80 110

1.0 0 1.9 1.15 1.3 0 0.60

0 1.04 0 0 0 1.76 0

Prismatic

Fig. 8. Schematic showing the influence of non-basal slip and f1 0 1 2g extension twinning on pile-up during (0 0 0 1) indentation. Panel (a) shows the intersection of non-basal slip planes with the indented plane. Panel (b) indicates projections of the f1 0 1 2g twin planes on the indented plane. Panel (c) illustrates the manner in which f1 0 1 2g twinning contributes to the pile-up along ½1 0 1 0.

and twin systems that determine the surface morphology and not the overall strain hardening behavior. Hence, the observed surface morphology can be explained based on the mechanics of the crystallographic deformation modes activated and how they influence the material flow beneath and around the indenter. Wang et al. [17] reported pile-up patterns in copper single crystals indented along the [1 0 0], [1 1 0] and [1 1 1] orientations; pile-up occurred along the lines of intersection of the active f1 1 1g h1 1 0i slip planes with the indented plane. Hence, pile-up due to slip occurs along the lines of intersection of the active slip planes with the indented plane. Fig. 8a shows the intersecting lines of the various slip planes in Mg with the indented plane. From the figure, it can be deduced that slip on the prismatic hai and pyramidal hai slip planes produce pile-up along the h1 1 2 0i directions while slip on the pyramidal hc þ ai systems produce pile-up along the h1 0  1 0i directions. We explain the observed pile-up based on the strain accumulated in active slip/twin systems. From the preceding discussion (Fig. 7), the active modes of deformation in the indenter neighborhood are: basal slip (around), pyramidal hai (around) and prismatic hai (around), pyramidal hc þ ai (beneath and around), and f1 0  1 2gextension twins

(around). Of these, although the basal slip is triggered at small indentation depths, it is theoretically not expected to contribute to the pile-up as there is a zero Schmid factor on the basal plane. That notwithstanding, the regions in the immediate vicinity of the indenter undergo non-trivial lattice rotations, which may be sufficient to cause the Schmid factor on the basal plane to become non-zero, thereby activating it locally. Indeed, this is the process that drives the initial plastic flow in the present case. However, it is known that the accrual of pile-up due to slip is associated with the inability of the material to strain harden along active slip systems. Against that backdrop, we note that the basal slip exhibits a non-saturation-type hardening behavior even at large values of local shear strain on the basal slip system [43], which is also accounted for in our CP constitutive description. This may explain why, in the simulation result, no pile-up is observed until d  150 nm despite active basal-slip-induced plasticity right from a small indentation depth. In comparison, the prismatic and pyramidal hai slip systems exhibit saturation-type hardening responses, as illustrated in Fig. 7e and f. Moreover, the slip planes of both these systems also intersect the ð0 0 0 1Þ plane along h1 1  2 0i. This provides a logical reason for why the

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prismatic and pyramidal hai slip systems drive pile-up during this stage of loading. In Fig. 6, note that the pile-up height (h) is relatively much smaller compared to the indentation depth. Besides the possible oxidation-induced surface roughness, it may be difficult to observe such incipient pile-up at the resolution of an AFM, which could be why our experiment did not reveal the sixfold symmetry with peaks corresponding to h1 1 2 0i directions at d 6 200 nm (Fig. 5). 4.1.2. Stage 2: transition from anisotropic to isotropic pileup: prevalence of twinning With further indentation, the pile-up undergoes a gradual transition. Fig. 6b shows the contour plot indicating the nature of the pile-up at d  350 nm. A noteworthy aspect at

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this stage of indentation is that the initially discrete nature of the pile-up (Stage 1) no longer prevails. Instead, the pileup appears to be nearly uniform along the entire contact circumference of the indenter. In other words, the pile-up transitions from the initial anisotropic to an isotropic mode. This is remarkably consistent with the experimentally observed pile-up at d  296 nm (Fig. 5), which also indicates an isotropically distributed pile-up. In the simulation this isotropic nature of pile-up persists in the regime 250 nm 6 d 6 350 nm. We explain this transition as follows: at this stage of the indentation process, in addition to the mechanisms that drive pile-up along h1 1  2 0i (i.e. prismatic hai and pyramidal hai), the mechanisms that cause pile-up along h1 0  1 0i begin to contribute. The primary mechanisms that contribute to h1 0  1 0i pile-up are

Fig. 9. Extension twinning during (0 0 0 1) indentation. Panel (a) shows the predicted distribution of the f1 0 1 2g tension twin v.f., corresponding to d  930 nm (close to peak load) from the CPFE simulation. The sectional views exposing f1 1 2 0g planes indicate extension twinning around the indenter closer to the free surface. Panels (b–d) show evidence of twinning in our experiment: (b) the location of TEM sampling highlighted by the rectangular box in the AFM image; (c) TEM image of the region highlighted in (b) indicating evidence of twin (red rectangular box) near the free surface; and (d) SAED pattern of the material in the red rectangle confirming the occurrence of f1 0 1 2g twins in basal plane indentation (M and T in the SAED pattern stand for the parent and the twin, respectively). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Displacement in the indented direction (nm)

the pyramidal hc þ ai and f1 0  1 2g twinning; the way in which each of them contributes to pile-up is discussed below. The pyramidal hc þ ai slip planes, which intersect the basal plane along h1 0  1 0i, begin to accrue plastic strain while its slip resistance tends to saturate; this results in pileup along the planes of intersection of the slip plane with the indented plane (shown in Fig. 8a). With increased material accumulation along the h1 0  1 0i directions, the initially distinct directionality around the indenter gradually becomes diffuse. In addition to slip, pile-up along h1 0  1 0i is aided by the strain on extension twinning systems. The criterion for twin-induced lattice reorientation f ¼ f cr is satisfied at d  350 nm. Fig. 8b and c schematically explains the manner in which twin-induced lattice reorientation and the associated shear strain assists pile-up. The shear strain on the f1 0 1 2g twin systems pushes the material upward normal to the indented surface. The resultant strain manifests as pile-up along the projections of the twin planes

on the indented plane (Fig. 8b). As can be understood from Fig. 8b, the projection of the twin planes on the indented plane is sixfold symmetric extending along h1 0  1 0i. 4.1.3. Stage 3: re-emergence of a sixfold symmetric pile-up Following the transition from a discrete (Stage 1) to an isotropic (Stage 2) pile-up, further indentation results in a re-emergence of discrete pile-up regions, but with an important difference compared to Stage 1. Fig. 6c shows the pileup profile at d  600 nm, with distinct peaks ensuing from a sixfold symmetry. However, unlike Stage 1, where the peaks occurred along h1 1  2 0i, at this stage the peaks appear along h1 0  1 0i. Importantly, these locations of these peaks corroborate well with the experimental results at similar indentation depths (Fig. 5). As noted in the preceding paragraph, h1 0  1 0i correspond to the intersection of the pyramidal hc þ ai plane with the basal plane. The dominant pile-up along these directions underscore the hardening saturation on this slip system together with the rapid evolution of tensile twinning relative to the prismatic hai and pyramidal hai slip activity responsible for the pile-up along h1 1  2 0i. With progressive indentation, pile-up continues to preferentially evolve with the peaks along h1 0  1 0i. Fig. 9a shows the contour plot of the total v.f. of 1 2g twins at d  930 nm and the sectional views taken f1 0  along the f1 1  2 0g planes to corroborate the position and the direction of twin formation with the experiment (shown in Fig. 9b). To confirm the presence of twins, a TEM sample was extracted using the FIB process from the surface near the indentation impression for d  990 nm (the location of sectioning is indicated in Fig. 9c). The selected area electron diffraction (SAED) pattern shown in Fig. 9d ascertains the occurrence of extension twinning around the indenter, which concurs with the predicted result.

0 At peak load After unload

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10

12

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4.1.4. Stage 4: unloading The last stage of the indentation process is unloading, during which elastic recovery occurs. An important feature resulting from the unloading process is the change in the

Fig. 10. Predicted material pile-up along ½1 0 1 0 at peak load and after unloading in basal plane indentation.

60

20

Stage III - Pile-up in the <1010> direction Driven by tension twinning

Pile-up height (nm)

40

Stage II - Tranistion in pile-up directionality First observance of twin induced reorientation

Stage I - Pile-up in the <1120> direction Driven by prismatic and pyramidal slip

<1120> <1010>

Unloading

0 0

100

200

300

400

500

600

700

800

900 900

800

Indentation Depth (nm)

Fig. 11. Evolution of peak pile-up heights along the h1 1 2 0i and h1 0 1 0i directions during basal plane indentation. The figure also demarcates regimes that exhibit distinct processes that govern the pile-up phenomenon. Note the crossover in the 250 6 d 6 350 nm regime, which in concomitant to the transition in the dominant deformation mechanisms.

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Fig. 12. AFM images showing evolution of sink-in observed in the indentation experiment on the prismatic plane.

Fig. 13. Contour plots showing the evolution of pile-up in the prismatic plane indentation simulation.

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pile-up height due to elastic recovery. Fig. 10 shows the change in the surface morphology along the ½1 0  1 0 direction during unloading from the maximum indentation depth (d  930 nm) to complete unloading (d 815 nm). Importantly, this maximum residual pile-up remains alongh1 0 1 0i, which is again consistent with the experimental observation. The simulations indicate that the change in pile-up height (Dh) due to the elastic recovery is significant; the peak pile-up height during loading is hp  30 nm, whereas after complete unloading it is hr  57:5 nm   ¼ hr ¼ 1:92 . For isotropic polycrystalline materials, h hp

the change in the pile-up height due to elastic recovery depends on the n ¼ E=ry ratio, where E is the elastic modulus and ry is the yield stress [44]. The lower the ratio, the higher the elastic recovery, and vice versa. For the present case, taking E ¼ 61:6 GPa (c-axis) and substituting = 220 MPa, we obtain n  275. It is interesting ry ¼ shcþai c to note that Taljat and Pharr [44] reported  h  3 for an iso-

tropic material with n  100 (and with spherical indenter), which is in the same range despite the elastic and plastic anisotropies in the present work. Fig. 11 exemplifies the competition between the pile-up evolution along h1 1  2 0i and h1 0  1 0i as a function of the indentation depth in the loading and unloading stages. Clearly, while initially the h1 1  2 0i pile-up evolves faster for d  220 , a transition occurs at d  240 nm beyond which the pile-up along h1 0  1 0i increases at a faster rate than along h1 1  2 0i. Notably, this dominance of h1 0  1 0i-driven pile-up continues at peak load and even after complete unloading is evident. Summarizing, indentation of Mg single crystal on the basal plane produces a sixfold pile-up, and the direction of pile-up formation changes during the course of the loading process. The four slip modes and two twin modes listed in Section 3.2 are all found to be active. The presence of f1 0  1 2g extension twinning in the regions around the indenter is confirmed by both TEM and the simulation.

Fig. 14. (a) Schematic indicating locations where the first signs of yield are observed in CPFE simulations of prismatic plane indentation: basal slip (I) and f1 0  12 g extension twinning (II). Panels (b) and (c) show the evolution of ca at these points. Table 3 summarizes the positions of these points as a function of the current radius of contact (rc ) of the indenter.

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There is a significant amount of elastic recovery during the unloading process, and the same is corroborated using elastic–plastic contact analysis.

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the region directly under the indenter, there is compressive stress perpendicular to the c-axis, effectively causing c-axis extension . Hence, f1 0  1 2g extension twinning is expected underneath the indenter. Twin-induced lattice reorientation by  86 about the f1 0  1 2g plane has a significant effect on the indentation load as after reorientation the c-axis experiences near-direct contraction. Fig. 12 shows a series of AFM images taken during our ½1 1  2 0 indentation experiment. Unlike the pile-up phenomenon in the ½0 0 0 1 indentation scenario, here the material around the indenter has sunk in. The sink-in has a twofold symmetry and extends along the ½1  1 0 0 direction with

4.2. Indentation on ð1 1  2 0Þ prismatic plane The qualitative nature of the triaxial stress state while indenting along ½1 1 2 0 is different from the stress state observed while indenting along [0 0 0 1]. The schematic in Fig. 4b, indicating the deformation state in the substrate during ½1 1 2 0 indentation, contrasts with that in Fig. 4a shown earlier for the [0 0 0 1] indentation case. Here, in

Twinned region Secon plane

(0001)

Twinned region

(a)

(b)

(c)

(d)

Fig. 15. Simulated indentation of f1 1 2 0g plane indicates (a) the distribution of extension twin v.f. at d 575 nm. The TEM image in (c), taken from the highlighted rectangularregion in (b), shows the presence of twins. Panel (d) shows the SAED pattern from the highlighted region in (c), confirming the presence of 1 0 1 2 twins.

Fig. 16. Schematic indicating the mechanism of sink-in due to f1 0 1 2g extension twinning during indentation on the prismatic plane. The figure on the left shows the initial crystal orientation. The figure on the right shows the extension twin-induced lattice reorientation and the resulting material flow.

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increasing indentation depth. Fig. 13 shows the FE simulation result with excellent qualitative corroboration with the experimentally observed twofold sink-in along the ½1  1 0 0 direction. The important features of the indentationinduced plasticity are analyzed in two stages: loading and unloading.

Displacement in the indented direction (nm)

4.2.1. Loading response A distinct sink-in phenomenon is visible starting at d  30 nm (Fig. 13a), which conforms to the early appearance of sink-in in the experiment (Fig. 12). Further, Fig. 13b and c shows the evolution of the sink-in at d  150 nm and d  300 nm, respectively, with a twofold symmetry extending along the ½1  1 0 0 direction perpendicular to the indentation direction, which is also consistent with the experimental observation. The radial extent of sink-in continues to increase with increasing indentation depth and there is no change in directionality, unlike the case of pile-up that occurred in the indentation of the basal plane. Fig. 14a highlights the positions in the substrate where the first sign of yield occurs on particular deformation systems, and Fig. 14b and c shows the corresponding evolution of ca on the active slip/twin systems at those positions. Here, the onset of plasticity is due to the activation of f1 0 1 2g twinning at d  5 nm beneath the indenter (point I

At peak load After unload

0

-100

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-600 0

2

4

6

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10

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Distance along the surface (µm)

(Fig. 14b)) immediately followed by the activation of basal slip at d  10 nm (point II (Fig. 14c)). Table 3 quantifies the positions at which yield occurs; while basal slip yields in the vicinity of the indenter, extension twin yields away from the surface, underneath the indenter. Consistent with the schematic in Figs. 4b, and 15a reveals the occurrence of profuse extension twinning beneath the indenter. The corresponding experimental result (Fig. 15b–d) also clearly indicates the occurrence of f1 0  1 2g twins beneath the indenter. In materials that deform solely by dislocation slip, the sink-in phenomenon ensues from their high strain hardening capacity [41,42]. In comparison, for the case of indentation on the f1 1  2 0g plane, it is the strain associated with the 1 2g extension twinning beneath the indenter that gives f1 0  rise of the sink-in characteristic. This is explained with reference to extension Fig. 16. The sudden large lattice reorientation due to extension twinning under the indenter and the associated shear strain pushes the material away from the indented surface. The downward material flow due to this directional strain manifests as sink-in along the projections of the extension f1 0  1 2g planes on the indented surface. For the particular orientation considered 2 0), two of the six extension here (indentation along ½1 1  1 0 2Þ½ 1 1 0 1, ð 1 1 0 2Þ½1  1 0 1) are less twin systems (ð1  favored (smaller Schmid factor) while the other four are equally favorable. The cumulative effect of twinning on these four active planes results in the formation of the observed sink-in with a twofold symmetry. Note that the projections of the four active twin planes on the indented plane possess a twofold symmetry. Naturally, the twin v.f. distribution also exhibits a twofold symmetry (Fig. 15a). Focusing on the simulation result, as the indentation progresses, the material beneath the indenter continues to be compressed. However, the region that has already twinned now experiences c-axis due to contraction  86 lattice reorientation. Consequently, the plasticity in this region rapidly transitions from being twinning dominated to being slip dominated, with pyramidal hc þ ai being the governing mode. In Fig. 14b and c it can be seen that tension-twin-induced reorientation results in activation of pyramidal hc þ ai slip. Some f1 0  1 1gh1 0  1 2i contraction twinning also occurs in the twin-reoriented region, where

Fig. 17. Predicted sink-in profiles along ½1 1 0 0 at peak load and after unloading for prismatic plane indentation. Note that the elastic recovery effect is not as dramatic as in the case of basal plane indentation (cf. Fig. 10).

1200 Full model Reduced model

1000

Force (µN)

800

600

400

200

0 0

(a)

(b)

Fig. 18. Crystal symmetry about (a) the h0 0 0 1i direction and (b) the h1 1 2 0i direction.

50

100

150

200

250

Indentation depth (nm)

Fig. 19. P  d curve for basal plane indentation using the full model and the reduced model.

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Fig. 20. Comparison of total plastic strain distribution in the case of basal plane indentation between (a) the full model and (b) the reduced model (d ¼ 250 nm).

Fig. 21. Comparison of cumulative shear strain on the basal slip systems in the case of basal plane indentation between (a) the full model and (b) the reduced model (d ¼ 250 nm).

Fig. 22. Comparison of f1 0 1 2g extension twin v.f. fraction in the case of basal plane indentation between (a) the full model and (b) the reduced model (d ¼ 250 nm).

the contraction c-axis is under direct contraction (not shown here for brevity). However, the total contraction twin v.f. is two orders of magnitude lower than the volume fraction of extension twinning. Consequently, the contribution of contraction twinning to the surface morphology is expected to be negligible. Likewise, the average contribution of the prismatic contraction extensioncontraction contraction hai, pyramidal hai and pyramidal hc þ ai slip systems are one order of magnitude lower than the averaged strain on the basal slip and extension twin systems; therefore, their contribution to the sink-in is not appreciable.extension.

(d 500 nm). Note that in Fig. 17 the change in sink-in profile due to the elastic recovery is only marginal compared to in the ½0 0 0 1 case. Taking E ¼ 59:6 GPa (h1 1  2 0i direction) and ry ¼ sTT c = 20 MPa, n  3000. As noted in Section 4.1.4, such a large n is expected to result in very low elastic recovery. As a result, the residual sink-in depth is not significantly different from that corresponding to the maximum load, which is quite distinct from the [0 0 0 1] indentation case.

4.2.2. Unloading response Fig. 17 shows the surface morphologies along ½1  1 0 0 at peak load (d  575 nm) and upon complete unloading

As noted in Section 1, recent experiments of Catoor et al. [28] along the basal planes did not reveal any evidence of extension twinning, unlike in the present work, which

5. Further discussion

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showed the presence of tension twins. We posit that this may be due to the difference in the indenter geometry, which is expected to affect the stress distribution. Catoor et al. used a large spherical indenter (radius r ¼ 3:3 lm), compared to our cono-spherical indenter, with a much smaller tip radius. It is useful to note that our simulations also do not indicate twin-induced lattice reorientation in Stage 1 or even during the best part of Stage 2 (Section 4.1). In our simulations, twinning-induced reorientation occurs only at d  350 nm, which is just after the spherical part of the indenter has moved sufficiently into the material so that the conical portion of the indenter is in contact. Next, it is of interest to discuss the role of lattice reorientation on the pile-up and sink-in phenomena. Using Vickers indentation experiments on Cu and MoSi2 single crystals, Peralta et al. [18] indicated that small lattice rotations under the indenter produced pile-up while large lattice rotations caused sink-in. In the present work, lattice reorientations occur primarily due to f1 0  1 2g twinning, with elastic lattice rotations playing a secondary role. Against that backdrop, recall that the extension -twinning-induced lattice reorientation in the [0 0 0 1] indentation aids the overall pile-up that is driven by the saturation hardening effects of the slip. On the other hand, the extension twinning causes a sink-in effect in the ½1 1  2 0 indentation case. We explain this likely dichotomy as follows. Note that, in the former case, the shear strain due to extension twinning causes the material to flow toward the surface. In comparison, for the latter scenario, the twinning process causes a downward material flow that is directed away from the surface.

6. Summary and concluding remarks In this work, we investigated the indentation response of Mg single crystals with 99.9% purity. Nanoindentation experiments were performed on the ð0 0 0 1Þand ð1 1  2 0Þ planes using a cono-spherical indenter, thereby ensuring that the responses were characterized only by the crystallographic symmetries. TEM and AFM were used to identify twins and measure surface morphology, respectively. Three-dimensional FE simulations were performed using our CP model, which takes into account all the slip/twin modes observed in Mg with appropriate strain hardening laws. The important observations from our study are:

1. Nanoindentation experiments on Mg single crystals show strongly orientation-dependent characteristics. Indentation of the basal plane exhibits a harder load–depth response compared to that of the prismatic plane. Further, basal plane indentation shows a sixfold pile-up along h1 0 1 0i, 2 0Þ plane produces while indentation along ð1 1  1 0 0. a twofold sink-in along ½1  2. The CPFE simulations corroborate well with these experimental results. This demonstrates the capability of CPFE simulations to predict the orientation-dependent nanoindentation response of hcp single crystals. 3. The CPFE simulations of basal plane indentation indicate a transition in the pile-up from 2 0i to being sixfold along being sixfold along h1 1  1 0i with increasing indentation depth. Analh1 0  ysis of the underlying mechanisms reveals that

the pile-up along h1 1 2 0i is governed by the CRSS and hardening behaviors of the prismatic and pyramidal hai slip systems, while the pile-up along h1 0 1 0i is governed by the material characteristics of the pyramidal hc þ ai slip and f1 0 1 2gh1 0 1 1i twinning. In a broader sense, these results shed light on how the micromechanics of slip and twinning systems may be responsible for the observed nature of the pile-up phenomena. 4. The simulations also indicate that both the pileup (basal plane indentation) and sink-in (prismatic plane indentation) phenomena are influenced by f1 0 1 2gextension twinning, albeit in different ways. In the former, the extension twinning pushes the material toward the surface, while in the latter, it tends to push the material away from the surface. 5. The simulations indicate that elastic recovery during unloading leads to a significant increase in the residual pile-up in the case of (0 0 0 1) indentation, whereas the decrease in sink-in for the case of ð1 1 2 0Þ indentation is much smaller. The present work provides a useful insight into the rich interactions that occur between key deformation mechanisms in Mg under heterogeneous loading conditions. In particular, the evolution of pile-up/sink-in phenomena serve as a signature to understand the activation and prevalence of behavior of key deformation systems. Although the focus here is on single crystals, some of the characteristics may be relevant to highly textured Mg polycrystalline counterparts. There are several directions that may be pursued as extensions of this work. First, it is clear that the hardening parameters for slip and twinning will affect the overall load–depth characteristics, as well as the pile-up/ sink-in phenomena that occur. Our ongoing work this direction includes understanding the length-scale effects on the material parameters and the interactions between the slip and twin systems in modulating the macroscopic responses. It would also be worthwhile investigating the effect of the contact geometry on the evolution of slip and twinning, e.g. symmetric vs. non-symmetric indenters. Acknowledgments B.S. acknowledges an NUS Research Scholarship from the Ministry of Education, Singapore. S.P.J. acknowledges partial financial support from the Scientific and Engineering Research Council (SERC) – Singapore under the SERC-NSF Materials World Network Program (Grant No. R-365-000-373-305) and NUS-AcRF-Tier-1 Grant No. R-265-000-459-112. I.S.C. was supported by the KIST Institute Research Fund and KIST GRL program (Grant Nos. 2E24692 and 2Z04050). H.N.H. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013008806).

Appendix A. Validation of the half symmetry CPFE model The hcp crystal structure of Mg possesses sixfold symmetry about the h0 0 0 1i direction. Consequently, for the present problem, modeling one-sixth of the Mg substrate should suffice to elicit the crystallographic features for (0 0 0 1) indentation (Fig. 18a). On the other hand, it pos-

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sesses a twofold symmetry about the h1 1  2 0i direction, which requires modeling half of the substrate in order to correctly model the crystallographic features (Fig. 18b). With these considerations in mind, we modeled one-half of the substrate with the appropriate symmetry boundary conditions. We validated this reduced model by simulating the indentation on the full model up to a reasonable indentation depth (d 250 nm) and comparing the two results (Fig. 19). As can be seen from the figure, the two results match rather well. Figs. 20–22 further reinforce the validity of the reduced model. References [1] J. Christian, S. Mahajan, Deformation twinning, Prog. Mater. Sci. 39 (1) (1995) 1–157. [2] E. Kelley, W. Hosford, Plane-strain compression of magnesium and magnesium alloy crystals, Trans. Met. Soc. AIME 242 (1) (1968) 5–13. [3] J. Zhang, S. Joshi, Phenomenological crystal plasticity modeling and detailed micromechanical investigations of pure magnesium, J. Mech. Phys. Solids 60 (5) (2012) 945–972. [4] F. Fro¨hlich, P. Grau, W. Grellmann, Performance and analysis of recording microhardness tests, Phys. Status Solidi A 42 (1) (1977) 79–89. [5] G. Pharr, J. Strader, W. Oliver, Critical issues in making small-depth mechanical property measurements by nanoindentation with continuous stiffness measurement, J. Mater. Res. 24 (03) (2009) 653–666. [6] W. Oliver, G. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7 (6) (1992) 1564–1583. [7] W. Oliver, G. Pharr, Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology, J. Mater. Res. 19 (1) (2004) 3–20. [8] A. Krell, A new look at grain size and load effects in the hardness of ceramics, Mater. Sci. Eng. A 245 (2) (1998) 277–284. [9] J. Gong, Z. Guan, Load dependence of low-load knoop hardness in ceramics: A modified PSR model, Mater. Lett. 47 (3) (2001) 140–144. [10] J. Vlassak, M. Ciavarella, J. Barber, X. Wang, The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape, J. Mech. Phys. Solids 51 (9) (2003) 1701– 1721. [11] H. Lee, J. Haeng Lee, G. Pharr, A numerical approach to spherical indentation techniques for material property evaluation, J. Mech. Phys. Solids 53 (9) (2005) 2037–2069. [12] W. Nix, Elastic and plastic properties of thin films on substrates: Nanoindentation techniques, Mater. Sci. Eng. A 234 (1997) 37–44. [13] M. Doerner, W. Nix, A method for interpreting the data from depth-sensing indentation instruments, J. Mater. Res. 1 (04) (1986) 601–609. [14] I. Choi, R. Schwaiger, L. Kurmanaeva, O. Kraft, On the effect of Ag content on the deformation behavior of ultrafinegrained Pd-Ag alloys, Scr. Mater. 61 (1) (2009) 64–67. [15] I. Choi, M. Dao, S. Suresh, Mechanics of indentation of plastically graded materials-I: Analysis, J. Mech. Phys. Solids 56 (1) (2008) 157–171. [16] I. Choi, A. Detor, R. Schwaiger, M. Dao, C. Schuh, S. Suresh, Mechanics of indentation of plastically graded materials-II: Experiments on nanocrystalline alloys with grain size gradients, J. Mech. Phys. Solids 56 (1) (2008) 172–183. [17] Y. Wang, D. Raabe, C. Klu¨ber, F. Roters, Orientation dependence of nanoindentation pile-up patterns and of nanoindentation microtextures in copper single crystals, Acta Mater. 52 (8) (2004) 2229–2238.

375

[18] P. Peralta, R. Ledoux, M. Hakik, R. Dickerson, P. Dickerson, Characterization of surface deformation around vickers indents in monocrystalline materials, Metall. Mater. Trans. A 35 (8) (2004) 2247–2255. [19] N. Zaafarani, D. Raabe, R. Singh, F. Roters, S. Zaefferer, Three-dimensional investigation of the texture and microstructure below a nanoindent in a Cu single crystal using 3D EBSD and crystal plasticity finite element simulations, Acta Mater. 54 (7) (2006) 1863–1876. [20] O. Casals, J. Ocˇena´sˇek, J. Alcala, Crystal plasticity finite element simulations of pyramidal indentation in copper single crystals, Acta Mater. 55 (1) (2007) 55–68. [21] Y. Saito, M. Oztop, J. Kysar, Wedge indentation into elasticplastic single crystals. 2: Simulations for face-centered cubic crystals, Int. J. Plast. 28 (1) (2012) 70–87. [22] M. Biener, J. Biener, A. Hodge, A. Hamza, Dislocation nucleation in BCC Ta single crystals studied by nanoindentation, Phys. Rev. B 76 (16) (2007) 165422. [23] L. Zhang, T. Ohmura, Plasticity initiation and evolution during nanoindentation of an iron 3% silicon crystal, Phys. Rev. Lett. 112 (2014) 145504. [24] T. Britton, H. Liang, F. Dunne, A. Wilkinson, The effect of crystal orientation on the indentation response of commercially pure titanium: Experiments and simulations, Proc. R. Soc. A 466 (2115) (2010) 695–719. [25] C. Zambaldi, Y. Yang, T. Bieler, D. Raabe, Orientation informed nanoindentation of a-titanium: Indentation pileup in hexagonal metals deforming by prismatic slip, J. Mater. Res. 27 (01) (2012) 356–367. [26] R. Blish II., T. Vreeland Jr., Twinning and slip in zinc by indentation, J. Appl. Phys. 39 (5) (2003) 2290–2292. [27] J. Kwon, M. Brandes, P. Sudharshan Phani, A. Pilchak, Y. Gao, E. George, et al., Characterization of deformation anisotropies in an a-Ti alloy by nanoindentation and electron microscopy, Acta Mater. 61 (13) (2013) 4743–4756. [28] D. Catoor, Y. Gao, J. Geng, M. Prasad, E. Herbert, K. Kumar, et al., Incipient plasticity and deformation mechanisms in single-crystal Mg during spherical nanoindentation, Acta Mater. 61 (8) (2013) 2953–2965. [29] J. Shin, S. Kim, T. Ha, K. Oh, I. Choi, H. Han, Nanoindentation study for deformation twinning of magnesium single crystal, Scr. Mater. 68 (7) (2013) 483–486. [30] R. Sa´nchez-Martı´n, M. Pe´rez-Prado, J. Segurado, J. Bohlen, I. Gutie´rrez-Urrutia, J. Llorca, et al., Measuring the critical resolved shear stresses in Mg alloys by instrumented nanoindentation, Acta Mater. 71 (2014) 283–292. [31] P. Partridge, Distribution of twins around a microhardness indenter in close-packed hexagonal metals, Nature 203 (1964) 634–635. [32] M. Yoo, C. Wei, Twinning in zinc by indentation, J. Appl. Phys. 38 (7) (2004) 2974–2976. [33] H. Somekawa, C. Schuh, Nanoindentation behavior and deformed microstructures in coarse-grained magnesium alloys, Scr. Mater. 68 (6) (2013) 416–419. [34] A. Giannakopoulos, P. Larsson, R. Vestergaard, Analysis of vickers indentation, Int. J. Solids Struct. 31 (19) (1994) 2679– 2708. [35] M. Mata, J. Alcala, The role of friction on sharp indentation, J. Mech. Phys. Solids 52 (1) (2004) 145–165. [36] J. Alcala, O. Casals, J. Ocˇena´sˇek, Micromechanics of pyramidal indentation in FCC metals: Single crystal plasticity finite element analysis, J. Mech. Phys. Solids 56 (11) (2008) 3277–3303. [37] D. Syste´mes, Abaqus 6.9 Theory Manual, Dassault Syste´mes Simulia Corp, RI, USA 2009. [38] W. Nix, H. Gao, Indentation size effects in crystalline materials: A law for strain gradient plasticity, J. Mech. Phys. Solids 46 (3) (1998) 411–425. [39] Y. Huang, F. Zhang, K. Hwang, W. Nix, G. Pharr, G. Feng, A model of size effects in nano-indentation, J. Mech. Phys. Solids 54 (8) (2006) 1668–1686.

376

B. Selvarajou et al. / Acta Materialia 81 (2014) 358–376

[40] G. Feng, W. Nix, Indentation size effect in MgO, Scr. Mater. 51 (6) (2004) 599–603. [41] Y. Cheng, C. Cheng, Effects of ’sinking in’ and ’piling up’ on estimating the contact area under load in indentation, Philos. Mag. Lett. 78 (2) (1998) 115–120. [42] J. Alcala, A. Barone, M. Anglada, The influence of plastic hardening on surface deformation modes around vickers and spherical indents, Acta Mater. 48 (13) (2000) 3451–3464.

[43] F. Lavrentev, Y. Pokhil, Relation of dislocation density in different slip systems to work hardening parameters for magnesium crystals, Mater. Sci. Eng. 18 (2) (1975) 261–270. [44] B. Taljat, G. Pharr, Development of pile-up during spherical indentation of elastic-plastic solids, Int. J. Solids Struct. 41 (14) (2004) 3891–3904.